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Book.CircleFolding istorijaPaslėpti nežymius pakeitimus  Rodyti galutinio teksto pakeitimus 2016 birželio 19 d., 14:24
atliko 
Pakeistos 3340 eilutės iš
Circle folding also comes up, I think, in terms of a more general "spherefolding" where the sphere can be of any dimension N. į:
Circle folding also comes up, I think, in terms of a more general "spherefolding" where the sphere can be of any dimension N.  Bradford, here is a picture that I came across today of the Ford circles: https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_circle_method#/media/File:Circumfer%C3%A8ncies_de_Ford.svg So curious! 2016 birželio 19 d., 14:19
atliko 
Pridėtos 410 eilutės:
Bradford, thank you for writing about circle folding. I cut out a circle and folded it. I found some of your web pages: http://analyzer.depaul.edu/paperplate/Full%20Circle.htm http://wholemovement.com You are a wonderful artist, creative geometer and a true pioneer. I expect to fold more circles and to relate that to some insights I'm developing.  2016 birželio 19 d., 13:12
atliko 
Pridėtos 12 eilutės:
See: [[Math]] Pakeistos 26102 eilutės iš
Circle folding also comes up, I think, in terms of a more general "spherefolding" where the sphere can be of any dimension N. https://en.wikipedia.org/wiki/Simplex Look for the table of simplexes, see how it relates to Pascal's triangle, and note the unexplained 1 dimensional simplex. I note that the 1 simplex can be conceived as the unexpressed unique center which we can associate with each simplex. For example, it is the center of a point, the center of an edge, the center of a triangle, the center of a tetrahedron, and so on. We can also imagine the Totality of the simplex, which for a 2dimensional triangle would be the area, and for a 3dimensional tetrahedron would be the volume, whatever is defined by all of the vertices. So we can imagine that first the Center is all by itself, and then it generates vertices along with the Totality of these vertices, so that the Center expresses itself but is never expressed, always moving on as the new Center of the vertices it has created. The Center is never expressed because each vertex it creates is distinct from the rest and so the Center is ever new. Well, it turns out that this is just one of four "theological" models of how the Center can behave. We can think of the Center as modeling God and the Totality as modeling Everything. And this particular process generates the Simplexes, which are known as polytope family An because the symmetries of these simplexes are given by the Symmetric group. Which is to say, every vertex is linked to every other vertex, and the only symmetry is that the vertices could be relabelled or renumbered in any possible way. The lack of any additional symmetry also means that each vertex is uniquely "positive" as Kirby keeps saying. So this is a model that "God is good" in every direction. But there are four families of such polytopes: An: ndimensional simplexes Bn: ndimensional cubes Cn: ndimensional cross polytopes (such as the octahedron) Dn: ndimensional demicubes (demihypercubes) which I find are best thought of as halfcubes (hemicubes) with double edges. The symmetries of these four families are captured by their symmetry groups (Coxeter groups of reflections), which are the Weyl groups of the root systems, which are vector bases for the Lie algebras, whose vector "addition" is the analogue (by way of the exponential function) of the matrix "multiplication" in the Lie groups, which are the continuous groups, which are restricted by the possible, allowable "shortcuts" for inverting (undoing) a continuous action, so that there could be a continuous geometry in which nothing gets ripped apart. In other words, I think that the symmetries of the four families of polytopes dictate four geometries, such as Steve distinguishes. And it is interesting what geometries are yielded by the 5 exceptional Lie groups. There is a paper by Victor Kac http://arxiv.org/pdf/math/9912235.pdf "Classification of infinitedimensional simple groups of supersymmetries and quantum field theory" where in his final section (page 20) on "Speculations and Visions" he notes that "Each of the four types W, S, H, K of simple primitive Lie algebras (L, L0) correspond to the four most important types of geometries of manifolds: all manifolds, oriented manifolds, symplectic and contact manifolds." I found this quote through a link by John Baez https://golem.ph.utexas.edu/category/2007/10/geometric_representation_theor_3.html about a video lecture given by his coteacher James Dolan as part of their "Geometric Representation Theory Seminar" http://math.ucr.edu/home/baez/qgfall2007/ which discusses the relevance of the polytopes to logic and geometry. John Baez has a series of essays about these geometries which I suppose starts here: http://math.ucr.edu/home/baez/week181.html Note, however, that he swaps the letters Cn and Dn from the way I and many others use them. An yields projective geometry Bn yields conformal geometry Cn yields conformal geometry Dn yields symplectic conformal geometry. Basically, symplectic is relevant when two quantities "position" and "momentum" are related, as by time, energy. Symplectic is, I think, what happens when possiblities tracked in the "complex" quantum reality get manifested (through a natural or human "measurement") as a "real" actuality. Dn is where I imagine circlefolding is relevant, again. The Center and the Totality distinguish An, Bn, Cn, Dn in the following way. The An simplexes have both a Center and a Totality. For example, a tetrahedron has 1 Center (the center), 4 vertices, 6 edges, 4 faces and 1 Totality (the volume). Well, that's a row in Pascal's triangle. And we see that in each row there is always 1 Center and always 1 Totality at the opposite ends of Pascal's triangle. Let's consider next the "pascal triangle" which counts the pieces of the crosspolytopes (the orthoplexes, the octahedrons). Here in each row there is a power of two that keeps increasing as follows. An octahedron has 6 vertices, 12 edges and 8 faces. So the pascal triangle is: 1 x 2^0 3 x 2^1 3 x 2^2 1 x 2^3 In other words, it has 1 center, 3 x 2 vertices, 3 x 4 edges and 1 x 8 faces. And it has no Totality, no volume! This makes sense if we consider what the Center is doing here. It is creating two vertices at a time. We can think of them as "implicit opposites" because there is no way of telling them apart. However, later on, we will see how Dn labels them as "positive" and "negative". Anyways, each new dimension yields 2 new vertices that are connected to all of the previous vertices to keep them all distinct. The Center yields first two unconnected points, then a square, then the octahedron, then the tetracross (the 16cell). So each vertex is directly connected to every vertex except for its opposite. If you were to connect all of the opposites, then you would have a simplex (but the dimension would be twice as large because the simplex is generated one vertex at at time). Anyways, when all the vertices are related to each other, (except for their opposites!), then we have the 8 faces of the octahedron. That's why the octahedron has, by definition, as we can see from Pascal's triangle, no concept of totality, no volume. It can't because the opposites aren't allowed to be related. They don't need to be related because they are, by definition, opposite. So that's a rather typical example of where physical reality can be misleading or not the most helpful model. I first came to that conclusion when I tried to understand electricity in terms of "running water". Finally, a teacher explained to me that it can't be thought of that way because the relationship between voltage, amperage, resistance, etc. is simply different and the analogy fails spectacularly. Similarly, the spin of an electron doesn't relate to our physical models. And that seems to be the case with most of modern physics. In general, the Nunez/Lakoff view that mathematical thinking arose to match the experience of the human body seems to me very foreign. Instead, it seems natural to me that mathematical structures reflect our internal, spiritual modeling. Steve's focus on vision seems fruitful and then there arises the question as to whether our visual evolution was driven by our spiritual inner life or our practical outer life. And I'm curious what mathematical thinking is like for the blind and for those conscious in the womb. But I think the latter could be "playing" with the type of philosophical Center and Totality issues that I'm discussing. It's enough to abstractly think two or three abstract "things" and the various ways they may relate. Now the cubes Bn are just like the crosspolytopes but with the rows of the "pascal triangle" reversed. A cube has 8 points, 12 edges, 6 faces. This means, by analogy above, that is has a Totality but it has no Center! So I will explain what seems to be happening. Let us start with a Totality and imagine it as an unfolding mirror. The Totality is itself the one initial mirror. We can think of it as defining two opposite directions. Now imagine that mirror dividing itself into two and moving out into those two opposite directions. Imagine that it is thereby opening up a mirror in the perpendicular direction. Now we have a second dimension which also defines opposite directions. Now imagine the totality of all the mirrors dividing to reveal a new dimension with a new mirror. As this process continues, we get 2^N quadrants, that is, N pairs of "implicit opposites". I see that we really don't get vertices, edges, faces. Indeed, the quadrants keeps getting refined at each step. We end up drawing them as "vertices" but it's not like the simplex or octahedron where we add new vertices at each step. Rather, we multiply each quadrant by 2 at each step. Also, in counting the "pieces", we have to be mindful of the different "paths" or "sequences" by which, for example, the faces of a cube arose. We have a Totality, but there isn't any Center because, frankly, there aren't any vertices, either. It's just pairs of mirrors sliding out (to infinity?) and opening up more pairs of mirrors. Now, theologically, we can imagine that we have the issues that Kirby raises which I suppose come up amongst his Martian and Earthling theologians. Namely, the Martians suppose that in every direction, "God is good". There is no concept of "bad", of negative. But Earthlings have polar valuations. At this point there isn't any moral distinction. It's like "up" and "down", "right" and "left", "forwards" and "backwards", without any moral preference (ignoring connotations!) In the case of the octahedrons (crosspolytopes), each dimension has a single, independent polarity and the Center keeps adding new dimensions. In the case of the cubes, the polarities build on each other, so that each "quadrant" participates in all of the polarities. We could flip our thinking around in the last couple of sentences because these are dual structures, that is, we could say that each triangular face in an octahedron involves a choice from the three polarities (dimensions) and thus defines a "quadrant". In every way these two families are dual to each other. The difference is that the Center is building bottomup whereas the Totality is building topdown. This distinction appears in the Simplex where the Center creates vertices whereas the Totality could be thought of as getting rid of vertices, in the opposite direction. Finally, we can explain the Dn polytopes. I have found that these are actually poorly or inconsistently defined. How could that be for such an important object? Well, partly, it seems that this math seems "peripheral", too concrete and simple yet messy for those mathematicians and physicists trying to make big discoveries without having to imagine a "big picture". But also, there is a great book by Imre Latakos, "Proofs and refutations", which shows how in his dialogue about Euler's characteristic formula how mathematicians make up their definitions as they go along, and then remake them as they feel they need, quite arbitrarily, in fact. So I've found inconsistent descriptions of these polytopes, especially as to whether they have single edges or double edges. It's just a philosophical distinction because the symmetry group is the same in each case, which is what matters to them. So I offer a philosophical answer. Typically, the Dn polytopes are defined as the demicubes (demihypercubes) which are gotten by starting with a cube and taking half of its vertices. In three dimensions, this means that a cube's vertices can be split into the vertices of two tetrahedrons, for example, if we color alternating vertices white and black. This can be done for an Ndimensional cube as well, in which case we take half of the vertices, say, the black ones, and relink them by adding new edges between "second nearest neighbors". The object we get will not be a regular polytope, that is, not all the faces will be the same, but there will be at least two different kinds, as with the cuboctahedron. However, every vertex will look the same, nevertheless, and so it will be a uniform polytope. And the symmetry group will be almost the same as for the cube. The symmetry group for the Ncube includes the symmetric group SN which relabels all of the "vertices" but also the group of reflections Z2^N because you can reflect the cube (or octahedron) in each dimension (they are built of opposites!) which is not true for the simplex (no sense of opposites). Well, for the Dn polytopes we only allow even reflection sequences of reflections. If you reflect once, then you have to reflect in some direction yet again, because we've trashed every other vertex, or so to say, we've prohibited odd sequences of reflections. Let's instead think of folding the cube! (or sphere!) In order to do this, we have to distinguish a "vertex" (a quadrant) in our cube. And we will identify it with the vertex which is opposite to it in every way, in every dimension. Now they are one and the same, and likewise, we identify in this direction each pair of vertices. So we get a spiky construct that looks like, and is, a coordinate system. Each edge is actually a double edge. Indeed, from the "pascal triangle" for the Dn polytopes, it is most elegant to have double edges instead of single edges. Then this pascal triangle is actually the sum of two pascal triangles, one for bottomup simplexes and the other for topdown cubes. These spiky constructs are similar to the "half cubes" (hemicubes). However, we have to add a double edge between the tips of each "vector" in the coordinate system. This makes each coordinate system into a simplex (all vertices are related!) with double edges and a distinguished "origin". The angles look 904545, though, because the origin is special. What are the the pieces of our coordinate system? There are two kinds because we have a double perspective: * We have 2^(N1) vertices because we fused opposite vertices. For each of these vertices we have our "coordinate system" which we made into a symplex and is understood to have all of the usual pieces of a simplex. * We also keep in mind the original cube, the original Totality, and its pieces (except for vertices, edges and twodimensional faces). So for the threedimensional D3polytope we have just the "volume cell", for the fourdimensional D4polytope we have one 4D cell and 4 3d faces, all as we would for the cube as a whole. Instead of imagining this as a "coordinate system" with the vector tips related (which reminds me of Kirby's "closing the lid" operator), we can also imagine these as cubes with double edges and also "reinforcements" on every 2dimensional face which connects the opposite corners, making an X of double edges on each face. That seems to yield the same construct. Even as I described it, it's messier than the others. I found an open access paper on it by R.M.Green, "Homology representations arising from the half cube" http://www.sciencedirect.com/science/article/pii/S0001870809001017 and there is also a paper by Daniel Pellicer on a very much related construct, "The Higher Dimensional Hemicubeoctahedron": https://books.google.lt/books?id=HarWCwAAQBAJ&printsec=frontcover#v=onepage&q&f=false The latter construct is very similar: Imagine the vertices of an ndimensional cube and make squares out of every pair of opposite edges (carving up the middle). Look at the "cube" as if it were an octahedron and place, alternating, half of the simplexes that you would need to cover it all. So there is a "cubic" inside and a "simplex" outside. Philosophically and even theologically, I think that Dn is the whole point of An, Bn, Cn. We started with An having both a Center and a Totality. We can think of it has having arisen from an initial tension in the original "implicit opposites", Center and Totality, before they found expression. Then we saw that instead, if we created in terms of such "opposites", we could generate a series Cn with a Center but no Totality, as with the octahedrons. Or we could generate a series Bn with a Totality but no Center. Now, understandably, Dn is the series with no Center and no Totality. That is, it has an AntiCenter and an AntiTotality. Each pair of opposite vertices can be fused together to create a Coordinate System, an Origin which is an AntiCenter. With regard to that coordinate system, suddenly every polarity becomes explicit. One direction is with the vectors of the coordinate system, and one direct is against. So we have "good" and "bad" made plain in every direction. But there is one direction which is above it all, namely the two fused vertices who formed the Origin. Apparently, they chose one corner, and from that corner everything emmanates as "good" towards the other corner, but "bad" in the opposite direction. Or as we say in math, "positive" and "negative". And, apparently, you can't try to get out of choosing, you can't just meet in the center, because you would then lose a dimension. The Totality insists on the space being full, one way or the other, and all of the Ncube's cells likewise must assert which way is the "good" way. Of course, which direction is "good" is completely arbitrary. But the Center is above it all. I expect that the point of this all is to model what I think is the big truth, which is that, "God doesn't have to be good. Life doesn't have to be fair." A related way to think about this all is in terms of Christopher Alexander's books "The Nature of Order". Here is a picture of a sequence of "wholeness preserving transformations" which very much bring to my mind what I've described for An, Bn, Cn... http://www.ms.lt/derlius/MatematikosGrozis/06.jpg Well, suppose we already live in a built environment. Then the question is, how can we restructure it? And that is what I think Dn is all about. So the Dn is a kind of "sphere folding". But I suppose what's important is also "sphere unfolding"? That is, abandoning our particular AntiCenter, our particular key dimension for orienting ourselves as to what is "good" and what is "bad". I think that is what I mean when I say: God doesn't have to be "good". I started writing this letter to share my experiences about circle folding. I hope I've at least suggested that I have reason to believe it can be informative about the most central issues in math and life. I've expressed that in ideas and language that I personally am more familiar with. I've ended up writing here about my own explorations. For my own sake, I'm encouraged that the kinds of models that I think are most basic for life seem to be at the very heart of mathematics and help us sense how and why it all gets generated. I wish to offer some pictures at some point. Thank you to all of us for giving us our various worlds. į:
Circle folding also comes up, I think, in terms of a more general "spherefolding" where the sphere can be of any dimension N. 2016 birželio 19 d., 13:11
atliko 
Pridėtos 1102 eilutės:
>>bgcolor=#EEEEEE<< Dear Bradford and all, Every so often I would like to do some circle folding and share my findings. Thank you, Bradford, for opening up this very new way of thinking for me. As I wrote, I sent my niece Ona a circle folding link for her birthday: http://wholemovement.com/howtofoldcircles I made a "sphere" with her name on it and I sent her this photo: http://www.ms.lt/derlius/HappyCirclesOna800.jpg As I mentioned, the surprise for me was that the sphere was so taut after I assembled the four circles with paper clips. Another surprise was that I would need to use paper clips. And then I was curious what I had actually built. I counted 14 holes. I realized that 8 of the holes had 3 sides and 6 of the holes had 4 sides. And they were laid out like the 6 faces and 8 corners of a cube. (Equivalently, the 8 faces and 6 corners of an octahedron). So my "sphere" is basically a cuboctahedron. Each circle (actually, it's a disk) contributed 2 corners (tetrahedrons). And then, apparently, it also contributed 1 1/2 corners of the "octahedron". They aren't cubical corners in the sense that if I try to mount them on a book, I see that the angles are too acute, they aren't "right angles". (I wonder if Kirby's Martians are lefthanded and they call the tetrahedron angle the "left angle".) The sphere can be thought of as two halfspheres and the plane in between is divided into 6 angles, which I see need to be equal, so these "left angles" are 360/6 = 60 degrees. They need to be equal because the side of each "square hole" matches the side of a "triangular hole" and the triangle at the top of the sphere is clearly equilateral. Putting together the 4 circles creates the 6 "square holes". Basically, each circle contributes 3 "quartersquares". So 6 x 3/4 = 4. So we see that the circles pull together two different worlds, the "intraworld" of the holes in the circles and the "interworld" created by the spaces between the circles. This distinction may be very meaningful in the field of "homology" which is a very abstract branch of mathematics that is basically the study of "holes". And it's a very difficult, abstract subject perhaps because the holes "are not there" from the point of view of set theory, so it's hard for set theory to talk about them. It's also very helpful to be able to build models of the threedimensional polytopes because I've been thinking a lot about polytopes as they turn out to be central to the distinctions I am looking for, many of which Kirby discusses. Theoretically, I can think of two ways in which circle folding makes tangible fundamental ideas in math. Folding the circle at once creates "implicit opposites" very much like i and j, the square roots of negative one. One point that I would make to Steve which I think isn't apparent in his exposition of Clifford Algebras is that there is, a priori, no distinction to be made between the two square roots of negative one as they are both indistinguishable in every way. The distinction i and i is artificial and misleading if they make us think that one rotation is more preferable or natural than another. Whereas the distinction between +1 and 1 is completely valid because 1 x 1 = 1 whereas 1 x 1 = 1 and so there is a real distinction to be made. So circle folding gives us practice with the implications of such implicit opposites. It models for us the complex plane. Then things like "complex conjugation" become natural. Or looking at the "upper half plane" of the complexes likewise, I imagine. Also, complex multiplication may make more sense. Perhaps we could try to draw a "unit circle" on the circle to consider how that multiplication works. Or perhaps, better yet, we could consider one side of the circle as the InSide (centered on 0) and the flip side of the paper circle as the OutSide (centered on infinity). Now we have a representation of the complex plane where 0 and infinity are naturally identifiable and motion away from the unit circle and towards the unit circle is, I expect, equivalent. The two operations involved in complex multiplication  moving towards the center or the edge  and rotating along the edge  then seem completely natural in a way that is unnatural in the "square" Euclidean plane. Circle folding also comes up, I think, in terms of a more general "spherefolding" where the sphere can be of any dimension N. In the letter that I am writing I am describing about a Center which generates four families of polytopes An, Bn, Cn, Dn. The Center is the 1 simplex, which is to say, it is the unexplained unique 1 simplex which should be accompanying as an "empty set" every ksimplex made up of 1dimensional vertices, 2dimensional edges, 3dimensional faces and such in kdimensions. https://en.wikipedia.org/wiki/Simplex Look for the table of simplexes, see how it relates to Pascal's triangle, and note the unexplained 1 dimensional simplex. I note that the 1 simplex can be conceived as the unexpressed unique center which we can associate with each simplex. For example, it is the center of a point, the center of an edge, the center of a triangle, the center of a tetrahedron, and so on. We can also imagine the Totality of the simplex, which for a 2dimensional triangle would be the area, and for a 3dimensional tetrahedron would be the volume, whatever is defined by all of the vertices. So we can imagine that first the Center is all by itself, and then it generates vertices along with the Totality of these vertices, so that the Center expresses itself but is never expressed, always moving on as the new Center of the vertices it has created. The Center is never expressed because each vertex it creates is distinct from the rest and so the Center is ever new. Well, it turns out that this is just one of four "theological" models of how the Center can behave. We can think of the Center as modeling God and the Totality as modeling Everything. And this particular process generates the Simplexes, which are known as polytope family An because the symmetries of these simplexes are given by the Symmetric group. Which is to say, every vertex is linked to every other vertex, and the only symmetry is that the vertices could be relabelled or renumbered in any possible way. The lack of any additional symmetry also means that each vertex is uniquely "positive" as Kirby keeps saying. So this is a model that "God is good" in every direction. But there are four families of such polytopes: An: ndimensional simplexes Bn: ndimensional cubes Cn: ndimensional cross polytopes (such as the octahedron) Dn: ndimensional demicubes (demihypercubes) which I find are best thought of as halfcubes (hemicubes) with double edges. The symmetries of these four families are captured by their symmetry groups (Coxeter groups of reflections), which are the Weyl groups of the root systems, which are vector bases for the Lie algebras, whose vector "addition" is the analogue (by way of the exponential function) of the matrix "multiplication" in the Lie groups, which are the continuous groups, which are restricted by the possible, allowable "shortcuts" for inverting (undoing) a continuous action, so that there could be a continuous geometry in which nothing gets ripped apart. In other words, I think that the symmetries of the four families of polytopes dictate four geometries, such as Steve distinguishes. And it is interesting what geometries are yielded by the 5 exceptional Lie groups. There is a paper by Victor Kac http://arxiv.org/pdf/math/9912235.pdf "Classification of infinitedimensional simple groups of supersymmetries and quantum field theory" where in his final section (page 20) on "Speculations and Visions" he notes that "Each of the four types W, S, H, K of simple primitive Lie algebras (L, L0) correspond to the four most important types of geometries of manifolds: all manifolds, oriented manifolds, symplectic and contact manifolds." I found this quote through a link by John Baez https://golem.ph.utexas.edu/category/2007/10/geometric_representation_theor_3.html about a video lecture given by his coteacher James Dolan as part of their "Geometric Representation Theory Seminar" http://math.ucr.edu/home/baez/qgfall2007/ which discusses the relevance of the polytopes to logic and geometry. John Baez has a series of essays about these geometries which I suppose starts here: http://math.ucr.edu/home/baez/week181.html Note, however, that he swaps the letters Cn and Dn from the way I and many others use them. An yields projective geometry Bn yields conformal geometry Cn yields conformal geometry Dn yields symplectic conformal geometry. Basically, symplectic is relevant when two quantities "position" and "momentum" are related, as by time, energy. Symplectic is, I think, what happens when possiblities tracked in the "complex" quantum reality get manifested (through a natural or human "measurement") as a "real" actuality. Dn is where I imagine circlefolding is relevant, again. The Center and the Totality distinguish An, Bn, Cn, Dn in the following way. The An simplexes have both a Center and a Totality. For example, a tetrahedron has 1 Center (the center), 4 vertices, 6 edges, 4 faces and 1 Totality (the volume). Well, that's a row in Pascal's triangle. And we see that in each row there is always 1 Center and always 1 Totality at the opposite ends of Pascal's triangle. Let's consider next the "pascal triangle" which counts the pieces of the crosspolytopes (the orthoplexes, the octahedrons). Here in each row there is a power of two that keeps increasing as follows. An octahedron has 6 vertices, 12 edges and 8 faces. So the pascal triangle is: 1 x 2^0 3 x 2^1 3 x 2^2 1 x 2^3 In other words, it has 1 center, 3 x 2 vertices, 3 x 4 edges and 1 x 8 faces. And it has no Totality, no volume! This makes sense if we consider what the Center is doing here. It is creating two vertices at a time. We can think of them as "implicit opposites" because there is no way of telling them apart. However, later on, we will see how Dn labels them as "positive" and "negative". Anyways, each new dimension yields 2 new vertices that are connected to all of the previous vertices to keep them all distinct. The Center yields first two unconnected points, then a square, then the octahedron, then the tetracross (the 16cell). So each vertex is directly connected to every vertex except for its opposite. If you were to connect all of the opposites, then you would have a simplex (but the dimension would be twice as large because the simplex is generated one vertex at at time). Anyways, when all the vertices are related to each other, (except for their opposites!), then we have the 8 faces of the octahedron. That's why the octahedron has, by definition, as we can see from Pascal's triangle, no concept of totality, no volume. It can't because the opposites aren't allowed to be related. They don't need to be related because they are, by definition, opposite. So that's a rather typical example of where physical reality can be misleading or not the most helpful model. I first came to that conclusion when I tried to understand electricity in terms of "running water". Finally, a teacher explained to me that it can't be thought of that way because the relationship between voltage, amperage, resistance, etc. is simply different and the analogy fails spectacularly. Similarly, the spin of an electron doesn't relate to our physical models. And that seems to be the case with most of modern physics. In general, the Nunez/Lakoff view that mathematical thinking arose to match the experience of the human body seems to me very foreign. Instead, it seems natural to me that mathematical structures reflect our internal, spiritual modeling. Steve's focus on vision seems fruitful and then there arises the question as to whether our visual evolution was driven by our spiritual inner life or our practical outer life. And I'm curious what mathematical thinking is like for the blind and for those conscious in the womb. But I think the latter could be "playing" with the type of philosophical Center and Totality issues that I'm discussing. It's enough to abstractly think two or three abstract "things" and the various ways they may relate. Now the cubes Bn are just like the crosspolytopes but with the rows of the "pascal triangle" reversed. A cube has 8 points, 12 edges, 6 faces. This means, by analogy above, that is has a Totality but it has no Center! So I will explain what seems to be happening. Let us start with a Totality and imagine it as an unfolding mirror. The Totality is itself the one initial mirror. We can think of it as defining two opposite directions. Now imagine that mirror dividing itself into two and moving out into those two opposite directions. Imagine that it is thereby opening up a mirror in the perpendicular direction. Now we have a second dimension which also defines opposite directions. Now imagine the totality of all the mirrors dividing to reveal a new dimension with a new mirror. As this process continues, we get 2^N quadrants, that is, N pairs of "implicit opposites". I see that we really don't get vertices, edges, faces. Indeed, the quadrants keeps getting refined at each step. We end up drawing them as "vertices" but it's not like the simplex or octahedron where we add new vertices at each step. Rather, we multiply each quadrant by 2 at each step. Also, in counting the "pieces", we have to be mindful of the different "paths" or "sequences" by which, for example, the faces of a cube arose. We have a Totality, but there isn't any Center because, frankly, there aren't any vertices, either. It's just pairs of mirrors sliding out (to infinity?) and opening up more pairs of mirrors. Now, theologically, we can imagine that we have the issues that Kirby raises which I suppose come up amongst his Martian and Earthling theologians. Namely, the Martians suppose that in every direction, "God is good". There is no concept of "bad", of negative. But Earthlings have polar valuations. At this point there isn't any moral distinction. It's like "up" and "down", "right" and "left", "forwards" and "backwards", without any moral preference (ignoring connotations!) In the case of the octahedrons (crosspolytopes), each dimension has a single, independent polarity and the Center keeps adding new dimensions. In the case of the cubes, the polarities build on each other, so that each "quadrant" participates in all of the polarities. We could flip our thinking around in the last couple of sentences because these are dual structures, that is, we could say that each triangular face in an octahedron involves a choice from the three polarities (dimensions) and thus defines a "quadrant". In every way these two families are dual to each other. The difference is that the Center is building bottomup whereas the Totality is building topdown. This distinction appears in the Simplex where the Center creates vertices whereas the Totality could be thought of as getting rid of vertices, in the opposite direction. Finally, we can explain the Dn polytopes. I have found that these are actually poorly or inconsistently defined. How could that be for such an important object? Well, partly, it seems that this math seems "peripheral", too concrete and simple yet messy for those mathematicians and physicists trying to make big discoveries without having to imagine a "big picture". But also, there is a great book by Imre Latakos, "Proofs and refutations", which shows how in his dialogue about Euler's characteristic formula how mathematicians make up their definitions as they go along, and then remake them as they feel they need, quite arbitrarily, in fact. So I've found inconsistent descriptions of these polytopes, especially as to whether they have single edges or double edges. It's just a philosophical distinction because the symmetry group is the same in each case, which is what matters to them. So I offer a philosophical answer. Typically, the Dn polytopes are defined as the demicubes (demihypercubes) which are gotten by starting with a cube and taking half of its vertices. In three dimensions, this means that a cube's vertices can be split into the vertices of two tetrahedrons, for example, if we color alternating vertices white and black. This can be done for an Ndimensional cube as well, in which case we take half of the vertices, say, the black ones, and relink them by adding new edges between "second nearest neighbors". The object we get will not be a regular polytope, that is, not all the faces will be the same, but there will be at least two different kinds, as with the cuboctahedron. However, every vertex will look the same, nevertheless, and so it will be a uniform polytope. And the symmetry group will be almost the same as for the cube. The symmetry group for the Ncube includes the symmetric group SN which relabels all of the "vertices" but also the group of reflections Z2^N because you can reflect the cube (or octahedron) in each dimension (they are built of opposites!) which is not true for the simplex (no sense of opposites). Well, for the Dn polytopes we only allow even reflection sequences of reflections. If you reflect once, then you have to reflect in some direction yet again, because we've trashed every other vertex, or so to say, we've prohibited odd sequences of reflections. Let's instead think of folding the cube! (or sphere!) In order to do this, we have to distinguish a "vertex" (a quadrant) in our cube. And we will identify it with the vertex which is opposite to it in every way, in every dimension. Now they are one and the same, and likewise, we identify in this direction each pair of vertices. So we get a spiky construct that looks like, and is, a coordinate system. Each edge is actually a double edge. Indeed, from the "pascal triangle" for the Dn polytopes, it is most elegant to have double edges instead of single edges. Then this pascal triangle is actually the sum of two pascal triangles, one for bottomup simplexes and the other for topdown cubes. These spiky constructs are similar to the "half cubes" (hemicubes). However, we have to add a double edge between the tips of each "vector" in the coordinate system. This makes each coordinate system into a simplex (all vertices are related!) with double edges and a distinguished "origin". The angles look 904545, though, because the origin is special. What are the the pieces of our coordinate system? There are two kinds because we have a double perspective: * We have 2^(N1) vertices because we fused opposite vertices. For each of these vertices we have our "coordinate system" which we made into a symplex and is understood to have all of the usual pieces of a simplex. * We also keep in mind the original cube, the original Totality, and its pieces (except for vertices, edges and twodimensional faces). So for the threedimensional D3polytope we have just the "volume cell", for the fourdimensional D4polytope we have one 4D cell and 4 3d faces, all as we would for the cube as a whole. Instead of imagining this as a "coordinate system" with the vector tips related (which reminds me of Kirby's "closing the lid" operator), we can also imagine these as cubes with double edges and also "reinforcements" on every 2dimensional face which connects the opposite corners, making an X of double edges on each face. That seems to yield the same construct. Even as I described it, it's messier than the others. I found an open access paper on it by R.M.Green, "Homology representations arising from the half cube" http://www.sciencedirect.com/science/article/pii/S0001870809001017 and there is also a paper by Daniel Pellicer on a very much related construct, "The Higher Dimensional Hemicubeoctahedron": https://books.google.lt/books?id=HarWCwAAQBAJ&printsec=frontcover#v=onepage&q&f=false The latter construct is very similar: Imagine the vertices of an ndimensional cube and make squares out of every pair of opposite edges (carving up the middle). Look at the "cube" as if it were an octahedron and place, alternating, half of the simplexes that you would need to cover it all. So there is a "cubic" inside and a "simplex" outside. Philosophically and even theologically, I think that Dn is the whole point of An, Bn, Cn. We started with An having both a Center and a Totality. We can think of it has having arisen from an initial tension in the original "implicit opposites", Center and Totality, before they found expression. Then we saw that instead, if we created in terms of such "opposites", we could generate a series Cn with a Center but no Totality, as with the octahedrons. Or we could generate a series Bn with a Totality but no Center. Now, understandably, Dn is the series with no Center and no Totality. That is, it has an AntiCenter and an AntiTotality. Each pair of opposite vertices can be fused together to create a Coordinate System, an Origin which is an AntiCenter. With regard to that coordinate system, suddenly every polarity becomes explicit. One direction is with the vectors of the coordinate system, and one direct is against. So we have "good" and "bad" made plain in every direction. But there is one direction which is above it all, namely the two fused vertices who formed the Origin. Apparently, they chose one corner, and from that corner everything emmanates as "good" towards the other corner, but "bad" in the opposite direction. Or as we say in math, "positive" and "negative". And, apparently, you can't try to get out of choosing, you can't just meet in the center, because you would then lose a dimension. The Totality insists on the space being full, one way or the other, and all of the Ncube's cells likewise must assert which way is the "good" way. Of course, which direction is "good" is completely arbitrary. But the Center is above it all. I expect that the point of this all is to model what I think is the big truth, which is that, "God doesn't have to be good. Life doesn't have to be fair." A related way to think about this all is in terms of Christopher Alexander's books "The Nature of Order". Here is a picture of a sequence of "wholeness preserving transformations" which very much bring to my mind what I've described for An, Bn, Cn... http://www.ms.lt/derlius/MatematikosGrozis/06.jpg Well, suppose we already live in a built environment. Then the question is, how can we restructure it? And that is what I think Dn is all about. So the Dn is a kind of "sphere folding". But I suppose what's important is also "sphere unfolding"? That is, abandoning our particular AntiCenter, our particular key dimension for orienting ourselves as to what is "good" and what is "bad". I think that is what I mean when I say: God doesn't have to be "good". I started writing this letter to share my experiences about circle folding. I hope I've at least suggested that I have reason to believe it can be informative about the most central issues in math and life. I've expressed that in ideas and language that I personally am more familiar with. I've ended up writing here about my own explorations. For my own sake, I'm encouraged that the kinds of models that I think are most basic for life seem to be at the very heart of mathematics and help us sense how and why it all gets generated. I wish to offer some pictures at some point. Thank you to all of us for giving us our various worlds. >><< 
CircleFoldingNaujausi pakeitimai 
Puslapis paskutinį kartą pakeistas 2016 birželio 19 d., 14:24
