Mintys.Simplex istorija
Paslėpti nežymius pakeitimus - Rodyti galutinio teksto pakeitimus
2016 birželio 19 d., 12:34
atliko -
Pakeistos 1-165 eilutės iš
Žr. [[Matematika]]
>>bgcolor=#FFFFC0<<
* How are the demihypercubes generated?
* What does the end of the Dynkin diagram mean?
* Why is it that 3-cycles lead us to higher dimensions in the Dynkin diagrams?
* How should the demihypercube binomial triangle be interpreted?
* What is the q-analogue for the binomial triangles of Bn,Cn,Dn?
* What is the role of opposites and implicit math in the classical groups?
* What does it mean that F1 is {0,infinity}?
* Does the end of the dynkin diagram code for opposites as they are drawn from the center?
Investigation
* Look at coordinates
* Look at binomial theorem
Įsivaizduoti centrą (žvilgsnį iš viršaus) ir apskaičiuoti centro daugiaprasmiškumą (naudojant q?)
Understand the generator point and mirror construction here https://en.m.wikipedia.org/wiki/Coxeter–Dynkin_diagram and the ringed nodes
* study stanley's proof of the volume of a hypersimplex. alexander postnikov and thomas lam alcoved polytopes
* What does it mean if the center forms 3 new nodes at each step? (We get a hexagon with 3 vertices and 3 edges each. This could be mounted on a cube with a hole. Then we get 6 vertices with 6 edges each, and so on.)
* What does it mean if the center forms k new nodes at each step?
* What is the polytope which G2 gives the symmetries of?
>><<
http://math.ucr.edu/home/baez/week186.html dynkin diagram and q polynomial
'''Coxeter group Dn and hemicubes'''
Hemicubeoctahedron is half-cube (on the inside) and half-orthogon (on the outside). It is thus without a center (?) and without a volume (?)
Note that it wants to choose half of the orthogon. But which half? this would depend on the coordinates. But we are working coordinate-free. Thus the natural answer is ambiguity, as in a mixed quantum state. It is the same ambiguity that is in the internal tension of the original center, "those things are which show themselves to be" - "to be or not to be?" So it is modeling that ambiguity. And that ambiguity is modeled locally at all points. And that is how the simplex is adorned by local growth.
Dn demihypercube construction
[[http://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html#Half | Half measure polytope]]
* Also known as a demihypercube in n-space, n>2. The uniform polytope constructed by completely truncating the alternate vertices of a measure polytope, that is, truncating half its vertices by hyperplanes that pass through a truncated vertex’s edge-neighboring vertices. If n=2, this truncation of a square leaves only the square’s diagonal, which is not a polygon. But when n=3, this truncation of a cube produces a regular tetrahedron, and when n=4, it produces a regular hexadecachoron from a tesseract. For n>4, demihypercubes are no longer regular, only uniform, with two different kinds of facets: 2n (n–1)-dimensional demihypercubes and 2n–1 (n–1)-dimensional simplexes. The names of the demihypercubes are constructed by prefixing demi- to the name of a measure polytope: demipenteract, demihexeract, demihepteract, and so forth. As noted above, a demicube is a regular tetrahedron, and a demitesseract is a regular hexadecachoron.
* The vertex figure of a half measure polytope H in n-space is a rectified (n–1)-dimensional simplex (the simplex has n vertices and n facets). The facets of the rectified simplex are (1) n (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional simplex facets of H, and (2) n rectified (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional demihypercube facets of H. Thus, for example, the vertex figure of the demicube is a rectified triangle, which is a smaller triangle, the vertex figure of a tetrahedron (the demicube). The vertex figure of the demitesseract is a rectified tetrahedron, which is an octahedron, the vertex figure of a regular hexadecachoron (the demitesseract). And so on.
* Euclidean n-space can always be uniformly honeycombed by half measure polytopes and cross polytopes: In the regular honeycomb of n-space by measure polytopes, remove alternate vertices, thereby transforming each measure polytope into a half measure polytope, and fill the gaps with n-dimensional cross polytopes (the vertex figures of the measure polytope honeycomb). In the plane, this changes the checkerboard tiling into another checkerboard tiling rotated 45° to the original; in 3-space this produces the uniform honeycomb of regular tetrahedra and octahedra; and in 4-space this produces the regular honeycomb of hexadecachora
* Vertices: Take those coordinates (of a cube) which have an even number of minus signs.
* Edges: ?
The polytopes are irregular because the Dynkin diagram is not a path but has a branch. Thus there should be not one center or volume but rather the "generation" of the polytope should derive from two perspectives. These two perspectives may relate to the bipartition of the demicube which itself is a bipartition of the cube. The two perspectives are related by double edges.
There should be two different, complementary, flags of faces. And they should relate to the cross polytopes - one flag should be like the cross polytopes and the other flag should be the opposite - and together they should form a cube.
The pascal triangle can be explained by noting that we have the fusion of two paths in the Dynkin diagram, one of length 3 and one that grows without bound. The first nontrivial example is the 16 cell.
The purpose is to split the perspective. The human perspective is given by the short path.
[[https://books.google.lt/books?id=HarWCwAAQBAJ&printsec=frontcover#v=onepage&q&f=false | The Higher Dimensional Hemicubeoctahedron]] Daniel Pellicer, in: Symmetries of Graphs, Maps and Polytopes. 2014.
http://www.sciencedirect.com/science/article/pii/S0001870809001017 Homology representations arising from the half cube
http://math.ucr.edu/home/baez/week187.html The Coxeter group for Dn is the subgroup of the symmetries of the n-dimensional cube generated by permutations of the coordinate axes and reflections along ''pairs'' of coordinate axes.
http://link.springer.com/chapter/10.1007/978-3-642-04295-9_1 geometry of cuts and metrics
http://arxiv.org/abs/1506.06702 paper about 16 cube
nim and sierpinski demihypercube
[[https://plus.google.com/117663015413546257905/posts/ZCyDfcc8gRQ | John Baez on demicubes]]
http://link.springer.com/chapter/10.1007/978-3-642-21590-2_15 nth roots of pitch class inversion
'''Coxeter groups and polytopes'''
* dynkin diagram coxeter group An reflections are encoded by the sequence 1, q2, q3, ..., q*(n-1)
* symmetry group is Sn
* Koch snowflake is an illustration of An for all n.
* Bn hypercube construction given by having the volume grow by having the volume be conceived as a mirror from which two mirror images arise on either sides. Thus mirror points are linked by edges.
* Cn cross-polytope construction given by having two points extend from the center in opposite directions. Each of the points links to all of the existing vertices. The old volume gives way to a new volume of one dimension higher. The initial C is given by two unconnected points. And these two points came from the center, which is why they are unconnected.
* Thus we have duality of adding two opposing vertices or adding a hyperplane with two reflections, of growing the center or the volume. We have the duality of vertices and facets.
Symmetry groups
* An: symmetric group
* Bn: A deck of n cards where card j has j written on one side and -j on the other side. All possible arrangements of stacks of cards with orientation.
* Dn: corresponds to all arrangements of stacks of n cards with orientation with an even number of cards turned over (pairs of reflections).
'''Exceptional groups'''
Also this suggests that the exceptional groups are modeling interactions of limited human perspectives.
'''Generative center'''
Centras: apibendrinimas. Išrašus jį atsiranda matas.
Projective space. All lines that go through the origin. Natural origin = "center" of simplex. Natural infinity = fold out the next point = so vertices are halfway between Center and Infinity.
center is what gives 3 rather than 2. center is what links involutions in the dynkin diagram.
the linked involutions, the edges, are numbered and are added as the normal form which is preserved.
An add the center along with edges from the center to the other vertices. Then you can revision in a higher dimension.
'''Q-analogue'''
q-characteristic
0=q=infinity jų požiūriu
Visa šeima įvairių q - iš esmės panašūs - bet mes esame keisti q=1
0=1=infinity mes esame keisti.
All of the nodes chip in a weight "q" to allow the new weight to be distinguished from the rest "q^k". They all go down by one (temporarily?) to stay distinguished, to meet their previous obligations.
[[https://qchu.wordpress.com/2009/06/11/young-diagrams-q-analogues-and-one-of-my-favorite-proofs/#more-266 | Young diagrams - q-analogue Chu]]
[[https://www.bowdoin.edu/faculty/f/fisk/dissemination/homology-projective.pdf | q-analogue of a simplex is a projective space over a finite field]] homology
'''Theological interpretation'''
Would like to believe that God is good. Eternal life, eternal learning is driven by the building up that God inspires. And all learning is dependent. But at any height there is the dual: we can unlearn (or tear down) whatever we learn (or build up). Unlearning is independent like the cube. But in between is the space of eternal life. It is based on the idea that God doesn't have to be good, life doesn't have to be fair. We can unlearn in order to learn. And ultimately we hope that truly God is good, that truly we can learn without unlearning. That is what we are demonstrating. So it is related to John Harland's quest about learning.
'''Field with one element'''
David Corfield's post
https://golem.ph.utexas.edu/category/2007/04/the_field_with_one_element.html
about Nikolai Durov's book
http://arxiv.org/abs/0704.2030
John Baez: This fits nicely with my own intuitions about linear algebra over the field with one element. A pointed set acts like a ‘vector space over the field with one element’; a set acts like a projective space over the field with one element.
* http://math.ucr.edu/home/baez/week185.html
* http://math.ucr.edu/home/baez/week186.html
* http://math.ucr.edu/home/baez/week187.html
'''Coxeter groups'''
[[http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html | Lecture notes by Federico Ardila]]
Videos
* Lecture 10 properties of Bruhat order
* Lecture 15 Mobius function Eulerian characteristic Eulerian posets
* Lecture 20 Dihedral group D5
* Lecture 25 Contragradient action Coxeter groups are automatic Root poset
* Lecture 30 Root systems for An and Bn
* Lecture 31 Bn. Root system Coxeter group
* Lecture 32 Crystallographic root systems Lie groups Cartan matrix
* Lecture 33 Coxeter matrix mij and aij aji
* Lecture 34 Rank 2 Dynkin diagrams
* Lecture 35 Coxeter group reflection group bipartite
* Lecture 36 Weyl group is finite if bilinear form is positive definite
* Lecture 37 Group representation sum of irreducibles
* Lecture 38 Weyl group finite iff bilinear form is positive definite
* Lecture 39 Classification of finite Coxeter groups
* Lecture 40 Proof
* Lecture 41 Proof
* Lecture 42 Classification of regular polytopes
https://ecommons.cornell.edu/handle/1813/3206 catalan and coxeter
https://ecommons.cornell.edu/handle/1813/17339 coxeter dynkin interview
https://en.wikipedia.org/wiki/Conway_polyhedron_notation
polyhedron operators https://en.wikipedia.org/wiki/Alternation_(geometry)
https://upload.wikimedia.org/wikipedia/commons/6/67/Wythoffian_construction_diagram.png
>>bgcolor=#FFFFC0<<
* How are the demihypercubes generated?
* What does the end of the Dynkin diagram mean?
* Why is it that 3-cycles lead us to higher dimensions in the Dynkin diagrams?
* How should the demihypercube binomial triangle be interpreted?
* What is the q-analogue for the binomial triangles of Bn,Cn,Dn?
* What is the role of opposites and implicit math in the classical groups?
* What does it mean that F1 is {0,infinity}?
* Does the end of the dynkin diagram code for opposites as they are drawn from the center?
Investigation
* Look at coordinates
* Look at binomial theorem
Įsivaizduoti centrą (žvilgsnį iš viršaus) ir apskaičiuoti centro daugiaprasmiškumą (naudojant q?)
Understand the generator point and mirror construction here https://en.m.wikipedia.org/wiki/Coxeter–Dynkin_diagram and the ringed nodes
* study stanley's proof of the volume of a hypersimplex. alexander postnikov and thomas lam alcoved polytopes
* What does it mean if the center forms 3 new nodes at each step? (We get a hexagon with 3 vertices and 3 edges each. This could be mounted on a cube with a hole. Then we get 6 vertices with 6 edges each, and so on.)
* What does it mean if the center forms k new nodes at each step?
* What is the polytope which G2 gives the symmetries of?
>><<
http://math.ucr.edu/home/baez/week186.html dynkin diagram and q polynomial
'''Coxeter group Dn and hemicubes'''
Hemicubeoctahedron is half-cube (on the inside) and half-orthogon (on the outside). It is thus without a center (?) and without a volume (?)
Note that it wants to choose half of the orthogon. But which half? this would depend on the coordinates. But we are working coordinate-free. Thus the natural answer is ambiguity, as in a mixed quantum state. It is the same ambiguity that is in the internal tension of the original center, "those things are which show themselves to be" - "to be or not to be?" So it is modeling that ambiguity. And that ambiguity is modeled locally at all points. And that is how the simplex is adorned by local growth.
Dn demihypercube construction
[[http://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html#Half | Half measure polytope]]
* Also known as a demihypercube in n-space, n>2. The uniform polytope constructed by completely truncating the alternate vertices of a measure polytope, that is, truncating half its vertices by hyperplanes that pass through a truncated vertex’s edge-neighboring vertices. If n=2, this truncation of a square leaves only the square’s diagonal, which is not a polygon. But when n=3, this truncation of a cube produces a regular tetrahedron, and when n=4, it produces a regular hexadecachoron from a tesseract. For n>4, demihypercubes are no longer regular, only uniform, with two different kinds of facets: 2n (n–1)-dimensional demihypercubes and 2n–1 (n–1)-dimensional simplexes. The names of the demihypercubes are constructed by prefixing demi- to the name of a measure polytope: demipenteract, demihexeract, demihepteract, and so forth. As noted above, a demicube is a regular tetrahedron, and a demitesseract is a regular hexadecachoron.
* The vertex figure of a half measure polytope H in n-space is a rectified (n–1)-dimensional simplex (the simplex has n vertices and n facets). The facets of the rectified simplex are (1) n (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional simplex facets of H, and (2) n rectified (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional demihypercube facets of H. Thus, for example, the vertex figure of the demicube is a rectified triangle, which is a smaller triangle, the vertex figure of a tetrahedron (the demicube). The vertex figure of the demitesseract is a rectified tetrahedron, which is an octahedron, the vertex figure of a regular hexadecachoron (the demitesseract). And so on.
* Euclidean n-space can always be uniformly honeycombed by half measure polytopes and cross polytopes: In the regular honeycomb of n-space by measure polytopes, remove alternate vertices, thereby transforming each measure polytope into a half measure polytope, and fill the gaps with n-dimensional cross polytopes (the vertex figures of the measure polytope honeycomb). In the plane, this changes the checkerboard tiling into another checkerboard tiling rotated 45° to the original; in 3-space this produces the uniform honeycomb of regular tetrahedra and octahedra; and in 4-space this produces the regular honeycomb of hexadecachora
* Vertices: Take those coordinates (of a cube) which have an even number of minus signs.
* Edges: ?
The polytopes are irregular because the Dynkin diagram is not a path but has a branch. Thus there should be not one center or volume but rather the "generation" of the polytope should derive from two perspectives. These two perspectives may relate to the bipartition of the demicube which itself is a bipartition of the cube. The two perspectives are related by double edges.
There should be two different, complementary, flags of faces. And they should relate to the cross polytopes - one flag should be like the cross polytopes and the other flag should be the opposite - and together they should form a cube.
The pascal triangle can be explained by noting that we have the fusion of two paths in the Dynkin diagram, one of length 3 and one that grows without bound. The first nontrivial example is the 16 cell.
The purpose is to split the perspective. The human perspective is given by the short path.
[[https://books.google.lt/books?id=HarWCwAAQBAJ&printsec=frontcover#v=onepage&q&f=false | The Higher Dimensional Hemicubeoctahedron]] Daniel Pellicer, in: Symmetries of Graphs, Maps and Polytopes. 2014.
http://www.sciencedirect.com/science/article/pii/S0001870809001017 Homology representations arising from the half cube
http://math.ucr.edu/home/baez/week187.html The Coxeter group for Dn is the subgroup of the symmetries of the n-dimensional cube generated by permutations of the coordinate axes and reflections along ''pairs'' of coordinate axes.
http://link.springer.com/chapter/10.1007/978-3-642-04295-9_1 geometry of cuts and metrics
http://arxiv.org/abs/1506.06702 paper about 16 cube
nim and sierpinski demihypercube
[[https://plus.google.com/117663015413546257905/posts/ZCyDfcc8gRQ | John Baez on demicubes]]
http://link.springer.com/chapter/10.1007/978-3-642-21590-2_15 nth roots of pitch class inversion
'''Coxeter groups and polytopes'''
* dynkin diagram coxeter group An reflections are encoded by the sequence 1, q2, q3, ..., q*(n-1)
* symmetry group is Sn
* Koch snowflake is an illustration of An for all n.
* Bn hypercube construction given by having the volume grow by having the volume be conceived as a mirror from which two mirror images arise on either sides. Thus mirror points are linked by edges.
* Cn cross-polytope construction given by having two points extend from the center in opposite directions. Each of the points links to all of the existing vertices. The old volume gives way to a new volume of one dimension higher. The initial C is given by two unconnected points. And these two points came from the center, which is why they are unconnected.
* Thus we have duality of adding two opposing vertices or adding a hyperplane with two reflections, of growing the center or the volume. We have the duality of vertices and facets.
Symmetry groups
* An: symmetric group
* Bn: A deck of n cards where card j has j written on one side and -j on the other side. All possible arrangements of stacks of cards with orientation.
* Dn: corresponds to all arrangements of stacks of n cards with orientation with an even number of cards turned over (pairs of reflections).
'''Exceptional groups'''
Also this suggests that the exceptional groups are modeling interactions of limited human perspectives.
'''Generative center'''
Centras: apibendrinimas. Išrašus jį atsiranda matas.
Projective space. All lines that go through the origin. Natural origin = "center" of simplex. Natural infinity = fold out the next point = so vertices are halfway between Center and Infinity.
center is what gives 3 rather than 2. center is what links involutions in the dynkin diagram.
the linked involutions, the edges, are numbered and are added as the normal form which is preserved.
An add the center along with edges from the center to the other vertices. Then you can revision in a higher dimension.
'''Q-analogue'''
q-characteristic
0=q=infinity jų požiūriu
Visa šeima įvairių q - iš esmės panašūs - bet mes esame keisti q=1
0=1=infinity mes esame keisti.
All of the nodes chip in a weight "q" to allow the new weight to be distinguished from the rest "q^k". They all go down by one (temporarily?) to stay distinguished, to meet their previous obligations.
[[https://qchu.wordpress.com/2009/06/11/young-diagrams-q-analogues-and-one-of-my-favorite-proofs/#more-266 | Young diagrams - q-analogue Chu]]
[[https://www.bowdoin.edu/faculty/f/fisk/dissemination/homology-projective.pdf | q-analogue of a simplex is a projective space over a finite field]] homology
'''Theological interpretation'''
Would like to believe that God is good. Eternal life, eternal learning is driven by the building up that God inspires. And all learning is dependent. But at any height there is the dual: we can unlearn (or tear down) whatever we learn (or build up). Unlearning is independent like the cube. But in between is the space of eternal life. It is based on the idea that God doesn't have to be good, life doesn't have to be fair. We can unlearn in order to learn. And ultimately we hope that truly God is good, that truly we can learn without unlearning. That is what we are demonstrating. So it is related to John Harland's quest about learning.
'''Field with one element'''
David Corfield's post
https://golem.ph.utexas.edu/category/2007/04/the_field_with_one_element.html
about Nikolai Durov's book
http://arxiv.org/abs/0704.2030
John Baez: This fits nicely with my own intuitions about linear algebra over the field with one element. A pointed set acts like a ‘vector space over the field with one element’; a set acts like a projective space over the field with one element.
* http://math.ucr.edu/home/baez/week185.html
* http://math.ucr.edu/home/baez/week186.html
* http://math.ucr.edu/home/baez/week187.html
'''Coxeter groups'''
[[http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html | Lecture notes by Federico Ardila]]
Videos
* Lecture 10 properties of Bruhat order
* Lecture 15 Mobius function Eulerian characteristic Eulerian posets
* Lecture 20 Dihedral group D5
* Lecture 25 Contragradient action Coxeter groups are automatic Root poset
* Lecture 30 Root systems for An and Bn
* Lecture 31 Bn. Root system Coxeter group
* Lecture 32 Crystallographic root systems Lie groups Cartan matrix
* Lecture 33 Coxeter matrix mij and aij aji
* Lecture 34 Rank 2 Dynkin diagrams
* Lecture 35 Coxeter group reflection group bipartite
* Lecture 36 Weyl group is finite if bilinear form is positive definite
* Lecture 37 Group representation sum of irreducibles
* Lecture 38 Weyl group finite iff bilinear form is positive definite
* Lecture 39 Classification of finite Coxeter groups
* Lecture 40 Proof
* Lecture 41 Proof
* Lecture 42 Classification of regular polytopes
https://ecommons.cornell.edu/handle/1813/3206 catalan and coxeter
https://ecommons.cornell.edu/handle/1813/17339 coxeter dynkin interview
https://en.wikipedia.org/wiki/Conway_polyhedron_notation
polyhedron operators https://en.wikipedia.org/wiki/Alternation_(geometry)
https://upload.wikimedia.org/wikipedia/commons/6/67/Wythoffian_construction_diagram.png
į:
Žr. [[Book/Simplex]]
2016 birželio 15 d., 12:34
atliko -
Pridėtos 28-30 eilutės:
http://math.ucr.edu/home/baez/week186.html dynkin diagram and q polynomial
2016 birželio 10 d., 12:01
atliko -
Pridėtos 32-33 eilutės:
Note that it wants to choose half of the orthogon. But which half? this would depend on the coordinates. But we are working coordinate-free. Thus the natural answer is ambiguity, as in a mixed quantum state. It is the same ambiguity that is in the internal tension of the original center, "those things are which show themselves to be" - "to be or not to be?" So it is modeling that ambiguity. And that ambiguity is modeled locally at all points. And that is how the simplex is adorned by local growth.
2016 birželio 10 d., 11:40
atliko -
Pridėtos 30-32 eilutės:
Hemicubeoctahedron is half-cube (on the inside) and half-orthogon (on the outside). It is thus without a center (?) and without a volume (?)
2016 birželio 10 d., 09:43
atliko -
Pakeistos 29-30 eilutės iš
į:
'''Coxeter group Dn and hemicubes'''
Dn demihypercube construction
[[http://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html#Half | Half measure polytope]]
* Also known as a demihypercube in n-space, n>2. The uniform polytope constructed by completely truncating the alternate vertices of a measure polytope, that is, truncating half its vertices by hyperplanes that pass through a truncated vertex’s edge-neighboring vertices. If n=2, this truncation of a square leaves only the square’s diagonal, which is not a polygon. But when n=3, this truncation of a cube produces a regular tetrahedron, and when n=4, it produces a regular hexadecachoron from a tesseract. For n>4, demihypercubes are no longer regular, only uniform, with two different kinds of facets: 2n (n–1)-dimensional demihypercubes and 2n–1 (n–1)-dimensional simplexes. The names of the demihypercubes are constructed by prefixing demi- to the name of a measure polytope: demipenteract, demihexeract, demihepteract, and so forth. As noted above, a demicube is a regular tetrahedron, and a demitesseract is a regular hexadecachoron.
* The vertex figure of a half measure polytope H in n-space is a rectified (n–1)-dimensional simplex (the simplex has n vertices and n facets). The facets of the rectified simplex are (1) n (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional simplex facets of H, and (2) n rectified (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional demihypercube facets of H. Thus, for example, the vertex figure of the demicube is a rectified triangle, which is a smaller triangle, the vertex figure of a tetrahedron (the demicube). The vertex figure of the demitesseract is a rectified tetrahedron, which is an octahedron, the vertex figure of a regular hexadecachoron (the demitesseract). And so on.
* Euclidean n-space can always be uniformly honeycombed by half measure polytopes and cross polytopes: In the regular honeycomb of n-space by measure polytopes, remove alternate vertices, thereby transforming each measure polytope into a half measure polytope, and fill the gaps with n-dimensional cross polytopes (the vertex figures of the measure polytope honeycomb). In the plane, this changes the checkerboard tiling into another checkerboard tiling rotated 45° to the original; in 3-space this produces the uniform honeycomb of regular tetrahedra and octahedra; and in 4-space this produces the regular honeycomb of hexadecachora
* Vertices: Take those coordinates (of a cube) which have an even number of minus signs.
* Edges: ?
The polytopes are irregular because the Dynkin diagram is not a path but has a branch. Thus there should be not one center or volume but rather the "generation" of the polytope should derive from two perspectives. These two perspectives may relate to the bipartition of the demicube which itself is a bipartition of the cube. The two perspectives are related by double edges.
There should be two different, complementary, flags of faces. And they should relate to the cross polytopes - one flag should be like the cross polytopes and the other flag should be the opposite - and together they should form a cube.
The pascal triangle can be explained by noting that we have the fusion of two paths in the Dynkin diagram, one of length 3 and one that grows without bound. The first nontrivial example is the 16 cell.
The purpose is to split the perspective. The human perspective is given by the short path.
Dn demihypercube construction
[[http://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html#Half | Half measure polytope]]
* Also known as a demihypercube in n-space, n>2. The uniform polytope constructed by completely truncating the alternate vertices of a measure polytope, that is, truncating half its vertices by hyperplanes that pass through a truncated vertex’s edge-neighboring vertices. If n=2, this truncation of a square leaves only the square’s diagonal, which is not a polygon. But when n=3, this truncation of a cube produces a regular tetrahedron, and when n=4, it produces a regular hexadecachoron from a tesseract. For n>4, demihypercubes are no longer regular, only uniform, with two different kinds of facets: 2n (n–1)-dimensional demihypercubes and 2n–1 (n–1)-dimensional simplexes. The names of the demihypercubes are constructed by prefixing demi- to the name of a measure polytope: demipenteract, demihexeract, demihepteract, and so forth. As noted above, a demicube is a regular tetrahedron, and a demitesseract is a regular hexadecachoron.
* The vertex figure of a half measure polytope H in n-space is a rectified (n–1)-dimensional simplex (the simplex has n vertices and n facets). The facets of the rectified simplex are (1) n (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional simplex facets of H, and (2) n rectified (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional demihypercube facets of H. Thus, for example, the vertex figure of the demicube is a rectified triangle, which is a smaller triangle, the vertex figure of a tetrahedron (the demicube). The vertex figure of the demitesseract is a rectified tetrahedron, which is an octahedron, the vertex figure of a regular hexadecachoron (the demitesseract). And so on.
* Euclidean n-space can always be uniformly honeycombed by half measure polytopes and cross polytopes: In the regular honeycomb of n-space by measure polytopes, remove alternate vertices, thereby transforming each measure polytope into a half measure polytope, and fill the gaps with n-dimensional cross polytopes (the vertex figures of the measure polytope honeycomb). In the plane, this changes the checkerboard tiling into another checkerboard tiling rotated 45° to the original; in 3-space this produces the uniform honeycomb of regular tetrahedra and octahedra; and in 4-space this produces the regular honeycomb of hexadecachora
* Vertices: Take those coordinates (of a cube) which have an even number of minus signs.
* Edges: ?
The polytopes are irregular because the Dynkin diagram is not a path but has a branch. Thus there should be not one center or volume but rather the "generation" of the polytope should derive from two perspectives. These two perspectives may relate to the bipartition of the demicube which itself is a bipartition of the cube. The two perspectives are related by double edges.
There should be two different, complementary, flags of faces. And they should relate to the cross polytopes - one flag should be like the cross polytopes and the other flag should be the opposite - and together they should form a cube.
The pascal triangle can be explained by noting that we have the fusion of two paths in the Dynkin diagram, one of length 3 and one that grows without bound. The first nontrivial example is the 16 cell.
The purpose is to split the perspective. The human perspective is given by the short path.
Pridėtos 51-84 eilutės:
http://www.sciencedirect.com/science/article/pii/S0001870809001017 Homology representations arising from the half cube
http://math.ucr.edu/home/baez/week187.html The Coxeter group for Dn is the subgroup of the symmetries of the n-dimensional cube generated by permutations of the coordinate axes and reflections along ''pairs'' of coordinate axes.
http://link.springer.com/chapter/10.1007/978-3-642-04295-9_1 geometry of cuts and metrics
http://arxiv.org/abs/1506.06702 paper about 16 cube
nim and sierpinski demihypercube
[[https://plus.google.com/117663015413546257905/posts/ZCyDfcc8gRQ | John Baez on demicubes]]
http://link.springer.com/chapter/10.1007/978-3-642-21590-2_15 nth roots of pitch class inversion
'''Coxeter groups and polytopes'''
* dynkin diagram coxeter group An reflections are encoded by the sequence 1, q2, q3, ..., q*(n-1)
* symmetry group is Sn
* Koch snowflake is an illustration of An for all n.
* Bn hypercube construction given by having the volume grow by having the volume be conceived as a mirror from which two mirror images arise on either sides. Thus mirror points are linked by edges.
* Cn cross-polytope construction given by having two points extend from the center in opposite directions. Each of the points links to all of the existing vertices. The old volume gives way to a new volume of one dimension higher. The initial C is given by two unconnected points. And these two points came from the center, which is why they are unconnected.
* Thus we have duality of adding two opposing vertices or adding a hyperplane with two reflections, of growing the center or the volume. We have the duality of vertices and facets.
Symmetry groups
* An: symmetric group
* Bn: A deck of n cards where card j has j written on one side and -j on the other side. All possible arrangements of stacks of cards with orientation.
* Dn: corresponds to all arrangements of stacks of n cards with orientation with an even number of cards turned over (pairs of reflections).
'''Exceptional groups'''
Also this suggests that the exceptional groups are modeling interactions of limited human perspectives.
'''Generative center'''
http://math.ucr.edu/home/baez/week187.html The Coxeter group for Dn is the subgroup of the symmetries of the n-dimensional cube generated by permutations of the coordinate axes and reflections along ''pairs'' of coordinate axes.
http://link.springer.com/chapter/10.1007/978-3-642-04295-9_1 geometry of cuts and metrics
http://arxiv.org/abs/1506.06702 paper about 16 cube
nim and sierpinski demihypercube
[[https://plus.google.com/117663015413546257905/posts/ZCyDfcc8gRQ | John Baez on demicubes]]
http://link.springer.com/chapter/10.1007/978-3-642-21590-2_15 nth roots of pitch class inversion
'''Coxeter groups and polytopes'''
* dynkin diagram coxeter group An reflections are encoded by the sequence 1, q2, q3, ..., q*(n-1)
* symmetry group is Sn
* Koch snowflake is an illustration of An for all n.
* Bn hypercube construction given by having the volume grow by having the volume be conceived as a mirror from which two mirror images arise on either sides. Thus mirror points are linked by edges.
* Cn cross-polytope construction given by having two points extend from the center in opposite directions. Each of the points links to all of the existing vertices. The old volume gives way to a new volume of one dimension higher. The initial C is given by two unconnected points. And these two points came from the center, which is why they are unconnected.
* Thus we have duality of adding two opposing vertices or adding a hyperplane with two reflections, of growing the center or the volume. We have the duality of vertices and facets.
Symmetry groups
* An: symmetric group
* Bn: A deck of n cards where card j has j written on one side and -j on the other side. All possible arrangements of stacks of cards with orientation.
* Dn: corresponds to all arrangements of stacks of n cards with orientation with an even number of cards turned over (pairs of reflections).
'''Exceptional groups'''
Also this suggests that the exceptional groups are modeling interactions of limited human perspectives.
'''Generative center'''
Pridėtos 89-96 eilutės:
center is what gives 3 rather than 2. center is what links involutions in the dynkin diagram.
the linked involutions, the edges, are numbered and are added as the normal form which is preserved.
An add the center along with edges from the center to the other vertices. Then you can revision in a higher dimension.
'''Q-analogue'''
the linked involutions, the edges, are numbered and are added as the normal form which is preserved.
An add the center along with edges from the center to the other vertices. Then you can revision in a higher dimension.
'''Q-analogue'''
Pakeistos 105-167 eilutės iš
symmetry group is Sn
center is what gives 3 rather than 2. center is what links involutions in the dynkin diagram
An add the center along with edges from the center to the other vertices
Cn cross-polytope construction given by having two points extend from the center in opposite directions. Each of the points links to all of the existing vertices. The old volume gives way to a new volume of one dimension higher. The initial C is given by two unconnected points. And these two points came from the center, which is why they are unconnected.
Bn hypercube construction given by having the volume grow by having the volume be conceived as a mirror from which two mirror images arise on either sides. Thus mirror points are linked by edges
Thus we have duality of adding two opposing vertices or adding a hyperplane with two reflections, of growing the center or the volume
Dn demihypercube construction
Koch snowflake is an illustration of An for all n.
[[http://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html#Half | Half measure polytope]]
* Also known as a demihypercube in n-space, n>2. The uniform polytope constructed by completely truncating the alternate vertices of a measure polytope, that is, truncating half its vertices by hyperplanes that pass through a truncated vertex’s edge-neighboring vertices. If n=2, this truncation of a square leaves only the square’s diagonal, which is not a polygon. But when n=3, this truncation of a cube produces a regular tetrahedron, and when n=4, it produces a regular hexadecachoron from a tesseract. For n>4, demihypercubes are no longer regular, only uniform, with two different kinds of facets: 2n (n–1)-dimensional demihypercubes and 2n–1 (n–1)-dimensional simplexes. The names of the demihypercubes are constructed by prefixing demi- to the name of a measure polytope: demipenteract, demihexeract, demihepteract, and so forth. As noted above, a demicube is a regular tetrahedron, and a demitesseract is a regular hexadecachoron.
* The vertex figure of a half measure polytope H in n-space is a rectified (n–1)-dimensional simplex (the simplex has n vertices and n facets). The facets of the rectified simplex are (1) n (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional simplex facets of H, and (2) n rectified (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional demihypercube facets of H. Thus, for example, the vertex figure of the demicube is a rectified triangle, which is a smaller triangle, the vertex figure of a tetrahedron (the demicube). The vertex figure of the demitesseract is a rectified tetrahedron, which is an octahedron, the vertex figure of a regular hexadecachoron (the demitesseract). And so on.
* Euclidean n-space can always be uniformly honeycombed by half measure polytopes and cross polytopes: In the regular honeycomb of n-space by measure polytopes, remove alternate vertices, thereby transforming each measure polytope into a half measure polytope, and fill the gaps with n-dimensional cross polytopes (the vertex figures of the measure polytope honeycomb). In the plane, this changes the checkerboard tiling into another checkerboard tiling rotated 45° to the original; in 3-space this produces the uniform honeycomb of regular tetrahedra and octahedra; and in 4-space this produces the regular honeycomb of hexadecachora
* An: symmetric group
* Bn: A deck of n cards where card j has j written on one side and -j on the other side. All possible arrangements of stacks of cards with orientation.
* Dn: corresponds to all arrangements of stacks of n cards with orientation with an even number of cards turned over (pairs of reflections).
'''Demicubes'''
* Vertices: Take those coordinates (of a cube) which have an even number of minus signs.
* Edges: ?
The polytopes are irregular because the Dynkin diagram is not a path but has a branch. Thus there should be not one center or volume but rather the "generation" of the polytope should derive from two perspectives. These two perspectives may relate to the bipartition of the demicube which itself is a bipartition of the cube. The two perspectives are related by double edges.
There should be two different, complementary, flags of faces. And they should relate to the cross polytopes - one flag should be like the cross polytopes and the other flag should be the opposite - and together they should form a cube.
The pascal triangle can be explained by noting that we have the fusion of two paths in the Dynkin diagram, one of length 3 and one that grows without bound. The first nontrivial example is the 16 cell.
The purpose is to split the perspective. The human perspective is given by the short path.
Also this suggests that the exceptional groups are modeling interactions of limited human perspectives.
http://www.sciencedirect.com/science/article/pii/S0001870809001017 Homology representations arising from the half cube
http://math.ucr.edu/home/baez/week187.html The Coxeter group for Dn is the subgroup of the symmetries of the n-dimensional cube generated by permutations of the coordinate axes and reflections along ''pairs'' of coordinate axes.
http://link.springer.com/chapter/10.1007/978-3-642-04295-9_1 geometry of cuts and metrics
http://arxiv.org/abs/1506.06702 paper about 16 cube
nim and sierpinski demihypercube
[[https://plus.google.com/117663015413546257905/posts/ZCyDfcc8gRQ | John Baez on demicubes]]
http://link.springer.com/chapter/10.1007/978-3-642-21590-2_15 nth roots of pitch class inversion
į:
All of the nodes chip in a weight "q" to allow the new weight to be distinguished from the rest "q^k". They all go down by one (temporarily?) to stay distinguished, to meet their previous obligations.
[[https://qchu.wordpress.com/2009/06/11/young-diagrams-q-analogues-and-one-of-my-favorite-proofs/#more-266 | Young diagrams - q-analogue Chu]]
[[https://www.bowdoin.edu/faculty/f/fisk/dissemination/homology-projective.pdf | q-analogue of a simplex is a projective space over a finite field]] homology
'''Theological interpretation'''
Would like to believe that God is good. Eternal life, eternal learning is driven by the building up that God inspires. And all learning is dependent. But at any height there is the dual: we can unlearn (or tear down) whatever we learn (or build up). Unlearning is independent like the cube. But in between is the space of eternal life. It is based on the idea that God doesn't have to be good, life doesn't have to be fair. We can unlearn in order to learn. And ultimately we hope that truly God is good, that truly we can learn without unlearning. That is what we are demonstrating. So it is related to John Harland's quest about learning.
[[https://qchu.wordpress.com/2009/06/11/young-diagrams-q-analogues-and-one-of-my-favorite-proofs/#more-266 | Young diagrams - q-analogue Chu]]
[[https://www.bowdoin.edu/faculty/f/fisk/dissemination/homology-projective.pdf | q-analogue of a simplex is a projective space over a finite field]] homology
'''Theological interpretation'''
Would like to believe that God is good. Eternal life, eternal learning is driven by the building up that God inspires. And all learning is dependent. But at any height there is the dual: we can unlearn (or tear down) whatever we learn (or build up). Unlearning is independent like the cube. But in between is the space of eternal life. It is based on the idea that God doesn't have to be good, life doesn't have to be fair. We can unlearn in order to learn. And ultimately we hope that truly God is good, that truly we can learn without unlearning. That is what we are demonstrating. So it is related to John Harland's quest about learning.
Ištrintos 157-161 eilutės:
[[https://qchu.wordpress.com/2009/06/11/young-diagrams-q-analogues-and-one-of-my-favorite-proofs/#more-266 | Young diagrams - q-analogue Chu]]
[[https://www.bowdoin.edu/faculty/f/fisk/dissemination/homology-projective.pdf | q-analogue of a simplex is a projective space over a finite field]] homology
-------------------
2016 birželio 09 d., 20:51
atliko -
2016 birželio 09 d., 20:16
atliko -
Pridėtos 28-29 eilutės:
Would like to believe that God is good. Eternal life, eternal learning is driven by the building up that God inspires. And all learning is dependent. But at any height there is the dual: we can unlearn (or tear down) whatever we learn (or build up). Unlearning is independent like the cube. But in between is the space of eternal life. It is based on the idea that God doesn't have to be good, life doesn't have to be fair. We can unlearn in order to learn. And ultimately we hope that truly God is good, that truly we can learn without unlearning. That is what we are demonstrating. So it is related to John Harland's quest about learning.
2016 birželio 09 d., 16:25
atliko -
Pridėtos 28-29 eilutės:
[[https://books.google.lt/books?id=HarWCwAAQBAJ&printsec=frontcover#v=onepage&q&f=false | The Higher Dimensional Hemicubeoctahedron]] Daniel Pellicer, in: Symmetries of Graphs, Maps and Polytopes. 2014.
2016 birželio 09 d., 16:12
atliko -
Pakeistos 22-25 eilutės iš
study stanley's proof of the volume of a hypersimplex. alexander postnikov and thomas lam alcoved polytopes
į:
* study stanley's proof of the volume of a hypersimplex. alexander postnikov and thomas lam alcoved polytopes
* What does it mean if the center forms 3 new nodes at each step? (We get a hexagon with 3 vertices and 3 edges each. This could be mounted on a cube with a hole. Then we get 6 vertices with 6 edges each, and so on.)
* What does it mean if the center forms k new nodes at each step?
* What is the polytope which G2 gives the symmetries of?
* What does it mean if the center forms 3 new nodes at each step? (We get a hexagon with 3 vertices and 3 edges each. This could be mounted on a cube with a hole. Then we get 6 vertices with 6 edges each, and so on.)
* What does it mean if the center forms k new nodes at each step?
* What is the polytope which G2 gives the symmetries of?
2016 birželio 09 d., 13:47
atliko -
Pakeista 22 eilutė iš:
study stanley's proof of the volume of a hypersimplex
į:
study stanley's proof of the volume of a hypersimplex. alexander postnikov and thomas lam alcoved polytopes
2016 birželio 09 d., 13:44
atliko -
Pridėtos 21-22 eilutės:
study stanley's proof of the volume of a hypersimplex
2016 birželio 08 d., 00:06
atliko -
Pridėtos 19-20 eilutės:
Understand the generator point and mirror construction here https://en.m.wikipedia.org/wiki/Coxeter–Dynkin_diagram and the ringed nodes
2016 birželio 07 d., 21:25
atliko -
Pridėtos 142-143 eilutės:
[[https://www.bowdoin.edu/faculty/f/fisk/dissemination/homology-projective.pdf | q-analogue of a simplex is a projective space over a finite field]] homology
Ištrintos 144-359 eilutės:
[+Laiškas+]
Dear mathematicians,
I contribute my own perspective in the spirit of Harvey Friedman's invitation "...I am trying to get a dialog going on the FOM and in these other forums as to "what foundations of mathematics are, ought to be, and what purpose they serve"."
http://www.cs.nyu.edu/pipermail/fom/2016-April/019724.html
I propose to distinguish implicit math (which our minds intuit and interpret) and explicit math (which is expressed as written symbols). I apply insights from implicit math to yield results in explicit math, specifically, interpretations of the -1 simplex, the Gaussian binomial coefficients, the generation of the regular polytopes An, Bn, Cn, as well as ideas as to how we might think about F1, the field with one element. These results may or may not be new but I hope to persuade you that both explicit math and implicit math may benefit from a mutual conversation. In particular, I pose a question about the generation of the irregular polytopes Dn. I also suggest the development of a set theory based on finite fields of characteristic q such that q is interpreted as infinity.
-------------------------------------------
The distinct advantages of
explicit and implicit math
-------------------------------------------
Harvey Friedman has noted: "Among the greatest mathematical events of all time (different than greatest mathematics of all time) are results of the form: this or that mathematical question cannot be proved or refuted using accepted mathematical standards for rigorous proofs." Referring, in particular, to Goedel's Incompleteness Theorems. And he continues, "We need a general purpose foundation for mathematics where mathematicians readily see that ANYTHING that is done throughout the whole of rigorous mathematics is OBVIOUSLY incorporated in that general purpose foundation for mathematics."
http://www.cs.nyu.edu/pipermail/fom/2016-April/019727.html
Taking his point more broadly, and his word "cannot" more broadly, some of the greatest mathematics is that which axiomatizes conditions so rigorously that it is able to prove what CANNOT be done:
* circling the square
* trisecting the angle
* enumerating the real numbers
All statements to be taken under the relevant conditions. Similarly, great results include the classification of Platonic solids, Lie groups, simple groups, and so on, where it is shown that one CANNOT construct additional objects.
These are all magnificent accomplishments of explicit mathematics. Moreover, millions of people are able to use explicit math to work together, to write, read and check rigorous thousand-page proofs and even computer-assisted proofs. Furthermore, Harvey Friedman notes that this all depends on a robust consistency, a lack of contradiction. This grows in impressiveness along with the mathematical community.
The distinction between explicit and implicit math comes up in Harvey Friedman's exchanges with John Baez. John writes about the roles of explicitly stated rules and implicitly active mathematical taste: "However, some rules are more interesting than others, and it's one of the highest expressions of mathematical taste to be able to choose interesting rules to work with."
http://www.cs.nyu.edu/pipermail/fom/2016-April/019664.html
I suppose there are even fewer university courses, if any, on "mathematical taste" than on "foundations of mathematics". But this unspoken notion is at play in a range of mathematical discoveries which I think are just as significant but open up what mathematicians CAN do. These include "introducing" and "playing" with:
* irrational numbers
* imaginary numbers
* infinitesimals
* group actions
* infinite series
* fractals
"Sound" axiomatic systems typically arose decades after much experimentation led by mathematical inquiry and taste. Implicit math, if developed as a science, would seek and find principles for play, inquiry, taste and so on.
Furthermore, implicit math can consider questions that make sense in a dynamic cognitive environment but not in a static document. In the human mind, the logical assumptions can change. Logic can be dynamic. Indeed, the human mind can analyze counterfactuals, can consider logical alternatives. We can even take the state of contradiction as a starting point and consider how a noncontradictory system might evolve from it. This type of "experienced" logic might yield a novel intepretation of Goedel's Incompleteness Theorem as regards what it means for one who actually lives in an unfolding system. Explicit math might usefully model part of this but I think not absolutely all of it.
My own interest is to know everything and apply that knowledge usefully. I aspire to model the limits of our mind. We experience our lives internally, not externally. I feel that I have been able to model much of that internal life. In order to share that, I want to show that it relates usefully to math and physics. I have had some success with some aspects of implicit math. But, in my experience, mathematicians avoid considering implicit math. I am thus sharing some initial results which I think are relevant for explicit math.
-------------------------------------------
A challenge for explicit math:
The field with one element
-------------------------------------------
In defending classical set thory (ZFC=Zermelo-Fraenkel axioms along with the Axiom of Choice), Harvey Friedman asked if there was any issue in mathematics which it is not addressing successfully but another foundations might.
http://www.cs.nyu.edu/pipermail/fom/2016-April/019656.html
Currently, there is a mathematical object which has attracted attention but is not yet grounded in set theory and perhaps might never be. I am thinking of F1, the field with one element, which Jacques Tits proposed in 1956 as he introduced his theory of buildings in algebraic geometry.
https://en.wikipedia.org/wiki/Field_with_one_element
There are many areas of math where such a field suggests itself:
https://ncatlab.org/nlab/show/field+with+one+element
Conferences have been devoted to this object. It appears as #14 in Richard Stanley's survey of most intricate and beautiful mathematical structures.
http://mathoverflow.net/questions/49151/most-intricate-and-most-beautiful-structures-in-mathematics
However, this object does not exist! There is no finite field in which 0 and 1 would be the same number. That is, as of yet, there is no settled way to make sense of such a field-like object, which suggests itself in so many areas of math.
Here is a discussion
https://golem.ph.utexas.edu/category/2007/04/the_field_with_one_element.html
of a 586 page paper where Nikolai Durov describes it as "the free algebraic monad generated by one constant or the universal generalized ring with zero".
http://arxiv.org/abs/0704.2030
I present my own approach to show it is simple but fruitful.
-------------------------------------------
Examples of implicit math:
A "mind game" and opposites.
-------------------------------------------
I will start by giving an example of reasoning that occurs in implicit math. Here is a "mind game" which serves to define the concepts "one", "all" and "many":
Search for constancy. Either you find "one" example of constancy or "all" is constantly unconstant. But in searching thus, you need to know that in each case, what you select is the same as what you inspect, that is, "many" are constant.
Thus we define "singularly constant", "universally constant" and "multiply constant". This kind of reasoning is standard in recursive function theory. Note that it did not require any explicit symbols. Sure, I used words, but it is the concepts that matter. It could be thought without any words at all. It is the kind of reasoning that I imagine takes place even in the womb. This "mind game" is a participatory example of how to define primitives without appealing to even simpler primitives. They are defined relative to each other through an activity which evokes them. And they are actually defined more precisely, I think, then they are ever defined in set theory. The concept "one", as defined above, supposes that a search algorithm halts, and the concept "all" supposes that it does not. Immanuel Kant tried to derive these same three concepts as "categories" by appealing to the form of a logical implication, which however, others never took up. I simply want to suggest that such reasoning is possible, and that such metaphysical, pre-mathematical thinking can ground implicit math and ultimately even explicit math, physics and other domains of understanding.
Now I want to point out three different kinds of "opposites" that are relevant for implicit math and can be observed in explicit math as well.
By "implicit opposites", I mean that I may be presented with two choices which I cannot distinguish in any way except that one is not the other. We may call them "unlabelled opposites". Of course, this is difficult to talk about because we end up introducing distinctions. Still, there is a sense in which we can be absolutely indifferent as to "clockwise" and "counterclockwise". In math, there are two square roots of -1 and we have no way to distinguish them. We call one of them i, but which one? Do you call the same one i that I do? It shouldn't matter. And this makes clear to me, as I never understood before, why complex conjugation is so important, because that is simply reminding us that on some level it shouldn't matter which is which. So the opposition here between the two square roots is mathematically implicit, in our mind.
By "explicit opposites", I mean that the two choices are expressed and one is primary, "unmarked", and the other is secondary, "marked". Linguistically, we may have the opposites "happy" and "unhappy", and the derivation "unhappy" is marked with the negation "un". Mathematically, we have 1 and -1. Generally, in my thinking, it's not so much the actual symbols that are most important but rather the way we think about them. Be that as it may, mathematically, there are natural reasons to distinguish 1 and -1, for example, 1 times itself is 1 whereas -1 times itself is 1. Indeed, it is reasonable for 1 to be unmarked and -1 to be marked. However, I think it is very unhelpful that the square roots of -1 are written as i and -i rather than, say, i and j. Implicit, unlabelled opposites are being presented as explicit, unmarked-marked opposites. I have a Ph.D. in math and yet only recently did I realize that I have a completely baseless prejudice by which I think of i as more basic than -i. It is as if I thought that "clockwise" was more clocklike than "counterclockwise" or "lefthandness" was sinister and "righthandness" was righteous.
By "mixed opposites", I mean that an implicit opposite is opposed by an explicit opposite. In other words, an undefined notion is opposed by a defined notion. This is especially common when we oppose content and the form or symbol that conveys it. For example, a "nonaction" may be written down as the identity element e. The emptiness of a box may be referenced by the number 0 or the empty set.
I continue by noting why I find the "field with one element" interesting. It must be similar to both 0 and 1, to both the empty set and its empty content. Thus it may relate to my own insights into God and Everything as well as the kinds of opposites that I described above.
---------------------------------
The -1 Simplex
is the center of a simplex
---------------------------------
I will share my exploration of a simple example where the "field with one element" comes up. Namely, the (n,k) entry of the Gaussian binomial coefficients is a q-polynomial which counts the number of k-dimensional subspaces of an n-dimensional vector space over a finite field of characteristic q. When q is set to 1, then we recover the number of subsets of size k of a set of size n. These latter subsets of a set can thus be imagined as 1-dimensional subspaces of an n-dimensional vector space over the (mysterious or nonexistent) finite field of characteristic 1.
In trying to understand this, I learned that the binomial coefficients (n,k) count (among so many other things) the number of k-simplexes within an n-simplex. A 2-simplex is an equilateral triangle, a 3-simplex is a tetrahedron, and there is a nice table of vertices, edges, faces at the Wikipedia article:
https://en.wikipedia.org/wiki/Simplex
From the article it is striking that the 1-simplex is an edge, a 0-simplex is a point and from the binomial coefficients there ought to be an interpretation for a -1-simplex. There should always be one -1-simplex in any k-simplex. Wikipedia explains that the -1 simplex is sometimes defined as the empty set. I imagined that there should be a more informative interpretation which might be a first step in interpreting F1.
We can consider the n-simplexes and the k-simplexes inside of them to be generated as the terms of the expansion:
(chosen + unchosen)^n
Thus the -1 simplex can be interpreted as the unique way of choosing no vertices.
My solution is that the -1 simplex is the unique "center" of the k-simplex in question. And that "center" is defined implicitly, that is, we must imagine it with our minds. It is not explicitly indicated as are the vertices and edges. So I think it is a great example of a well-defined mathematical object which, however, is not defined by set theory in any natural, helpful or faithful way. It is very much an implicit object. Let me show you how the simplexes unfold from it!
First we have just the "center".
Next, the implicit center finds expression as an explicit point. They are a mixed opposite. They look like a fish eye. It is seeing zero dimensions.
We can think of the point as a constant and the center as a sliding variable. The center can thus define a dimension. And it can express itself as a point along that dimension. Thus we have a line segment. And it also has a center, I suppose, the same center.
Now we can imagine that center as a variable, let us say, sliding up and down. It is as if we are seeing a vertical triangle from above, so that it looks like a line segment to us. Let the center find expression as a point. Then indeed we have a triangle.
Now the triangle has a center. Again we can imagine it as sliding up and down a dimension. Of course, we can picture it in 2-dimensions or in 3-dimensions as well, but in either case, when the center finds expression as a point, then we have a tetrahedron. (If we are in 2-dimensions, then we could unfold the new faces and imagine the tetrahedron and subsequent k-simplexes as a tiling of the plane.)
And likewise the tetrahedron has a center. And if we draw that center then we can imagine the 4-simplex in 4 dimensions. We can practically see it, the 5 volumes, 10 faces, 10 edges and 5 vertices, how they must be organized! If we like, then to visualize this we can invert the tetrahedrons so that the 4 new tetrahedrons are on the faces of the old one.
This is a wonderful thing, this link between implicit and explicit math. In thinking one center and drawing five points, I feel the solemnity of the center, the spirited gaze of that first point, the companionship in the second point, the humor when our third point opens up like a tent, the beauty of the pyramid that surveys our spatial imagination, and the power in the fifth point at its center which has us transcend the world we know.
We can define the q-analogue by:
* giving the first vertex weight 1
* multiplying the weight of each successive vertex by q, so that they are 1, q, q^2, q^3...
* balancing the new vertex of weight q^k with k new edges, each of weight 1/q.
Then for each k-simplex, if we multiply the weights of its vertices and edges, we get exactly the terms of the Gaussian binomial coefficients. This can be proven by recursion or bijectively.
Note that the weight of the center is simply 1. Also, the center has a dual, the entire volume, whose weight is always 1, as is clear from the construction. The fact that the weights of the k-simplexes are polynomials in q means that their vertices outweigh their edges.
The beauty of this construction suggests that it is quite "correct" and even in harmony with whatever may be "absolutely correct". This kind of thinking is usual in algebraic combinatorics. Indeed, I studied it because I thought of it as the "basement" of mathematics where it is most convenient to glimpse how mathematical objects arise. However, my impression was that combinatorics is generally not considered "real math" because its results and techniques are too concrete. But precisely for this reason it is a most convenient domain for grappling with philosophical questions of interpretation. Even the analytic combinatorialists Philippe Flajolet and Robert Sedgewick write excitedly about "universality phenomenon". Are such fundamental, practical issues avoided in the foundations of mathematics and the philosophy of mathematics? Is it because we reject the notion of "interpretation" and the world of "implicit math" or even "absolute truth"? If we establish a very narrow scope of "rigor" are we truly safe from anything grander by deeming it as "not mathematics"? Or is there not a fish eye lurking at every point?
-------------------------------------------
Gaussian Binomial Coefficients
define what it means to count
-------------------------------------------
Let us now interpret the Gaussian binomial coefficients as the number of k-dimensional subspaces of an n-dimensional vector space over Fq, a field of characteristic q. I will explain how to interpret q-analogues of the integers n, n! and n-choose-k because the details are informative.
The q-analogue of the number n counts the number of 1-dimensional subspaces of an n-dimensional vector space V over Fq. That subspace is determined by any single nonzero element of the vector space. Thus there are q^(n–1) such elements. However, we are overcounting, because multiplying by a nonzero scalar gives us the same subspace. Thus, dividing by the number of nonzero scalars q-1, we get (q^n – 1) / (q-1) = 1 + q + q^2 + .... + q^(n-1). And if q=1, then we get n as promised.
Alternatively and significantly, we can fix a basis of V: e1, e2, ..., en–1. Then e1 defines a 1-dimensional subspace. But so does any combination f1 * e1 + e2 where f1 is any one of the elements of Fq including zero. And a 1-dimensional subspace is also defined by any combination f1 * e1 + f2 * e2 + e3. In this manner, the possible 1-dimensional subspaces grow as 1 + q + q^2 etc.
Now the q-analogue of n! counts the number of total flags, that is, the ways of adding 1-dimensional subspaces until they yield all of V. But that is rather clear because we keep looking at the remainder of V, which keeps shrinking by 1-dimension, and so we multiply the q-analogues of n x (n-1) x (n-2) ... = n!
Finally, we want to get the q-analogue of n-choose-k = n!/(k!*(n-k!) which is the number of k-dimensional subspaces of V. We use the construction above to remove (n-k) 1-dimensional subspaces from V yielding n!/k! k-dimensional subspaces. We need to know how much we have overcounted by, that is, how many times a subspace repeats. But that is given by the number of flags of (n-k) dimensions, which is to say, the number of ways our subspace K could be built up to give V is also the number of ways that V could be deconstructed to yield K.
We can now draw an analogy between the q-analogies for the simplexes and the vector subspaces. The vertices are weighted 1 + q + q^2 + ... These weights arise in the construction of the 1-dimensional vector space because each new basis element ek participates in a chain of weights f1 * e1 + f2 * e2 + ... + fk-1 * ek-1 + ek where each fi represents q choices and so ek has weight q^(k-1). The (k-1) edges are weighted 1/q. But that means that every time we extend the chain the weight grows by q except for the very first basis element.
So what does this mean when we set q=1 ?
Let us recall how we generated the simplexes and what we gained by considering q-analogues. The simplex is generated by relating the "new" vertex to all of the previous vertices. Thus all of the vertices are related to each other by edges. It is as if we are constructing the natural numbers. However, the construction simply emphasizes by way of the edges that the new vertex is different from each of the existing vertices. But otherwise all of the existing vertices are indistinguishable. The center "knows" which one is the "new" vertex. But then it is forgotten, that is, it is unlabelled, like the rest. Although they all know they are different from each other as expressed by the edges. In a sense, this is what "counting" is all about, except that there is no "answer" stored. There is a growing set or "bag" of indistinguishables, which is to say, it's not a traditional set.
When we add the q-weights, then each vertex gets a distinct label q^k which is the "cost" introduced to distinguish that vertex from all of the others. Also, we can think of each of the k edges as fairly bearing part of that cost 1/q. Now we have a total order. We also have an operation by which the center gives each new vertex a weight by multiplying q times the weight of the last new vertex. Thus we have an operation k -> k+1.
We also see that the center "marks" or "points" an initial number (we should call it "0") which is different than all of the other numbers (1,2,3...) even while it participates in generating them (as if by relations 0 + k + 1 = k+1).
I may not be describing this perfectly. But my point is that there seems to be a natural process at work here. It is as if we are literally seeing the God of math construct the "natural numbers" or rather, different versions of them. The most "natural" of the "numbers", if it is fair to call them that, are "pointed multisets" built from one element, "unmarked", where the Center points to the new element, like a hand placing a marble in a bag. Or a boson in a bag. In the q-analogue, all of the vertices get labels, as do all of the faces, and we have a total order, as well as a record of the operation k -> k+1. We have a set of fermions. These types of distinctions are natural in combinatorics as in the "twelvefold way":
https://en.wikipedia.org/wiki/Twelvefold_way
Again, set theory doesn't approach these naturally, implicitly, but rather requires that all knowledge be explicit and then removes it using equivalence classes. Which is to say, set theory destroys this entire way of thinking.
So what is happening when q=1 is that all of the "totally ordered" numbers become indistinguishable except for whichever one is in our hand, so to speak.
Note that the original vertex (or number) costs nothing (it has weight 1) and the reason is that it is not compared to any other vertex. Which is to say, the distinction of the center and the vertex comes for free. But when we add more vertices we need to keep them all distinct. Each relationship (each edge) bears the same weight (1/q) and so the weights (or costs) of the vertices progressively increases. Similarly, in the case of the vector space, the initial basis element e1 comes for free. But the subsequent basis elements come at an increasing cost. It is again the cost of choice q^(k-1), both the cost of being the k-th basis element, distinct from the others, and the cost of knowing (storing) the value of each of the scalars fi associated with the basis elements ei, where 0<=i<k.
Now we can also imagine what the q stands for as we generate the simplex. q is the number of choices (the colors) that we have to choose from for each facet.
q-analogue defines counting: an ordered set - a list. But here we don't have labels - we're not counting anything. Typically we write out finite sets in a particular order, although that order is
and when q=1 we have bagging: a bag
-------------------------------------------
The Field with One Element
Zero equals infinity
-------------------------------------------
This is why we get polynomials in q. However, there are for each n and k certain k-simplexes within the n-simplex which have weight exactly 1, that is, the vertices and the edges balance out. These are precisely those k-simplexes which are built from the k+1 vertices of smallest weight, namely: 1, q, q^2, q^3..., q^k.
If we set
"implicit opposites"
0 x 0 = 0
infinity x infinity = infinity
0 x infinity = undefined
undefined/0 = inverse of 0 = infinity
undefined/infinity = inverse of infinity = 0
0 + 0 = 0
0 + infinity = infinity
infinity + infinity = infinity
----------------------------------
Notes:
* Dn should come from BOTH center and volume. An comes from EITHER (building up and tearing down). Bn comes from VOLUME and Cn comes from CENTER.
* Tensorlikeness of An duality. Covariant and contravariant.
* An defines a total order - the integers - and also the relationship n->n+1. The former relationship arises directly (all are related to each other, and specifically, the newest to the rest), the latter relationship arises in the q-analogue, where we can compare the order.
* The total order shows that the set of relationships is natural, as in a category. And the set of subsets is natural.
* Edges can be interpreted as directed or nondirected.
* The duality or nonduality of vertices and edges. Vertices can label the edges (bottom-up) or edges can label the objects (top-down). Shows limitations (or not) of category theory (and set theory).
* Imre Latakos, definitions are not set in stone:
https://en.wikipedia.org/wiki/Proofs_and_Refutations
* Hypercube tiling - cross polytope tiling (invert) = demicube tiling (think of periodic tiling, then invert) Tile space with cross polytopes and the gaps are demicubes? Or tile space with n-cubes, transform to demicubes, the gaps are cross polytopes.
* Can reverse the weight construction: Volume has weight one. Remove a vertex. The subwhole has the weight of the missing vertex.
* The center is the manifest possibility of a distinct new vertex.
* Perhaps the new vertex gets a weight that shows the cost it has in being related to all of the other vertices and the edges show how that cost is born.
Questions:
* How to define the k-faces of the n-demicubes?
* Do the demicubes have single edges or double edges?
* What is special about the vertices, edges, faces of the demicubes that makes them distinct from the higher dimension faces?
* How to conceive of the demicubes as having k-demicubes + k-simplexes as in Dn,k = Cn,k + 2^(n-1-k) Cn,k+1 where Cn is for the cube
* How to picture a recurrence relation for demicubes, perhaps related to the relations with cubes: Dn = 1/2 Cn = (Cn–1)^2
* How to imagine the generation of demicubes?
* Why does a 4–cycle linked to a chain of 3-cycles make for independent Euclidean dimensions?
Figure out what the 4-cycle means in the Bn and Cn cases.
Figure out what it means to have a branching in the Coxeter diagram.
Interpret the four diagonals of the Dn triangle as the short path in the Coxeter diagram.
2016 birželio 07 d., 21:23
atliko -
Pakeistos 141-142 eilutės iš
į:
[[https://qchu.wordpress.com/2009/06/11/young-diagrams-q-analogues-and-one-of-my-favorite-proofs/#more-266 | Young diagrams - q-analogue Chu]]
2016 birželio 07 d., 19:03
atliko -
Pakeistos 139-359 eilutės iš
https://upload.wikimedia.org/wikipedia/commons/6/67/Wythoffian_construction_diagram.png
į:
https://upload.wikimedia.org/wikipedia/commons/6/67/Wythoffian_construction_diagram.png
-------------------
[+Laiškas+]
Dear mathematicians,
I contribute my own perspective in the spirit of Harvey Friedman's invitation "...I am trying to get a dialog going on the FOM and in these other forums as to "what foundations of mathematics are, ought to be, and what purpose they serve"."
http://www.cs.nyu.edu/pipermail/fom/2016-April/019724.html
I propose to distinguish implicit math (which our minds intuit and interpret) and explicit math (which is expressed as written symbols). I apply insights from implicit math to yield results in explicit math, specifically, interpretations of the -1 simplex, the Gaussian binomial coefficients, the generation of the regular polytopes An, Bn, Cn, as well as ideas as to how we might think about F1, the field with one element. These results may or may not be new but I hope to persuade you that both explicit math and implicit math may benefit from a mutual conversation. In particular, I pose a question about the generation of the irregular polytopes Dn. I also suggest the development of a set theory based on finite fields of characteristic q such that q is interpreted as infinity.
-------------------------------------------
The distinct advantages of
explicit and implicit math
-------------------------------------------
Harvey Friedman has noted: "Among the greatest mathematical events of all time (different than greatest mathematics of all time) are results of the form: this or that mathematical question cannot be proved or refuted using accepted mathematical standards for rigorous proofs." Referring, in particular, to Goedel's Incompleteness Theorems. And he continues, "We need a general purpose foundation for mathematics where mathematicians readily see that ANYTHING that is done throughout the whole of rigorous mathematics is OBVIOUSLY incorporated in that general purpose foundation for mathematics."
http://www.cs.nyu.edu/pipermail/fom/2016-April/019727.html
Taking his point more broadly, and his word "cannot" more broadly, some of the greatest mathematics is that which axiomatizes conditions so rigorously that it is able to prove what CANNOT be done:
* circling the square
* trisecting the angle
* enumerating the real numbers
All statements to be taken under the relevant conditions. Similarly, great results include the classification of Platonic solids, Lie groups, simple groups, and so on, where it is shown that one CANNOT construct additional objects.
These are all magnificent accomplishments of explicit mathematics. Moreover, millions of people are able to use explicit math to work together, to write, read and check rigorous thousand-page proofs and even computer-assisted proofs. Furthermore, Harvey Friedman notes that this all depends on a robust consistency, a lack of contradiction. This grows in impressiveness along with the mathematical community.
The distinction between explicit and implicit math comes up in Harvey Friedman's exchanges with John Baez. John writes about the roles of explicitly stated rules and implicitly active mathematical taste: "However, some rules are more interesting than others, and it's one of the highest expressions of mathematical taste to be able to choose interesting rules to work with."
http://www.cs.nyu.edu/pipermail/fom/2016-April/019664.html
I suppose there are even fewer university courses, if any, on "mathematical taste" than on "foundations of mathematics". But this unspoken notion is at play in a range of mathematical discoveries which I think are just as significant but open up what mathematicians CAN do. These include "introducing" and "playing" with:
* irrational numbers
* imaginary numbers
* infinitesimals
* group actions
* infinite series
* fractals
"Sound" axiomatic systems typically arose decades after much experimentation led by mathematical inquiry and taste. Implicit math, if developed as a science, would seek and find principles for play, inquiry, taste and so on.
Furthermore, implicit math can consider questions that make sense in a dynamic cognitive environment but not in a static document. In the human mind, the logical assumptions can change. Logic can be dynamic. Indeed, the human mind can analyze counterfactuals, can consider logical alternatives. We can even take the state of contradiction as a starting point and consider how a noncontradictory system might evolve from it. This type of "experienced" logic might yield a novel intepretation of Goedel's Incompleteness Theorem as regards what it means for one who actually lives in an unfolding system. Explicit math might usefully model part of this but I think not absolutely all of it.
My own interest is to know everything and apply that knowledge usefully. I aspire to model the limits of our mind. We experience our lives internally, not externally. I feel that I have been able to model much of that internal life. In order to share that, I want to show that it relates usefully to math and physics. I have had some success with some aspects of implicit math. But, in my experience, mathematicians avoid considering implicit math. I am thus sharing some initial results which I think are relevant for explicit math.
-------------------------------------------
A challenge for explicit math:
The field with one element
-------------------------------------------
In defending classical set thory (ZFC=Zermelo-Fraenkel axioms along with the Axiom of Choice), Harvey Friedman asked if there was any issue in mathematics which it is not addressing successfully but another foundations might.
http://www.cs.nyu.edu/pipermail/fom/2016-April/019656.html
Currently, there is a mathematical object which has attracted attention but is not yet grounded in set theory and perhaps might never be. I am thinking of F1, the field with one element, which Jacques Tits proposed in 1956 as he introduced his theory of buildings in algebraic geometry.
https://en.wikipedia.org/wiki/Field_with_one_element
There are many areas of math where such a field suggests itself:
https://ncatlab.org/nlab/show/field+with+one+element
Conferences have been devoted to this object. It appears as #14 in Richard Stanley's survey of most intricate and beautiful mathematical structures.
http://mathoverflow.net/questions/49151/most-intricate-and-most-beautiful-structures-in-mathematics
However, this object does not exist! There is no finite field in which 0 and 1 would be the same number. That is, as of yet, there is no settled way to make sense of such a field-like object, which suggests itself in so many areas of math.
Here is a discussion
https://golem.ph.utexas.edu/category/2007/04/the_field_with_one_element.html
of a 586 page paper where Nikolai Durov describes it as "the free algebraic monad generated by one constant or the universal generalized ring with zero".
http://arxiv.org/abs/0704.2030
I present my own approach to show it is simple but fruitful.
-------------------------------------------
Examples of implicit math:
A "mind game" and opposites.
-------------------------------------------
I will start by giving an example of reasoning that occurs in implicit math. Here is a "mind game" which serves to define the concepts "one", "all" and "many":
Search for constancy. Either you find "one" example of constancy or "all" is constantly unconstant. But in searching thus, you need to know that in each case, what you select is the same as what you inspect, that is, "many" are constant.
Thus we define "singularly constant", "universally constant" and "multiply constant". This kind of reasoning is standard in recursive function theory. Note that it did not require any explicit symbols. Sure, I used words, but it is the concepts that matter. It could be thought without any words at all. It is the kind of reasoning that I imagine takes place even in the womb. This "mind game" is a participatory example of how to define primitives without appealing to even simpler primitives. They are defined relative to each other through an activity which evokes them. And they are actually defined more precisely, I think, then they are ever defined in set theory. The concept "one", as defined above, supposes that a search algorithm halts, and the concept "all" supposes that it does not. Immanuel Kant tried to derive these same three concepts as "categories" by appealing to the form of a logical implication, which however, others never took up. I simply want to suggest that such reasoning is possible, and that such metaphysical, pre-mathematical thinking can ground implicit math and ultimately even explicit math, physics and other domains of understanding.
Now I want to point out three different kinds of "opposites" that are relevant for implicit math and can be observed in explicit math as well.
By "implicit opposites", I mean that I may be presented with two choices which I cannot distinguish in any way except that one is not the other. We may call them "unlabelled opposites". Of course, this is difficult to talk about because we end up introducing distinctions. Still, there is a sense in which we can be absolutely indifferent as to "clockwise" and "counterclockwise". In math, there are two square roots of -1 and we have no way to distinguish them. We call one of them i, but which one? Do you call the same one i that I do? It shouldn't matter. And this makes clear to me, as I never understood before, why complex conjugation is so important, because that is simply reminding us that on some level it shouldn't matter which is which. So the opposition here between the two square roots is mathematically implicit, in our mind.
By "explicit opposites", I mean that the two choices are expressed and one is primary, "unmarked", and the other is secondary, "marked". Linguistically, we may have the opposites "happy" and "unhappy", and the derivation "unhappy" is marked with the negation "un". Mathematically, we have 1 and -1. Generally, in my thinking, it's not so much the actual symbols that are most important but rather the way we think about them. Be that as it may, mathematically, there are natural reasons to distinguish 1 and -1, for example, 1 times itself is 1 whereas -1 times itself is 1. Indeed, it is reasonable for 1 to be unmarked and -1 to be marked. However, I think it is very unhelpful that the square roots of -1 are written as i and -i rather than, say, i and j. Implicit, unlabelled opposites are being presented as explicit, unmarked-marked opposites. I have a Ph.D. in math and yet only recently did I realize that I have a completely baseless prejudice by which I think of i as more basic than -i. It is as if I thought that "clockwise" was more clocklike than "counterclockwise" or "lefthandness" was sinister and "righthandness" was righteous.
By "mixed opposites", I mean that an implicit opposite is opposed by an explicit opposite. In other words, an undefined notion is opposed by a defined notion. This is especially common when we oppose content and the form or symbol that conveys it. For example, a "nonaction" may be written down as the identity element e. The emptiness of a box may be referenced by the number 0 or the empty set.
I continue by noting why I find the "field with one element" interesting. It must be similar to both 0 and 1, to both the empty set and its empty content. Thus it may relate to my own insights into God and Everything as well as the kinds of opposites that I described above.
---------------------------------
The -1 Simplex
is the center of a simplex
---------------------------------
I will share my exploration of a simple example where the "field with one element" comes up. Namely, the (n,k) entry of the Gaussian binomial coefficients is a q-polynomial which counts the number of k-dimensional subspaces of an n-dimensional vector space over a finite field of characteristic q. When q is set to 1, then we recover the number of subsets of size k of a set of size n. These latter subsets of a set can thus be imagined as 1-dimensional subspaces of an n-dimensional vector space over the (mysterious or nonexistent) finite field of characteristic 1.
In trying to understand this, I learned that the binomial coefficients (n,k) count (among so many other things) the number of k-simplexes within an n-simplex. A 2-simplex is an equilateral triangle, a 3-simplex is a tetrahedron, and there is a nice table of vertices, edges, faces at the Wikipedia article:
https://en.wikipedia.org/wiki/Simplex
From the article it is striking that the 1-simplex is an edge, a 0-simplex is a point and from the binomial coefficients there ought to be an interpretation for a -1-simplex. There should always be one -1-simplex in any k-simplex. Wikipedia explains that the -1 simplex is sometimes defined as the empty set. I imagined that there should be a more informative interpretation which might be a first step in interpreting F1.
We can consider the n-simplexes and the k-simplexes inside of them to be generated as the terms of the expansion:
(chosen + unchosen)^n
Thus the -1 simplex can be interpreted as the unique way of choosing no vertices.
My solution is that the -1 simplex is the unique "center" of the k-simplex in question. And that "center" is defined implicitly, that is, we must imagine it with our minds. It is not explicitly indicated as are the vertices and edges. So I think it is a great example of a well-defined mathematical object which, however, is not defined by set theory in any natural, helpful or faithful way. It is very much an implicit object. Let me show you how the simplexes unfold from it!
First we have just the "center".
Next, the implicit center finds expression as an explicit point. They are a mixed opposite. They look like a fish eye. It is seeing zero dimensions.
We can think of the point as a constant and the center as a sliding variable. The center can thus define a dimension. And it can express itself as a point along that dimension. Thus we have a line segment. And it also has a center, I suppose, the same center.
Now we can imagine that center as a variable, let us say, sliding up and down. It is as if we are seeing a vertical triangle from above, so that it looks like a line segment to us. Let the center find expression as a point. Then indeed we have a triangle.
Now the triangle has a center. Again we can imagine it as sliding up and down a dimension. Of course, we can picture it in 2-dimensions or in 3-dimensions as well, but in either case, when the center finds expression as a point, then we have a tetrahedron. (If we are in 2-dimensions, then we could unfold the new faces and imagine the tetrahedron and subsequent k-simplexes as a tiling of the plane.)
And likewise the tetrahedron has a center. And if we draw that center then we can imagine the 4-simplex in 4 dimensions. We can practically see it, the 5 volumes, 10 faces, 10 edges and 5 vertices, how they must be organized! If we like, then to visualize this we can invert the tetrahedrons so that the 4 new tetrahedrons are on the faces of the old one.
This is a wonderful thing, this link between implicit and explicit math. In thinking one center and drawing five points, I feel the solemnity of the center, the spirited gaze of that first point, the companionship in the second point, the humor when our third point opens up like a tent, the beauty of the pyramid that surveys our spatial imagination, and the power in the fifth point at its center which has us transcend the world we know.
We can define the q-analogue by:
* giving the first vertex weight 1
* multiplying the weight of each successive vertex by q, so that they are 1, q, q^2, q^3...
* balancing the new vertex of weight q^k with k new edges, each of weight 1/q.
Then for each k-simplex, if we multiply the weights of its vertices and edges, we get exactly the terms of the Gaussian binomial coefficients. This can be proven by recursion or bijectively.
Note that the weight of the center is simply 1. Also, the center has a dual, the entire volume, whose weight is always 1, as is clear from the construction. The fact that the weights of the k-simplexes are polynomials in q means that their vertices outweigh their edges.
The beauty of this construction suggests that it is quite "correct" and even in harmony with whatever may be "absolutely correct". This kind of thinking is usual in algebraic combinatorics. Indeed, I studied it because I thought of it as the "basement" of mathematics where it is most convenient to glimpse how mathematical objects arise. However, my impression was that combinatorics is generally not considered "real math" because its results and techniques are too concrete. But precisely for this reason it is a most convenient domain for grappling with philosophical questions of interpretation. Even the analytic combinatorialists Philippe Flajolet and Robert Sedgewick write excitedly about "universality phenomenon". Are such fundamental, practical issues avoided in the foundations of mathematics and the philosophy of mathematics? Is it because we reject the notion of "interpretation" and the world of "implicit math" or even "absolute truth"? If we establish a very narrow scope of "rigor" are we truly safe from anything grander by deeming it as "not mathematics"? Or is there not a fish eye lurking at every point?
-------------------------------------------
Gaussian Binomial Coefficients
define what it means to count
-------------------------------------------
Let us now interpret the Gaussian binomial coefficients as the number of k-dimensional subspaces of an n-dimensional vector space over Fq, a field of characteristic q. I will explain how to interpret q-analogues of the integers n, n! and n-choose-k because the details are informative.
The q-analogue of the number n counts the number of 1-dimensional subspaces of an n-dimensional vector space V over Fq. That subspace is determined by any single nonzero element of the vector space. Thus there are q^(n–1) such elements. However, we are overcounting, because multiplying by a nonzero scalar gives us the same subspace. Thus, dividing by the number of nonzero scalars q-1, we get (q^n – 1) / (q-1) = 1 + q + q^2 + .... + q^(n-1). And if q=1, then we get n as promised.
Alternatively and significantly, we can fix a basis of V: e1, e2, ..., en–1. Then e1 defines a 1-dimensional subspace. But so does any combination f1 * e1 + e2 where f1 is any one of the elements of Fq including zero. And a 1-dimensional subspace is also defined by any combination f1 * e1 + f2 * e2 + e3. In this manner, the possible 1-dimensional subspaces grow as 1 + q + q^2 etc.
Now the q-analogue of n! counts the number of total flags, that is, the ways of adding 1-dimensional subspaces until they yield all of V. But that is rather clear because we keep looking at the remainder of V, which keeps shrinking by 1-dimension, and so we multiply the q-analogues of n x (n-1) x (n-2) ... = n!
Finally, we want to get the q-analogue of n-choose-k = n!/(k!*(n-k!) which is the number of k-dimensional subspaces of V. We use the construction above to remove (n-k) 1-dimensional subspaces from V yielding n!/k! k-dimensional subspaces. We need to know how much we have overcounted by, that is, how many times a subspace repeats. But that is given by the number of flags of (n-k) dimensions, which is to say, the number of ways our subspace K could be built up to give V is also the number of ways that V could be deconstructed to yield K.
We can now draw an analogy between the q-analogies for the simplexes and the vector subspaces. The vertices are weighted 1 + q + q^2 + ... These weights arise in the construction of the 1-dimensional vector space because each new basis element ek participates in a chain of weights f1 * e1 + f2 * e2 + ... + fk-1 * ek-1 + ek where each fi represents q choices and so ek has weight q^(k-1). The (k-1) edges are weighted 1/q. But that means that every time we extend the chain the weight grows by q except for the very first basis element.
So what does this mean when we set q=1 ?
Let us recall how we generated the simplexes and what we gained by considering q-analogues. The simplex is generated by relating the "new" vertex to all of the previous vertices. Thus all of the vertices are related to each other by edges. It is as if we are constructing the natural numbers. However, the construction simply emphasizes by way of the edges that the new vertex is different from each of the existing vertices. But otherwise all of the existing vertices are indistinguishable. The center "knows" which one is the "new" vertex. But then it is forgotten, that is, it is unlabelled, like the rest. Although they all know they are different from each other as expressed by the edges. In a sense, this is what "counting" is all about, except that there is no "answer" stored. There is a growing set or "bag" of indistinguishables, which is to say, it's not a traditional set.
When we add the q-weights, then each vertex gets a distinct label q^k which is the "cost" introduced to distinguish that vertex from all of the others. Also, we can think of each of the k edges as fairly bearing part of that cost 1/q. Now we have a total order. We also have an operation by which the center gives each new vertex a weight by multiplying q times the weight of the last new vertex. Thus we have an operation k -> k+1.
We also see that the center "marks" or "points" an initial number (we should call it "0") which is different than all of the other numbers (1,2,3...) even while it participates in generating them (as if by relations 0 + k + 1 = k+1).
I may not be describing this perfectly. But my point is that there seems to be a natural process at work here. It is as if we are literally seeing the God of math construct the "natural numbers" or rather, different versions of them. The most "natural" of the "numbers", if it is fair to call them that, are "pointed multisets" built from one element, "unmarked", where the Center points to the new element, like a hand placing a marble in a bag. Or a boson in a bag. In the q-analogue, all of the vertices get labels, as do all of the faces, and we have a total order, as well as a record of the operation k -> k+1. We have a set of fermions. These types of distinctions are natural in combinatorics as in the "twelvefold way":
https://en.wikipedia.org/wiki/Twelvefold_way
Again, set theory doesn't approach these naturally, implicitly, but rather requires that all knowledge be explicit and then removes it using equivalence classes. Which is to say, set theory destroys this entire way of thinking.
So what is happening when q=1 is that all of the "totally ordered" numbers become indistinguishable except for whichever one is in our hand, so to speak.
Note that the original vertex (or number) costs nothing (it has weight 1) and the reason is that it is not compared to any other vertex. Which is to say, the distinction of the center and the vertex comes for free. But when we add more vertices we need to keep them all distinct. Each relationship (each edge) bears the same weight (1/q) and so the weights (or costs) of the vertices progressively increases. Similarly, in the case of the vector space, the initial basis element e1 comes for free. But the subsequent basis elements come at an increasing cost. It is again the cost of choice q^(k-1), both the cost of being the k-th basis element, distinct from the others, and the cost of knowing (storing) the value of each of the scalars fi associated with the basis elements ei, where 0<=i<k.
Now we can also imagine what the q stands for as we generate the simplex. q is the number of choices (the colors) that we have to choose from for each facet.
q-analogue defines counting: an ordered set - a list. But here we don't have labels - we're not counting anything. Typically we write out finite sets in a particular order, although that order is
and when q=1 we have bagging: a bag
-------------------------------------------
The Field with One Element
Zero equals infinity
-------------------------------------------
This is why we get polynomials in q. However, there are for each n and k certain k-simplexes within the n-simplex which have weight exactly 1, that is, the vertices and the edges balance out. These are precisely those k-simplexes which are built from the k+1 vertices of smallest weight, namely: 1, q, q^2, q^3..., q^k.
If we set
"implicit opposites"
0 x 0 = 0
infinity x infinity = infinity
0 x infinity = undefined
undefined/0 = inverse of 0 = infinity
undefined/infinity = inverse of infinity = 0
0 + 0 = 0
0 + infinity = infinity
infinity + infinity = infinity
----------------------------------
Notes:
* Dn should come from BOTH center and volume. An comes from EITHER (building up and tearing down). Bn comes from VOLUME and Cn comes from CENTER.
* Tensorlikeness of An duality. Covariant and contravariant.
* An defines a total order - the integers - and also the relationship n->n+1. The former relationship arises directly (all are related to each other, and specifically, the newest to the rest), the latter relationship arises in the q-analogue, where we can compare the order.
* The total order shows that the set of relationships is natural, as in a category. And the set of subsets is natural.
* Edges can be interpreted as directed or nondirected.
* The duality or nonduality of vertices and edges. Vertices can label the edges (bottom-up) or edges can label the objects (top-down). Shows limitations (or not) of category theory (and set theory).
* Imre Latakos, definitions are not set in stone:
https://en.wikipedia.org/wiki/Proofs_and_Refutations
* Hypercube tiling - cross polytope tiling (invert) = demicube tiling (think of periodic tiling, then invert) Tile space with cross polytopes and the gaps are demicubes? Or tile space with n-cubes, transform to demicubes, the gaps are cross polytopes.
* Can reverse the weight construction: Volume has weight one. Remove a vertex. The subwhole has the weight of the missing vertex.
* The center is the manifest possibility of a distinct new vertex.
* Perhaps the new vertex gets a weight that shows the cost it has in being related to all of the other vertices and the edges show how that cost is born.
Questions:
* How to define the k-faces of the n-demicubes?
* Do the demicubes have single edges or double edges?
* What is special about the vertices, edges, faces of the demicubes that makes them distinct from the higher dimension faces?
* How to conceive of the demicubes as having k-demicubes + k-simplexes as in Dn,k = Cn,k + 2^(n-1-k) Cn,k+1 where Cn is for the cube
* How to picture a recurrence relation for demicubes, perhaps related to the relations with cubes: Dn = 1/2 Cn = (Cn–1)^2
* How to imagine the generation of demicubes?
* Why does a 4–cycle linked to a chain of 3-cycles make for independent Euclidean dimensions?
Figure out what the 4-cycle means in the Bn and Cn cases.
Figure out what it means to have a branching in the Coxeter diagram.
Interpret the four diagonals of the Dn triangle as the short path in the Coxeter diagram.
-------------------
[+Laiškas+]
Dear mathematicians,
I contribute my own perspective in the spirit of Harvey Friedman's invitation "...I am trying to get a dialog going on the FOM and in these other forums as to "what foundations of mathematics are, ought to be, and what purpose they serve"."
http://www.cs.nyu.edu/pipermail/fom/2016-April/019724.html
I propose to distinguish implicit math (which our minds intuit and interpret) and explicit math (which is expressed as written symbols). I apply insights from implicit math to yield results in explicit math, specifically, interpretations of the -1 simplex, the Gaussian binomial coefficients, the generation of the regular polytopes An, Bn, Cn, as well as ideas as to how we might think about F1, the field with one element. These results may or may not be new but I hope to persuade you that both explicit math and implicit math may benefit from a mutual conversation. In particular, I pose a question about the generation of the irregular polytopes Dn. I also suggest the development of a set theory based on finite fields of characteristic q such that q is interpreted as infinity.
-------------------------------------------
The distinct advantages of
explicit and implicit math
-------------------------------------------
Harvey Friedman has noted: "Among the greatest mathematical events of all time (different than greatest mathematics of all time) are results of the form: this or that mathematical question cannot be proved or refuted using accepted mathematical standards for rigorous proofs." Referring, in particular, to Goedel's Incompleteness Theorems. And he continues, "We need a general purpose foundation for mathematics where mathematicians readily see that ANYTHING that is done throughout the whole of rigorous mathematics is OBVIOUSLY incorporated in that general purpose foundation for mathematics."
http://www.cs.nyu.edu/pipermail/fom/2016-April/019727.html
Taking his point more broadly, and his word "cannot" more broadly, some of the greatest mathematics is that which axiomatizes conditions so rigorously that it is able to prove what CANNOT be done:
* circling the square
* trisecting the angle
* enumerating the real numbers
All statements to be taken under the relevant conditions. Similarly, great results include the classification of Platonic solids, Lie groups, simple groups, and so on, where it is shown that one CANNOT construct additional objects.
These are all magnificent accomplishments of explicit mathematics. Moreover, millions of people are able to use explicit math to work together, to write, read and check rigorous thousand-page proofs and even computer-assisted proofs. Furthermore, Harvey Friedman notes that this all depends on a robust consistency, a lack of contradiction. This grows in impressiveness along with the mathematical community.
The distinction between explicit and implicit math comes up in Harvey Friedman's exchanges with John Baez. John writes about the roles of explicitly stated rules and implicitly active mathematical taste: "However, some rules are more interesting than others, and it's one of the highest expressions of mathematical taste to be able to choose interesting rules to work with."
http://www.cs.nyu.edu/pipermail/fom/2016-April/019664.html
I suppose there are even fewer university courses, if any, on "mathematical taste" than on "foundations of mathematics". But this unspoken notion is at play in a range of mathematical discoveries which I think are just as significant but open up what mathematicians CAN do. These include "introducing" and "playing" with:
* irrational numbers
* imaginary numbers
* infinitesimals
* group actions
* infinite series
* fractals
"Sound" axiomatic systems typically arose decades after much experimentation led by mathematical inquiry and taste. Implicit math, if developed as a science, would seek and find principles for play, inquiry, taste and so on.
Furthermore, implicit math can consider questions that make sense in a dynamic cognitive environment but not in a static document. In the human mind, the logical assumptions can change. Logic can be dynamic. Indeed, the human mind can analyze counterfactuals, can consider logical alternatives. We can even take the state of contradiction as a starting point and consider how a noncontradictory system might evolve from it. This type of "experienced" logic might yield a novel intepretation of Goedel's Incompleteness Theorem as regards what it means for one who actually lives in an unfolding system. Explicit math might usefully model part of this but I think not absolutely all of it.
My own interest is to know everything and apply that knowledge usefully. I aspire to model the limits of our mind. We experience our lives internally, not externally. I feel that I have been able to model much of that internal life. In order to share that, I want to show that it relates usefully to math and physics. I have had some success with some aspects of implicit math. But, in my experience, mathematicians avoid considering implicit math. I am thus sharing some initial results which I think are relevant for explicit math.
-------------------------------------------
A challenge for explicit math:
The field with one element
-------------------------------------------
In defending classical set thory (ZFC=Zermelo-Fraenkel axioms along with the Axiom of Choice), Harvey Friedman asked if there was any issue in mathematics which it is not addressing successfully but another foundations might.
http://www.cs.nyu.edu/pipermail/fom/2016-April/019656.html
Currently, there is a mathematical object which has attracted attention but is not yet grounded in set theory and perhaps might never be. I am thinking of F1, the field with one element, which Jacques Tits proposed in 1956 as he introduced his theory of buildings in algebraic geometry.
https://en.wikipedia.org/wiki/Field_with_one_element
There are many areas of math where such a field suggests itself:
https://ncatlab.org/nlab/show/field+with+one+element
Conferences have been devoted to this object. It appears as #14 in Richard Stanley's survey of most intricate and beautiful mathematical structures.
http://mathoverflow.net/questions/49151/most-intricate-and-most-beautiful-structures-in-mathematics
However, this object does not exist! There is no finite field in which 0 and 1 would be the same number. That is, as of yet, there is no settled way to make sense of such a field-like object, which suggests itself in so many areas of math.
Here is a discussion
https://golem.ph.utexas.edu/category/2007/04/the_field_with_one_element.html
of a 586 page paper where Nikolai Durov describes it as "the free algebraic monad generated by one constant or the universal generalized ring with zero".
http://arxiv.org/abs/0704.2030
I present my own approach to show it is simple but fruitful.
-------------------------------------------
Examples of implicit math:
A "mind game" and opposites.
-------------------------------------------
I will start by giving an example of reasoning that occurs in implicit math. Here is a "mind game" which serves to define the concepts "one", "all" and "many":
Search for constancy. Either you find "one" example of constancy or "all" is constantly unconstant. But in searching thus, you need to know that in each case, what you select is the same as what you inspect, that is, "many" are constant.
Thus we define "singularly constant", "universally constant" and "multiply constant". This kind of reasoning is standard in recursive function theory. Note that it did not require any explicit symbols. Sure, I used words, but it is the concepts that matter. It could be thought without any words at all. It is the kind of reasoning that I imagine takes place even in the womb. This "mind game" is a participatory example of how to define primitives without appealing to even simpler primitives. They are defined relative to each other through an activity which evokes them. And they are actually defined more precisely, I think, then they are ever defined in set theory. The concept "one", as defined above, supposes that a search algorithm halts, and the concept "all" supposes that it does not. Immanuel Kant tried to derive these same three concepts as "categories" by appealing to the form of a logical implication, which however, others never took up. I simply want to suggest that such reasoning is possible, and that such metaphysical, pre-mathematical thinking can ground implicit math and ultimately even explicit math, physics and other domains of understanding.
Now I want to point out three different kinds of "opposites" that are relevant for implicit math and can be observed in explicit math as well.
By "implicit opposites", I mean that I may be presented with two choices which I cannot distinguish in any way except that one is not the other. We may call them "unlabelled opposites". Of course, this is difficult to talk about because we end up introducing distinctions. Still, there is a sense in which we can be absolutely indifferent as to "clockwise" and "counterclockwise". In math, there are two square roots of -1 and we have no way to distinguish them. We call one of them i, but which one? Do you call the same one i that I do? It shouldn't matter. And this makes clear to me, as I never understood before, why complex conjugation is so important, because that is simply reminding us that on some level it shouldn't matter which is which. So the opposition here between the two square roots is mathematically implicit, in our mind.
By "explicit opposites", I mean that the two choices are expressed and one is primary, "unmarked", and the other is secondary, "marked". Linguistically, we may have the opposites "happy" and "unhappy", and the derivation "unhappy" is marked with the negation "un". Mathematically, we have 1 and -1. Generally, in my thinking, it's not so much the actual symbols that are most important but rather the way we think about them. Be that as it may, mathematically, there are natural reasons to distinguish 1 and -1, for example, 1 times itself is 1 whereas -1 times itself is 1. Indeed, it is reasonable for 1 to be unmarked and -1 to be marked. However, I think it is very unhelpful that the square roots of -1 are written as i and -i rather than, say, i and j. Implicit, unlabelled opposites are being presented as explicit, unmarked-marked opposites. I have a Ph.D. in math and yet only recently did I realize that I have a completely baseless prejudice by which I think of i as more basic than -i. It is as if I thought that "clockwise" was more clocklike than "counterclockwise" or "lefthandness" was sinister and "righthandness" was righteous.
By "mixed opposites", I mean that an implicit opposite is opposed by an explicit opposite. In other words, an undefined notion is opposed by a defined notion. This is especially common when we oppose content and the form or symbol that conveys it. For example, a "nonaction" may be written down as the identity element e. The emptiness of a box may be referenced by the number 0 or the empty set.
I continue by noting why I find the "field with one element" interesting. It must be similar to both 0 and 1, to both the empty set and its empty content. Thus it may relate to my own insights into God and Everything as well as the kinds of opposites that I described above.
---------------------------------
The -1 Simplex
is the center of a simplex
---------------------------------
I will share my exploration of a simple example where the "field with one element" comes up. Namely, the (n,k) entry of the Gaussian binomial coefficients is a q-polynomial which counts the number of k-dimensional subspaces of an n-dimensional vector space over a finite field of characteristic q. When q is set to 1, then we recover the number of subsets of size k of a set of size n. These latter subsets of a set can thus be imagined as 1-dimensional subspaces of an n-dimensional vector space over the (mysterious or nonexistent) finite field of characteristic 1.
In trying to understand this, I learned that the binomial coefficients (n,k) count (among so many other things) the number of k-simplexes within an n-simplex. A 2-simplex is an equilateral triangle, a 3-simplex is a tetrahedron, and there is a nice table of vertices, edges, faces at the Wikipedia article:
https://en.wikipedia.org/wiki/Simplex
From the article it is striking that the 1-simplex is an edge, a 0-simplex is a point and from the binomial coefficients there ought to be an interpretation for a -1-simplex. There should always be one -1-simplex in any k-simplex. Wikipedia explains that the -1 simplex is sometimes defined as the empty set. I imagined that there should be a more informative interpretation which might be a first step in interpreting F1.
We can consider the n-simplexes and the k-simplexes inside of them to be generated as the terms of the expansion:
(chosen + unchosen)^n
Thus the -1 simplex can be interpreted as the unique way of choosing no vertices.
My solution is that the -1 simplex is the unique "center" of the k-simplex in question. And that "center" is defined implicitly, that is, we must imagine it with our minds. It is not explicitly indicated as are the vertices and edges. So I think it is a great example of a well-defined mathematical object which, however, is not defined by set theory in any natural, helpful or faithful way. It is very much an implicit object. Let me show you how the simplexes unfold from it!
First we have just the "center".
Next, the implicit center finds expression as an explicit point. They are a mixed opposite. They look like a fish eye. It is seeing zero dimensions.
We can think of the point as a constant and the center as a sliding variable. The center can thus define a dimension. And it can express itself as a point along that dimension. Thus we have a line segment. And it also has a center, I suppose, the same center.
Now we can imagine that center as a variable, let us say, sliding up and down. It is as if we are seeing a vertical triangle from above, so that it looks like a line segment to us. Let the center find expression as a point. Then indeed we have a triangle.
Now the triangle has a center. Again we can imagine it as sliding up and down a dimension. Of course, we can picture it in 2-dimensions or in 3-dimensions as well, but in either case, when the center finds expression as a point, then we have a tetrahedron. (If we are in 2-dimensions, then we could unfold the new faces and imagine the tetrahedron and subsequent k-simplexes as a tiling of the plane.)
And likewise the tetrahedron has a center. And if we draw that center then we can imagine the 4-simplex in 4 dimensions. We can practically see it, the 5 volumes, 10 faces, 10 edges and 5 vertices, how they must be organized! If we like, then to visualize this we can invert the tetrahedrons so that the 4 new tetrahedrons are on the faces of the old one.
This is a wonderful thing, this link between implicit and explicit math. In thinking one center and drawing five points, I feel the solemnity of the center, the spirited gaze of that first point, the companionship in the second point, the humor when our third point opens up like a tent, the beauty of the pyramid that surveys our spatial imagination, and the power in the fifth point at its center which has us transcend the world we know.
We can define the q-analogue by:
* giving the first vertex weight 1
* multiplying the weight of each successive vertex by q, so that they are 1, q, q^2, q^3...
* balancing the new vertex of weight q^k with k new edges, each of weight 1/q.
Then for each k-simplex, if we multiply the weights of its vertices and edges, we get exactly the terms of the Gaussian binomial coefficients. This can be proven by recursion or bijectively.
Note that the weight of the center is simply 1. Also, the center has a dual, the entire volume, whose weight is always 1, as is clear from the construction. The fact that the weights of the k-simplexes are polynomials in q means that their vertices outweigh their edges.
The beauty of this construction suggests that it is quite "correct" and even in harmony with whatever may be "absolutely correct". This kind of thinking is usual in algebraic combinatorics. Indeed, I studied it because I thought of it as the "basement" of mathematics where it is most convenient to glimpse how mathematical objects arise. However, my impression was that combinatorics is generally not considered "real math" because its results and techniques are too concrete. But precisely for this reason it is a most convenient domain for grappling with philosophical questions of interpretation. Even the analytic combinatorialists Philippe Flajolet and Robert Sedgewick write excitedly about "universality phenomenon". Are such fundamental, practical issues avoided in the foundations of mathematics and the philosophy of mathematics? Is it because we reject the notion of "interpretation" and the world of "implicit math" or even "absolute truth"? If we establish a very narrow scope of "rigor" are we truly safe from anything grander by deeming it as "not mathematics"? Or is there not a fish eye lurking at every point?
-------------------------------------------
Gaussian Binomial Coefficients
define what it means to count
-------------------------------------------
Let us now interpret the Gaussian binomial coefficients as the number of k-dimensional subspaces of an n-dimensional vector space over Fq, a field of characteristic q. I will explain how to interpret q-analogues of the integers n, n! and n-choose-k because the details are informative.
The q-analogue of the number n counts the number of 1-dimensional subspaces of an n-dimensional vector space V over Fq. That subspace is determined by any single nonzero element of the vector space. Thus there are q^(n–1) such elements. However, we are overcounting, because multiplying by a nonzero scalar gives us the same subspace. Thus, dividing by the number of nonzero scalars q-1, we get (q^n – 1) / (q-1) = 1 + q + q^2 + .... + q^(n-1). And if q=1, then we get n as promised.
Alternatively and significantly, we can fix a basis of V: e1, e2, ..., en–1. Then e1 defines a 1-dimensional subspace. But so does any combination f1 * e1 + e2 where f1 is any one of the elements of Fq including zero. And a 1-dimensional subspace is also defined by any combination f1 * e1 + f2 * e2 + e3. In this manner, the possible 1-dimensional subspaces grow as 1 + q + q^2 etc.
Now the q-analogue of n! counts the number of total flags, that is, the ways of adding 1-dimensional subspaces until they yield all of V. But that is rather clear because we keep looking at the remainder of V, which keeps shrinking by 1-dimension, and so we multiply the q-analogues of n x (n-1) x (n-2) ... = n!
Finally, we want to get the q-analogue of n-choose-k = n!/(k!*(n-k!) which is the number of k-dimensional subspaces of V. We use the construction above to remove (n-k) 1-dimensional subspaces from V yielding n!/k! k-dimensional subspaces. We need to know how much we have overcounted by, that is, how many times a subspace repeats. But that is given by the number of flags of (n-k) dimensions, which is to say, the number of ways our subspace K could be built up to give V is also the number of ways that V could be deconstructed to yield K.
We can now draw an analogy between the q-analogies for the simplexes and the vector subspaces. The vertices are weighted 1 + q + q^2 + ... These weights arise in the construction of the 1-dimensional vector space because each new basis element ek participates in a chain of weights f1 * e1 + f2 * e2 + ... + fk-1 * ek-1 + ek where each fi represents q choices and so ek has weight q^(k-1). The (k-1) edges are weighted 1/q. But that means that every time we extend the chain the weight grows by q except for the very first basis element.
So what does this mean when we set q=1 ?
Let us recall how we generated the simplexes and what we gained by considering q-analogues. The simplex is generated by relating the "new" vertex to all of the previous vertices. Thus all of the vertices are related to each other by edges. It is as if we are constructing the natural numbers. However, the construction simply emphasizes by way of the edges that the new vertex is different from each of the existing vertices. But otherwise all of the existing vertices are indistinguishable. The center "knows" which one is the "new" vertex. But then it is forgotten, that is, it is unlabelled, like the rest. Although they all know they are different from each other as expressed by the edges. In a sense, this is what "counting" is all about, except that there is no "answer" stored. There is a growing set or "bag" of indistinguishables, which is to say, it's not a traditional set.
When we add the q-weights, then each vertex gets a distinct label q^k which is the "cost" introduced to distinguish that vertex from all of the others. Also, we can think of each of the k edges as fairly bearing part of that cost 1/q. Now we have a total order. We also have an operation by which the center gives each new vertex a weight by multiplying q times the weight of the last new vertex. Thus we have an operation k -> k+1.
We also see that the center "marks" or "points" an initial number (we should call it "0") which is different than all of the other numbers (1,2,3...) even while it participates in generating them (as if by relations 0 + k + 1 = k+1).
I may not be describing this perfectly. But my point is that there seems to be a natural process at work here. It is as if we are literally seeing the God of math construct the "natural numbers" or rather, different versions of them. The most "natural" of the "numbers", if it is fair to call them that, are "pointed multisets" built from one element, "unmarked", where the Center points to the new element, like a hand placing a marble in a bag. Or a boson in a bag. In the q-analogue, all of the vertices get labels, as do all of the faces, and we have a total order, as well as a record of the operation k -> k+1. We have a set of fermions. These types of distinctions are natural in combinatorics as in the "twelvefold way":
https://en.wikipedia.org/wiki/Twelvefold_way
Again, set theory doesn't approach these naturally, implicitly, but rather requires that all knowledge be explicit and then removes it using equivalence classes. Which is to say, set theory destroys this entire way of thinking.
So what is happening when q=1 is that all of the "totally ordered" numbers become indistinguishable except for whichever one is in our hand, so to speak.
Note that the original vertex (or number) costs nothing (it has weight 1) and the reason is that it is not compared to any other vertex. Which is to say, the distinction of the center and the vertex comes for free. But when we add more vertices we need to keep them all distinct. Each relationship (each edge) bears the same weight (1/q) and so the weights (or costs) of the vertices progressively increases. Similarly, in the case of the vector space, the initial basis element e1 comes for free. But the subsequent basis elements come at an increasing cost. It is again the cost of choice q^(k-1), both the cost of being the k-th basis element, distinct from the others, and the cost of knowing (storing) the value of each of the scalars fi associated with the basis elements ei, where 0<=i<k.
Now we can also imagine what the q stands for as we generate the simplex. q is the number of choices (the colors) that we have to choose from for each facet.
q-analogue defines counting: an ordered set - a list. But here we don't have labels - we're not counting anything. Typically we write out finite sets in a particular order, although that order is
and when q=1 we have bagging: a bag
-------------------------------------------
The Field with One Element
Zero equals infinity
-------------------------------------------
This is why we get polynomials in q. However, there are for each n and k certain k-simplexes within the n-simplex which have weight exactly 1, that is, the vertices and the edges balance out. These are precisely those k-simplexes which are built from the k+1 vertices of smallest weight, namely: 1, q, q^2, q^3..., q^k.
If we set
"implicit opposites"
0 x 0 = 0
infinity x infinity = infinity
0 x infinity = undefined
undefined/0 = inverse of 0 = infinity
undefined/infinity = inverse of infinity = 0
0 + 0 = 0
0 + infinity = infinity
infinity + infinity = infinity
----------------------------------
Notes:
* Dn should come from BOTH center and volume. An comes from EITHER (building up and tearing down). Bn comes from VOLUME and Cn comes from CENTER.
* Tensorlikeness of An duality. Covariant and contravariant.
* An defines a total order - the integers - and also the relationship n->n+1. The former relationship arises directly (all are related to each other, and specifically, the newest to the rest), the latter relationship arises in the q-analogue, where we can compare the order.
* The total order shows that the set of relationships is natural, as in a category. And the set of subsets is natural.
* Edges can be interpreted as directed or nondirected.
* The duality or nonduality of vertices and edges. Vertices can label the edges (bottom-up) or edges can label the objects (top-down). Shows limitations (or not) of category theory (and set theory).
* Imre Latakos, definitions are not set in stone:
https://en.wikipedia.org/wiki/Proofs_and_Refutations
* Hypercube tiling - cross polytope tiling (invert) = demicube tiling (think of periodic tiling, then invert) Tile space with cross polytopes and the gaps are demicubes? Or tile space with n-cubes, transform to demicubes, the gaps are cross polytopes.
* Can reverse the weight construction: Volume has weight one. Remove a vertex. The subwhole has the weight of the missing vertex.
* The center is the manifest possibility of a distinct new vertex.
* Perhaps the new vertex gets a weight that shows the cost it has in being related to all of the other vertices and the edges show how that cost is born.
Questions:
* How to define the k-faces of the n-demicubes?
* Do the demicubes have single edges or double edges?
* What is special about the vertices, edges, faces of the demicubes that makes them distinct from the higher dimension faces?
* How to conceive of the demicubes as having k-demicubes + k-simplexes as in Dn,k = Cn,k + 2^(n-1-k) Cn,k+1 where Cn is for the cube
* How to picture a recurrence relation for demicubes, perhaps related to the relations with cubes: Dn = 1/2 Cn = (Cn–1)^2
* How to imagine the generation of demicubes?
* Why does a 4–cycle linked to a chain of 3-cycles make for independent Euclidean dimensions?
Figure out what the 4-cycle means in the Bn and Cn cases.
Figure out what it means to have a branching in the Coxeter diagram.
Interpret the four diagonals of the Dn triangle as the short path in the Coxeter diagram.
2016 birželio 07 d., 10:21
atliko -
Pakeistos 135-137 eilutės iš
https://en.m.wikipedia.org/wiki/Conway_polyhedron_notation
polyhedron operators https://en.m.wikipedia.org/wiki/Alternation_(geometry)
polyhedron operators https://en.
į:
https://en.wikipedia.org/wiki/Conway_polyhedron_notation
polyhedron operators https://en.wikipedia.org/wiki/Alternation_(geometry)
https://upload.wikimedia.org/wikipedia/commons/6/67/Wythoffian_construction_diagram.png
polyhedron operators https://en.wikipedia.org/wiki/Alternation_(geometry)
https://upload.wikimedia.org/wikipedia/commons/6/67/Wythoffian_construction_diagram.png
2016 birželio 07 d., 10:14
atliko -
Pakeistos 133-137 eilutės iš
https://ecommons.cornell.edu/handle/1813/17339 coxeter dynkin interview
į:
https://ecommons.cornell.edu/handle/1813/17339 coxeter dynkin interview
https://en.m.wikipedia.org/wiki/Conway_polyhedron_notation
polyhedron operators https://en.m.wikipedia.org/wiki/Alternation_(geometry)
https://en.m.wikipedia.org/wiki/Conway_polyhedron_notation
polyhedron operators https://en.m.wikipedia.org/wiki/Alternation_(geometry)
2016 birželio 07 d., 09:33
atliko -
Pakeistos 131-133 eilutės iš
https://ecommons.cornell.edu/handle/1813/3206 catalan and coxeter
į:
https://ecommons.cornell.edu/handle/1813/3206 catalan and coxeter
https://ecommons.cornell.edu/handle/1813/17339 coxeter dynkin interview
https://ecommons.cornell.edu/handle/1813/17339 coxeter dynkin interview
2016 birželio 07 d., 09:04
atliko -
Pakeistos 129-131 eilutės iš
* Lecture 42 Classification of regular polytopes
į:
* Lecture 42 Classification of regular polytopes
https://ecommons.cornell.edu/handle/1813/3206 catalan and coxeter
https://ecommons.cornell.edu/handle/1813/3206 catalan and coxeter
2016 birželio 06 d., 10:58
atliko -
Pridėtos 87-88 eilutės:
http://link.springer.com/chapter/10.1007/978-3-642-04295-9_1 geometry of cuts and metrics
Pridėtos 94-95 eilutės:
http://link.springer.com/chapter/10.1007/978-3-642-21590-2_15 nth roots of pitch class inversion
2016 birželio 06 d., 10:56
atliko -
Pridėtos 82-83 eilutės:
http://www.sciencedirect.com/science/article/pii/S0001870809001017 Homology representations arising from the half cube
2016 birželio 06 d., 10:27
atliko -
Pridėtos 84-85 eilutės:
http://arxiv.org/abs/1506.06702 paper about 16 cube
2016 birželio 06 d., 10:17
atliko -
Pridėtos 78-81 eilutės:
The purpose is to split the perspective. The human perspective is given by the short path.
Also this suggests that the exceptional groups are modeling interactions of limited human perspectives.
2016 birželio 06 d., 10:15
atliko -
Pridėtos 76-77 eilutės:
The pascal triangle can be explained by noting that we have the fusion of two paths in the Dynkin diagram, one of length 3 and one that grows without bound. The first nontrivial example is the 16 cell.
2016 birželio 06 d., 09:58
atliko -
Pridėtos 74-75 eilutės:
There should be two different, complementary, flags of faces. And they should relate to the cross polytopes - one flag should be like the cross polytopes and the other flag should be the opposite - and together they should form a cube.
2016 birželio 06 d., 09:54
atliko -
Pridėtos 73-74 eilutės:
The polytopes are irregular because the Dynkin diagram is not a path but has a branch. Thus there should be not one center or volume but rather the "generation" of the polytope should derive from two perspectives. These two perspectives may relate to the bipartition of the demicube which itself is a bipartition of the cube. The two perspectives are related by double edges.
Pakeista 111 eilutė iš:
* Classification of regular polytopes
į:
* Lecture 42 Classification of regular polytopes
2016 birželio 06 d., 09:47
atliko -
Pridėtos 101-109 eilutės:
* Lecture 34 Rank 2 Dynkin diagrams
* Lecture 35 Coxeter group reflection group bipartite
* Lecture 36 Weyl group is finite if bilinear form is positive definite
* Lecture 37 Group representation sum of irreducibles
* Lecture 38 Weyl group finite iff bilinear form is positive definite
* Lecture 39 Classification of finite Coxeter groups
* Lecture 40 Proof
* Lecture 41 Proof
* Classification of regular polytopes
* Lecture 35 Coxeter group reflection group bipartite
* Lecture 36 Weyl group is finite if bilinear form is positive definite
* Lecture 37 Group representation sum of irreducibles
* Lecture 38 Weyl group finite iff bilinear form is positive definite
* Lecture 39 Classification of finite Coxeter groups
* Lecture 40 Proof
* Lecture 41 Proof
* Classification of regular polytopes
2016 birželio 06 d., 09:25
atliko -
Pridėtos 97-100 eilutės:
* Lecture 30 Root systems for An and Bn
* Lecture 31 Bn. Root system Coxeter group
* Lecture 32 Crystallographic root systems Lie groups Cartan matrix
* Lecture 33 Coxeter matrix mij and aij aji
* Lecture 31 Bn. Root system Coxeter group
* Lecture 32 Crystallographic root systems Lie groups Cartan matrix
* Lecture 33 Coxeter matrix mij and aij aji
2016 birželio 06 d., 09:16
atliko -
Pakeistos 93-96 eilutės iš
*
į:
* Lecture 10 properties of Bruhat order
* Lecture 15 Mobius function Eulerian characteristic Eulerian posets
* Lecture 20 Dihedral group D5
* Lecture 25 Contragradient action Coxeter groups are automatic Root poset
* Lecture 15 Mobius function Eulerian characteristic Eulerian posets
* Lecture 20 Dihedral group D5
* Lecture 25 Contragradient action Coxeter groups are automatic Root poset
2016 birželio 06 d., 09:08
atliko -
Pridėtos 89-90 eilutės:
[[http://math.sfsu.edu/federico/Clase/Coxeter/lectures.html | Lecture notes by Federico Ardila]]
2016 birželio 06 d., 09:06
atliko -
Pakeistos 86-91 eilutės iš
* http://math.ucr.edu/home/baez/week187.html
į:
* http://math.ucr.edu/home/baez/week187.html
'''Coxeter groups'''
Videos
*
'''Coxeter groups'''
Videos
*
2016 birželio 05 d., 22:38
atliko -
Pridėtos 17-18 eilutės:
Įsivaizduoti centrą (žvilgsnį iš viršaus) ir apskaičiuoti centro daugiaprasmiškumą (naudojant q?)
2016 birželio 05 d., 20:56
atliko -
Pakeistos 82-84 eilutės iš
http://math.ucr.edu/home/baez/week187.html
į:
* http://math.ucr.edu/home/baez/week185.html
* http://math.ucr.edu/home/baez/week186.html
* http://math.ucr.edu/home/baez/week187.html
* http://math.ucr.edu/home/baez/week186.html
* http://math.ucr.edu/home/baez/week187.html
2016 birželio 05 d., 20:52
atliko -
Pakeistos 73-82 eilutės iš
[[https://plus.google.com/117663015413546257905/posts/ZCyDfcc8gRQ | John Baez on demicubes]]
į:
[[https://plus.google.com/117663015413546257905/posts/ZCyDfcc8gRQ | John Baez on demicubes]]
'''Field with one element'''
David Corfield's post
https://golem.ph.utexas.edu/category/2007/04/the_field_with_one_element.html
about Nikolai Durov's book
http://arxiv.org/abs/0704.2030
John Baez: This fits nicely with my own intuitions about linear algebra over the field with one element. A pointed set acts like a ‘vector space over the field with one element’; a set acts like a projective space over the field with one element.
http://math.ucr.edu/home/baez/week187.html
'''Field with one element'''
David Corfield's post
https://golem.ph.utexas.edu/category/2007/04/the_field_with_one_element.html
about Nikolai Durov's book
http://arxiv.org/abs/0704.2030
John Baez: This fits nicely with my own intuitions about linear algebra over the field with one element. A pointed set acts like a ‘vector space over the field with one element’; a set acts like a projective space over the field with one element.
http://math.ucr.edu/home/baez/week187.html
2016 birželio 05 d., 10:52
atliko -
Pakeistos 60-61 eilutės iš
į:
Pridėtos 65-71 eilutės:
'''Demicubes'''
* Vertices: Take those coordinates (of a cube) which have an even number of minus signs.
* Edges: ?
http://math.ucr.edu/home/baez/week187.html The Coxeter group for Dn is the subgroup of the symmetries of the n-dimensional cube generated by permutations of the coordinate axes and reflections along ''pairs'' of coordinate axes.
2016 birželio 05 d., 09:16
atliko -
Pridėtos 14-17 eilutės:
Investigation
* Look at coordinates
* Look at binomial theorem
* Look at coordinates
* Look at binomial theorem
Pridėtos 65-66 eilutės:
[[https://plus.google.com/117663015413546257905/posts/ZCyDfcc8gRQ | John Baez on demicubes]]
2016 birželio 05 d., 08:46
atliko -
Pakeistos 5-12 eilutės iš
Does the end of the dynkin diagram code for opposites as they are drawn from the center?
į:
* How are the demihypercubes generated?
* What does the end of the Dynkin diagram mean?
* Why is it that 3-cycles lead us to higher dimensions in the Dynkin diagrams?
* How should the demihypercube binomial triangle be interpreted?
* What is the q-analogue for the binomial triangles of Bn,Cn,Dn?
* What is the role of opposites and implicit math in the classical groups?
* What does it mean that F1 is {0,infinity}?
* Does the end of the dynkin diagram code for opposites as they are drawn from the center?
* What does the end of the Dynkin diagram mean?
* Why is it that 3-cycles lead us to higher dimensions in the Dynkin diagrams?
* How should the demihypercube binomial triangle be interpreted?
* What is the q-analogue for the binomial triangles of Bn,Cn,Dn?
* What is the role of opposites and implicit math in the classical groups?
* What does it mean that F1 is {0,infinity}?
* Does the end of the dynkin diagram code for opposites as they are drawn from the center?
2016 birželio 04 d., 15:27
atliko -
Pakeistos 51-53 eilutės iš
į:
* An: symmetric group
* Bn: A deck of n cards where card j has j written on one side and -j on the other side. All possible arrangements of stacks of cards with orientation.
* Dn: corresponds to all arrangements of stacks of n cards with orientation with an even number of cards turned over (pairs of reflections).
* Bn: A deck of n cards where card j has j written on one side and -j on the other side. All possible arrangements of stacks of cards with orientation.
* Dn: corresponds to all arrangements of stacks of n cards with orientation with an even number of cards turned over (pairs of reflections).
2016 birželio 04 d., 14:02
atliko -
Pridėtos 47-51 eilutės:
http://math.ucr.edu/home/baez/week187.html The Coxeter group for Dn is the subgroup of the symmetries of the n-dimensional cube generated by permutations of the coordinate axes and reflections along ''pairs'' of coordinate axes.
2016 birželio 03 d., 17:16
atliko -
Pridėtos 41-46 eilutės:
[[http://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html#Half | Half measure polytope]]
* Also known as a demihypercube in n-space, n>2. The uniform polytope constructed by completely truncating the alternate vertices of a measure polytope, that is, truncating half its vertices by hyperplanes that pass through a truncated vertex’s edge-neighboring vertices. If n=2, this truncation of a square leaves only the square’s diagonal, which is not a polygon. But when n=3, this truncation of a cube produces a regular tetrahedron, and when n=4, it produces a regular hexadecachoron from a tesseract. For n>4, demihypercubes are no longer regular, only uniform, with two different kinds of facets: 2n (n–1)-dimensional demihypercubes and 2n–1 (n–1)-dimensional simplexes. The names of the demihypercubes are constructed by prefixing demi- to the name of a measure polytope: demipenteract, demihexeract, demihepteract, and so forth. As noted above, a demicube is a regular tetrahedron, and a demitesseract is a regular hexadecachoron.
* The vertex figure of a half measure polytope H in n-space is a rectified (n–1)-dimensional simplex (the simplex has n vertices and n facets). The facets of the rectified simplex are (1) n (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional simplex facets of H, and (2) n rectified (n–2)-dimensional simplexes, which are the vertex figures of the (n–1)-dimensional demihypercube facets of H. Thus, for example, the vertex figure of the demicube is a rectified triangle, which is a smaller triangle, the vertex figure of a tetrahedron (the demicube). The vertex figure of the demitesseract is a rectified tetrahedron, which is an octahedron, the vertex figure of a regular hexadecachoron (the demitesseract). And so on.
* Euclidean n-space can always be uniformly honeycombed by half measure polytopes and cross polytopes: In the regular honeycomb of n-space by measure polytopes, remove alternate vertices, thereby transforming each measure polytope into a half measure polytope, and fill the gaps with n-dimensional cross polytopes (the vertex figures of the measure polytope honeycomb). In the plane, this changes the checkerboard tiling into another checkerboard tiling rotated 45° to the original; in 3-space this produces the uniform honeycomb of regular tetrahedra and octahedra; and in 4-space this produces the regular honeycomb of hexadecachora
2016 gegužės 31 d., 08:39
atliko -
Pakeistos 32-33 eilutės iš
Cn cross-polytope construction given by having two points extend from the center in opposite directions. Each of the points links to all of the existing vertices. The old volume gives way to a new volume of one dimension higher.
į:
Cn cross-polytope construction given by having two points extend from the center in opposite directions. Each of the points links to all of the existing vertices. The old volume gives way to a new volume of one dimension higher. The initial C is given by two unconnected points. And these two points came from the center, which is why they are unconnected.
Ištrintos 38-39 eilutės:
2016 gegužės 31 d., 08:33
atliko -
Pakeistos 32-34 eilutės iš
Cn cross-polytope construction given by having two points extend from the center in opposite directions. The old volume gives way to a new volume of one dimension higher.
Bn hypercube construction given by having the volume grow by having the volume be conceived as a mirror from which two mirror images arise on either sides.
Bn hypercube construction given by having the
į:
Cn cross-polytope construction given by having two points extend from the center in opposite directions. Each of the points links to all of the existing vertices. The old volume gives way to a new volume of one dimension higher.
Bn hypercube construction given by having the volume grow by having the volume be conceived as a mirror from which two mirror images arise on either sides. Thus mirror points are linked by edges.
Bn hypercube construction given by having the volume grow by having the volume be conceived as a mirror from which two mirror images arise on either sides. Thus mirror points are linked by edges.
2016 gegužės 31 d., 08:31
atliko -
Pridėtos 29-30 eilutės:
An add the center along with edges from the center to the other vertices. Then you can revision in a higher dimension.
2016 gegužės 31 d., 08:23
atliko -
Pakeista 30 eilutė iš:
Cn cross-polytope construction given by having two points extend from the center in opposite directions.
į:
Cn cross-polytope construction given by having two points extend from the center in opposite directions. The old volume gives way to a new volume of one dimension higher.
2016 gegužės 31 d., 07:54
atliko -
Pridėtos 37-40 eilutės:
Koch snowflake is an illustration of An for all n.
2016 gegužės 31 d., 07:40
atliko -
Pakeista 34 eilutė iš:
Thus we have duality of adding two opposing vertices or adding a hyperplane with two reflections, of growing the center or the volume.
į:
Thus we have duality of adding two opposing vertices or adding a hyperplane with two reflections, of growing the center or the volume. We have the duality of vertices and facets.
2016 gegužės 31 d., 07:14
atliko -
Pridėta 25 eilutė:
Pakeistos 28-36 eilutės iš
the linked involutions, the edges, are numbered and are added as the normal form which is preserved.
į:
the linked involutions, the edges, are numbered and are added as the normal form which is preserved.
Cn cross-polytope construction given by having two points extend from the center in opposite directions.
Bn hypercube construction given by having the volume grow by having the volume be conceived as a mirror from which two mirror images arise on either sides.
Thus we have duality of adding two opposing vertices or adding a hyperplane with two reflections, of growing the center or the volume.
Dn demihypercube construction
Cn cross-polytope construction given by having two points extend from the center in opposite directions.
Bn hypercube construction given by having the volume grow by having the volume be conceived as a mirror from which two mirror images arise on either sides.
Thus we have duality of adding two opposing vertices or adding a hyperplane with two reflections, of growing the center or the volume.
Dn demihypercube construction
2016 gegužės 31 d., 06:14
atliko -
Pridėtos 2-7 eilutės:
>>bgcolor=#FFFFC0<<
Does the end of the dynkin diagram code for opposites as they are drawn from the center?
>><<
2016 gegužės 31 d., 06:01
atliko -
Pakeistos 17-21 eilutės iš
symmetry group is Sn
į:
symmetry group is Sn
center is what gives 3 rather than 2. center is what links involutions in the dynkin diagram.
the linked involutions, the edges, are numbered and are added as the normal form which is preserved.
center is what gives 3 rather than 2. center is what links involutions in the dynkin diagram.
the linked involutions, the edges, are numbered and are added as the normal form which is preserved.
2016 gegužės 31 d., 05:58
atliko -
Pakeistos 15-17 eilutės iš
dynkin diagram coxeter group An reflections are encoded by the sequence 1, q2, q3, ..., q*(n-1)
į:
dynkin diagram coxeter group An reflections are encoded by the sequence 1, q2, q3, ..., q*(n-1)
symmetry group is Sn
symmetry group is Sn
2016 gegužės 31 d., 05:53
atliko -
Pakeista 15 eilutė iš:
coxeter group An reflections are encoded by the sequence 1, q2, q3, ..., q*(n-1)
į:
dynkin diagram coxeter group An reflections are encoded by the sequence 1, q2, q3, ..., q*(n-1)
2016 gegužės 31 d., 05:52
atliko -
Pakeistos 13-15 eilutės iš
0=1=infinity mes esame keisti.
į:
0=1=infinity mes esame keisti.
coxeter group An reflections are encoded by the sequence 1, q2, q3, ..., q*(n-1)
coxeter group An reflections are encoded by the sequence 1, q2, q3, ..., q*(n-1)
2016 gegužės 30 d., 16:56
atliko -
Pakeistos 5-13 eilutės iš
Projective space. All lines that go through the origin. Natural origin = "center" of simplex. Natural infinity = fold out the next point = so vertices are halfway between Center and Infinity.
į:
Projective space. All lines that go through the origin. Natural origin = "center" of simplex. Natural infinity = fold out the next point = so vertices are halfway between Center and Infinity.
q-characteristic
0=q=infinity jų požiūriu
Visa šeima įvairių q - iš esmės panašūs - bet mes esame keisti q=1
0=1=infinity mes esame keisti.
q-characteristic
0=q=infinity jų požiūriu
Visa šeima įvairių q - iš esmės panašūs - bet mes esame keisti q=1
0=1=infinity mes esame keisti.
2016 gegužės 30 d., 16:54
atliko -
Pakeistos 3-5 eilutės iš
Centras: apibendrinimas. Išrašus jį atsiranda matas.
į:
Centras: apibendrinimas. Išrašus jį atsiranda matas.
Projective space. All lines that go through the origin. Natural origin = "center" of simplex. Natural infinity = fold out the next point = so vertices are halfway between Center and Infinity.
Projective space. All lines that go through the origin. Natural origin = "center" of simplex. Natural infinity = fold out the next point = so vertices are halfway between Center and Infinity.
2016 gegužės 30 d., 16:52
atliko -
Pridėtos 1-3 eilutės:
Žr. [[Matematika]]
Centras: apibendrinimas. Išrašus jį atsiranda matas.
Centras: apibendrinimas. Išrašus jį atsiranda matas.
Simplex
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