• ms@ms.lt
• +370 607 27 665
• My work is in the Public Domain for all to share freely.

Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

## Book.MathConnections istorija

2018 vasario 21 d., 13:58 atliko AndriusKulikauskas -
Pridėtos 81-82 eilutės:

* Try to understand Lawyere's eight-fold [[https://ncatlab.org/nlab/show/Hegelian%20taco | Hegelian taco]].
2016 gruodžio 13 d., 21:42 atliko AndriusKulikauskas -
Pakeista 43 eilutė iš:
* Grothendieck's [[six operations]]. Consider a function from one algebraic variety or scheme to another. Then we can define accordingly four functors from one category of sheaves to another such category. These functors are defined to make sense across a family of bases, that is, across base changes. Upper and lower star functors are like everything and nothing. Upper and lower shriek functors are like "fibers" within everything, thus: anything and something. The fiber may be identified with the category. Tensor and Hom are defined within the category of sheaves (thus within the input and also within the output). Tensor can be thought of as decreasing slack by filling it out. Hom can be thought of as increasing slack by creating multiplicity of functions. The four functors relate Hom and Tensor in the input category and in the output category. The six operations can be thought of as naturally defined within a higher order category of correspondences. The six operations can also be thought of as a generalization which grounds Poincare duality and its generalizations, Serre duality.
į:
* Grothendieck's [[six operations]]. Consider a function from one algebraic variety or scheme to another. Then we can define accordingly four functors from one category of sheaves to another such category. These functors are defined to make sense across a family of bases, that is, across base changes. Upper and lower star functors are like everything and nothing. Upper and lower shriek functors are like "fibers" within everything, thus: anything and something. The fiber may be identified with the category. Tensor and Hom are defined within the category of sheaves (thus within the input and also within the output). Tensor can be thought of as decreasing slack by filling it out. Hom can be thought of as increasing slack by creating multiplicity of functions. The four functors relate Hom and Tensor in the input category and in the output category. The six operations can be thought of as naturally defined within a higher order category of correspondences. The six operations can also be thought of as a generalization which grounds Poincare duality and its generalizations, Serre duality. Note that these two seem relate to the Snake lemma.
2016 gruodžio 13 d., 21:42 atliko AndriusKulikauskas -
Pridėtos 40-43 eilutės:

'''Representations'''

* Grothendieck's [[six operations]]. Consider a function from one algebraic variety or scheme to another. Then we can define accordingly four functors from one category of sheaves to another such category. These functors are defined to make sense across a family of bases, that is, across base changes. Upper and lower star functors are like everything and nothing. Upper and lower shriek functors are like "fibers" within everything, thus: anything and something. The fiber may be identified with the category. Tensor and Hom are defined within the category of sheaves (thus within the input and also within the output). Tensor can be thought of as decreasing slack by filling it out. Hom can be thought of as increasing slack by creating multiplicity of functions. The four functors relate Hom and Tensor in the input category and in the output category. The six operations can be thought of as naturally defined within a higher order category of correspondences. The six operations can also be thought of as a generalization which grounds Poincare duality and its generalizations, Serre duality.
2016 gruodžio 01 d., 23:07 atliko AndriusKulikauskas -
Pridėta 37 eilutė:
* String diagrams portray such exact sequences with divisions of the plane by way of objects.
2016 lapkričio 30 d., 21:29 atliko AndriusKulikauskas -
Pridėta 38 eilutė:
* Twosome: objects and morphisms
2016 lapkričio 27 d., 09:46 atliko AndriusKulikauskas -
Pakeistos 36-38 eilutės iš
į:
* Divisions of everything into N perspectives are given by finite exact sequences with N nonzero terms.
* Sixsome see: An Introduction to the K-theory of Banach Algebras
Pakeista 63 eilutė iš:
* I think that they are defined structurally by algebra and dynamically by analysis and they come together in the four geometries.
į:
* I think that they are defined structurally by algebira and dynamically by analysis and they come together in the four geometries.
2016 lapkričio 27 d., 07:20 atliko AndriusKulikauskas -
Pakeista 72 eilutė iš:
* https://en.m.wikipedia.org/wiki/Snake_lemma
į:
* https://en.m.wikipedia.org/wiki/Snake_lemma relate to sixsome
2016 lapkričio 27 d., 07:16 atliko AndriusKulikauskas -
Pridėta 72 eilutė:
* https://en.m.wikipedia.org/wiki/Snake_lemma
2016 lapkričio 24 d., 19:55 atliko AndriusKulikauskas -
Pridėtos 77-80 eilutės:

'''My philosophy'''

I think my philosophy may be an illuminating example for category theory. I mean that if we think of a functor F as going from a category C of our mental notions and association between them to a category D of linguistic expressions and continuations between them, then this particular application may also serve as a universally relevant interpretation and general foundation of category theory. It may indeed be meaningful to speak in category theory of a duality between paradigmatic application and universally relevant interpretation.
2016 lapkričio 22 d., 21:42 atliko AndriusKulikauskas -
Pakeistos 72-74 eilutės iš
*
į:
'''Equation of Life'''

* Spirit and structure are related by "duality", the operation +2.
* A set is the essence of the spirit, the free monoid, that it generates.
2016 lapkričio 22 d., 21:35 atliko AndriusKulikauskas -
Pakeistos 68-75 eilutės iš
į:
'''Eightfold Way'''

* The Eightfold Way relates a left exact sequence and a right exact sequence
* The Zermelo Frankel axioms of set theory seem to be structured by what I call the EightfoldWay.
*

Ištrintos 76-79 eilutės:

Septynerybė aštuonerybė
* triangle 1 unknown 3 vertices +3 edges +1 whole
* Aštuongubas kelias. SetTheory - The Zermelo Frankel axioms of set theory seem to be structured by what I call the EightfoldWay.
2016 lapkričio 22 d., 21:33 atliko AndriusKulikauskas -
Pakeista 6 eilutė iš:
* Understand [[God]] as the [[Center]] of polytopes (such as the [[Simplex]]) and [[Everything]] as the [[Totality]].
į:
* Understand [[God]] as the [[Center]] of polytopes (such as the [[Simplex]]) and Everything as the Totality.
Pakeista 27 eilutė iš:
* six visualizations ([[restructurings]] in terms of sequences, hierarchies and networks).
į:
* six visualizations (restructurings in terms of sequences, hierarchies and networks).
2016 lapkričio 22 d., 21:33 atliko AndriusKulikauskas -
Pakeistos 33-36 eilutės iš
'''Divisions of everything'''

Express eight [[Divisions | Divisions of everything]], if possible, in terms of [[Bott periodicity]]. There are eight [[Division | divisions]] of everything (from the nullsome to the sevensome) and there are three operations +1, +2, +3 which act on them cyclically. This brings to mind [[Bott periodicity]] and the clock shift of [[Clifford algebra | Clifford algebras]].
į:
'''[[Divisions | Divisions of everything]]'''

Express the eight divisions of everything, and the three operations +1, +2, +3, which act on them cyclically, in terms of [[Bott periodicity]] and the clock shift of [[Clifford algebra | Clifford algebras]].

'''Languages of argumentation
, verbalization and narration'''
Ištrintos 41-43 eilutės:

My philosophy is an exploration of everything, and especially, the conceptual limits of the mind. On this page I am overviewing how the various structures and concepts appear in the [[implicit math]] which we use in our mind to interpret [[explicit math]] that gets expressed in a language of mathematical symbols. That implicit math is especially evident in our mind's system of [[Math Discovery | ways of figuring things out in mathematics]].
Pridėtos 57-58 eilutės:

'''Perspectives'''
2016 lapkričio 22 d., 21:27 atliko AndriusKulikauskas -
Pakeista 24 eilutė iš:
* six [[transformations]] between the four geometries.
į:
* six [[specifications]] between the four geometries.
Pakeista 29 eilutė iš:
* six [[set theory axioms]].
į:
* six set theory axioms.
2016 lapkričio 22 d., 21:24 atliko AndriusKulikauskas -
Pakeista 24 eilutė iš:
* six transformations between the four geometries.
į:
* six [[transformations]] between the four geometries.
2016 lapkričio 22 d., 21:24 atliko AndriusKulikauskas -
Pakeistos 15-17 eilutės iš
Describe four levels of knowledge (whether, what, how, why).
į:
[+Structures to express+]

'''Four
levels of knowledge (whether, what, how, why).'''
Pakeista 23 eilutė iš:
Describe and relate various manifestations of six pairs of these four levels.
į:
'''Six pairs of these four levels.'''
Pakeistos 30-33 eilutės iš
* interpret them metalogically as six ways of relating two mental sheets.

Express eight [[Divisions | Divisions
of everything]], if possible, in terms of [[Bott periodicity]].
į:
* six bases of symmetric functions
*
six ways of relating two mental sheets, a logic and a metalogic.

'''Divisions
of everything'''

Express eight [[Divisions | Divisions
of everything]], if possible, in terms of [[Bott periodicity]]. There are eight [[Division | divisions]] of everything (from the nullsome to the sevensome) and there are three operations +1, +2, +3 which act on them cyclically. This brings to mind [[Bott periodicity]] and the clock shift of [[Clifford algebra | Clifford algebras]].
Pridėtos 43-44 eilutės:
'''God'''
Pakeista 52 eilutė iš:
* The simplexes An have a Center and a Totality. They are the basis for affine geometry where directions are preserved.
į:
* The simplexes An have a Center and a Totality. They are the basis for affine geometry where paths are preserved.
Ištrintos 58-65 eilutės:
I am looking for 6 transformations of perspective which link the structure of one geometry with the dynamics of another geometry. I think these 6 transformations relate to ways of interpreting multiplication, to restructurings of sequences, hierarchies and networks, and to the axioms of set theory which define sets.
* The (affine) whole is (projectively) recopied.
* The (affine) whole is (conformally) rescaled.
* The (projective) multiple is (conformally) rescaled.
* The (conformal) set is (symplectically) redistributed.
* The (projective) multiple is (symplectically) redistributed.
* The (affine) whole is (symplectically) redistributed.
Ištrinta 66 eilutė:
There are eight [[Division | divisions]] of everything (from the nullsome to the sevensome) and there are three operations +1, +2, +3 which act on them cyclically. This brings to mind [[Bott periodicity]] and the clock shift of [[Clifford algebra | Clifford algebras]].
2016 lapkričio 22 d., 21:19 atliko AndriusKulikauskas -
Pakeistos 21-27 eilutės iš
Discover six [[transformations]] between these four levels.
* collect and identify six ways of thinking about [[variables]]
.
* survey, specify and ground six transformations between the four geometries.
* relate them to
six way of thinking about [[multiplication]].
* relate them to six kinds of [[variables]]
.
* relate these six transformations to six visualizations ([[restructurings]] in terms of sequences, hierarchies and networks).
* relate them to six qualities of signs
.
į:
Describe and relate various manifestations of six pairs of these four levels.
* six transformations between the four geometries
.
* six ways of thinking about [[variables]].
* six ways of thinking about [[multiplication]].
*
six visualizations ([[restructurings]] in terms of sequences, hierarchies and networks).
* six qualities of signs.
*
six [[set theory axioms]].
Ištrinta 28 eilutė:
* relate them to six [[set theory axioms]].
2016 lapkričio 22 d., 21:16 atliko AndriusKulikauskas -
Ištrintos 0-1 eilutės:
'''Connections between math and my philosophy'''
Pakeistos 3-12 eilutės iš
I am expressing my philosopy's concepts in terms of mathematics. In particular, I wish to:

Understand [[Gods Dance | God's dance]] in terms of zero, infinity and one.
* understand [[God]] as the
[[Center]] of polytopes (such as the [[Simplex]]) and [[Everything]] as the [[Totality]].
* understand the
[[field with one element]], learn about the combinatorics and geometry of finite fields and understand F1n.
* overview examples of [[duality]] and understand their essence, how they relate one and zero, how they help map out the areas of math.
* understand how zero and
infinity get differentiated, how their symmetry gets variously broken.
* understand
how finite fields and the field with one element express infinity.
* express infinity in terms of geometries.
*
understand the kinds of [[Opposites]] and how they are expressed in mathematics.
į:
I am expressing my philosophy's concepts in terms of mathematics. In particular, I wish to:

* Express [[Gods Dance | God's dance]] in terms of the [[field with one element]] understood as zero, infinity and one.
* Understand
[[God]] as the [[Center]] of polytopes (such as the [[Simplex]]) and [[Everything]] as the [[Totality]].
* Express God's Ten Commandments as four geometries and six transformations between them
.

Some practical projects:
* Learn about the combinatorics and geometry of finite fields
and interpret F1n.
* Understand infinity. Understand how finite fields and the field with one element express
infinity. Understand how zero and infinity get differentiated, how their symmetry gets variously broken. Express infinity in terms of geometries.
* Overview examples of [[duality]] and understand their essence, how they relate one and zero, how they help map out the areas of math.
* Understand
the kinds of [[Opposites]] and how they are expressed in mathematics.
2016 lapkričio 22 d., 21:11 atliko AndriusKulikauskas -
Pridėtos 2-36 eilutės:

'''My philosopy's concepts in terms of math'''

I am expressing my philosopy's concepts in terms of mathematics. In particular, I wish to:

Understand [[Gods Dance | God's dance]] in terms of zero, infinity and one.
* understand [[God]] as the [[Center]] of polytopes (such as the [[Simplex]]) and [[Everything]] as the [[Totality]].
* understand the [[field with one element]], learn about the combinatorics and geometry of finite fields and understand F1n.
* overview examples of [[duality]] and understand their essence, how they relate one and zero, how they help map out the areas of math.
* understand how zero and infinity get differentiated, how their symmetry gets variously broken.
* understand how finite fields and the field with one element express infinity.
* express infinity in terms of geometries.
* understand the kinds of [[Opposites]] and how they are expressed in mathematics.

Describe four levels of knowledge (whether, what, how, why).
* survey, understand and distinguish between four geometries (affine, projective, conformal, symplectic).
* express them in terms of symmetric functions.
* review recursive function theory and describe the four levels in terms of related concepts such as the Yates Index theorem (the triple jump).
* understand intuitively, why there are four classical families of Lie groups/algebras and what four geometries they give rise to.

Discover six [[transformations]] between these four levels.
* collect and identify six ways of thinking about [[variables]].
* survey, specify and ground six transformations between the four geometries.
* relate them to six way of thinking about [[multiplication]].
* relate them to six kinds of [[variables]].
* relate these six transformations to six visualizations ([[restructurings]] in terms of sequences, hierarchies and networks).
* relate them to six qualities of signs.
* interpret them metalogically as six ways of relating two mental sheets.
* relate them to six [[set theory axioms]].

Express eight [[Divisions | Divisions of everything]], if possible, in terms of [[Bott periodicity]].

Relate mathematics to three cognitive languages by which things come to matter (argumentation), have meaning (verbalization) and take place (narration).
* Relate a hierarchy of six methods of proof to the prayer "Our Father" and a language of argumentation.
2016 birželio 23 d., 16:17 atliko AndriusKulikauskas -
Ištrintos 38-39 eilutės:
Roots of unity = divisions of everything?
Pakeistos 41-42 eilutės iš
į:
* Aštuongubas kelias. SetTheory - The Zermelo Frankel axioms of set theory seem to be structured by what I call the EightfoldWay.
Ištrintos 50-52 eilutės:

Aštuongubas kelias
* SetTheory - The Zermelo Frankel axioms of set theory seem to be structured by what I call the EightfoldWay.
2016 birželio 23 d., 16:12 atliko AndriusKulikauskas -
Pridėta 33 eilutė:
* Perspectives may be logical quantifiers.
2016 birželio 23 d., 16:11 atliko AndriusKulikauskas -
Pakeistos 34-35 eilutės iš
There are eight divisions of everything (from the nullsome to the sevensome) and there are three operations +1, +2, +3 which act on them cyclically. This brings to mind [[Bott periodicity]] and the clock shift of [[Clifford algebra | Clifford algebras]].
į:
There are eight [[Division | divisions]] of everything (from the nullsome to the sevensome) and there are three operations +1, +2, +3 which act on them cyclically. This brings to mind [[Bott periodicity]] and the clock shift of [[Clifford algebra | Clifford algebras]].
2016 birželio 23 d., 16:10 atliko AndriusKulikauskas -
Ištrintos 45-56 eilutės:

Ketverybė:
* Recursive functions - There is a jump hierarchy of recursive functions that (by the Yates index theorem) has one level be "conscious" of the level that is three levels below it, which is thus relevant for the foursome's role in consciousness.
* Reikėtų išmokti Yates Index Theorem, jinai pakankamai trumpa ir turbūt ne tokia sudėtinga. Ir paaiškinti ką jinai turi bendro su sąmoningumu.

Penkerybė:
* Analysis allows for work with limits.
* Eccentricity of conic sections - there are five eccentricities (for the circle, parabola, ellipse, hyperbola, line).

Septynerybė:
* Logic is the end result of structure, see the sevensome and Greimas' semiotic square.
* Reikėtų gerai išmokti aritmetinę hierarchiją ir bandyti ją taikyti kitur. Kaip jinai rūšiuoja sąvokas? Kaip ji siejasi su pirmos eilės, antros eilės ir kitokiomis logikomis? Kaip ji siejasi su tikrųjų skaičių ir kitokių skaičių tvėrimu? Kaip kvantoriai išsako septynerybę? Ar septynerybė išsako kvantorių ir neigimo derinius? Kaip jie susiję su požiūriais, požiūrių sudūrimu ir požiūrių grandinėmis, tad su kategorijų teorija ir požiūrių algebra?
2016 birželio 23 d., 16:09 atliko AndriusKulikauskas -
Ištrintos 42-48 eilutės:

Unmarked opposite
* turinys = raiška. "Those things are which show themselves to be." buvimo pagrindas
* inner 2-cycle, kurio paprastai nebūna.
* complex numbers i=j iš kurio atsiveria 1<>-1, i<>-i. Paprastai i -> j -> i ... banguoja, o šitą sustabdžius gaunasi +1 +1 +1 +1 amžinai ir atitinkamai -1-1-1-1 amžinai.
2016 birželio 23 d., 16:07 atliko AndriusKulikauskas -
Pakeistos 5-6 eilutės iš
God is a key concept for me. I am ever trying to imagine everything from God's point of view.
į:
[[God]] is a key concept for me. I am ever trying to imagine everything from God's point of view.
Ištrintos 37-40 eilutės:

God
* God may also be given by trivial tensor: T00. It is a zero-dimensional array, thus a scalar. An array is a perspective, and so it is having no perspective. A scalar is "spirit". Thus it is spirit with no perspective.
* Dievas išeina už savęs: 3 matai -> 2 matai -> (flip to dual) 1 matas -> 0 matas (taškas: gera širdis).
2016 birželio 23 d., 16:06 atliko AndriusKulikauskas -
Pakeistos 5-6 eilutės iš
God is a key concept for me. I am ever trying to imagine everything from God's point of view. I think that God gets expressed in math as the [[Center]] which generates the [[simplex | simplexes]]. Everything is then the dual of God, the Totality consisting of all of the vertices and all of the simplexes.
į:
God is a key concept for me. I am ever trying to imagine everything from God's point of view.

I think that the [[field with one element]] is a model of God's trinity. The sole element of the field can be interpreted as 0, ∞ and 1. 0 makes way for ∞ and 1 is their point of balance. God's trinity is the heart of God's dance. The various kinds of [[opposite]] are also I think important in driving God's dance.

I think that God gets expressed in math as the [[Center]] which generates the [[simplex | simplexes]]. Everything is then the dual of God, the Totality consisting of all of the vertices and all of the simplexes.
Pakeistos 17-18 eilutės iš
I think that the [[field with one element]] is a model of God's trinity. The sole element of the field can be interpreted as 0, ∞ and 1. 0 makes way for ∞ and 1 is their point of balance.
į:
The family Dn seems to model the equation of eternal life, namely, that God doesn't have to be good, life doesn't have to be fair.

I am looking for 6 transformations of perspective which link the structure of one geometry with the dynamics
of another geometry. I think these 6 transformations relate to ways of interpreting multiplication, to restructurings of sequences, hierarchies and networks, and to the axioms of set theory which define sets.
* The (affine) whole is (projectively) recopied.
* The (affine) whole is (conformally) rescaled.
* The (projective) multiple is (conformally) rescaled.
* The (conformal) set is (symplectically) redistributed.
* The (projective) multiple is (symplectically) redistributed.
* The (affine) whole is (symplectically) redistributed.

[[Perspective | Perspectives]] are important in my philosophy. There are several ways they appear in math.
* I think that they are defined structurally by algebra and dynamically by analysis and they come together in the four geometries.
* I think the 6 transformations link the 4 possible perspectives.
* I imagine that fields (or division rings) as scalars define perspectives, their freedom.
* The complex numbers offer a dual perspective as opposed to the real numbers' single perspective.
* Category theory perhaps defines perspectives and their composition
.
Ištrintos 56-63 eilutės:

Požiūriai:
* Complex numbers: dvimačiai: širdies tiesa. Real numbers: pasaulio tiesa.
* Kategorijų teorija.
* Kvantoriai ir septynerybė.
* Nėra quantum frequency. Fotonai yra "požiūriai". Bosonai yra "jėgų nešėjai", "santykiai", jie "neegzistuoja".
* Ar požiūriai yra neasociatyvūs?
* Perspectives are (multidimensional) arrays. The number of array dimensions is the number of divisions of everything.
2016 birželio 23 d., 15:40 atliko AndriusKulikauskas -
Pakeistos 15-17 eilutės iš
į:
There are eight divisions of everything (from the nullsome to the sevensome) and there are three operations +1, +2, +3 which act on them cyclically. This brings to mind [[Bott periodicity]] and the clock shift of [[Clifford algebra | Clifford algebras]].
Ištrintos 21-22 eilutės:
* Field of one element.
* [[https://www.youtube.com/watch?v=1XRna0vUYdo | field of one element video]]
Pakeistos 34-58 eilutės iš
* Bott periodicity [[http://math.ucr.edu/home/baez/week105.html | John Baez]]
* [[http://www.math.illinois.edu/K-theory/handbook/1-111-138.pdf | Max Karoubi vadovėlis apie Bott periodicity]]
* palyginti susijusias Lie grupes (ir jų ryšį su gaubliu) su požiūrių permainomis
* Bott periodicity turėtų būti susijęs su aštuonerybės sugriuvimu prieštaravimu
* susipažinti su Clifford algebra ir clock shift veiksmais
* Max Karoubi savo video paskaitoje paminėjo loop lygtį žiedams kurioje R,C,H,H' ir epsilon = +/-1 gaunasi 10 homotopy equivalences. Kodėl 10? 8+2=10? ar 6+4=10, dešimt Dievo įsakymų?
* Žiūrėk taip pat [[https://en.wikipedia.org/wiki/Hopf_fibration Hopf fibration]], [[https://en.wikipedia.org/wiki/Hopf_invariant | Hopf invariant]] ir Adam's theorem. [[https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres | Homotopy group of spheres]]. [[https://en.wikipedia.org/wiki/Clifford_parallel | Clifford paralells]] ir quaternions.
* [[https://golem.ph.utexas.edu/category/2007/10/higher_clifford_algebras.html | Higher Clifford Algebras]]

[[http://math.ucr.edu/home/baez/week82.html | Clifford algebra periodicity]]
* C0 R
* C1 C
* C2 H
* C3 H + H
* C4 H(2)
* C5 C(4)
* C6 R(8)
* C7 R(8) + R(8)
* C8 R(16)
''C_{n+8} consists of 16 x 16 matrices with entries in Cn ! For a proof you might try

2) H. Blaine Lawson, Jr. and Marie-Louise Michelson, "Spin Geometry", Princeton U. Press, Princeton, 1989. or 3) Dale Husemoller, "Fibre Bundles", Springer-Verlag, Berlin, 1994. These books also describe some of the amazing consequences of this periodicity phenomenon. The topology of n-dimensional manifolds is very similar to the topology of (n+8)-dimensional manifolds in some subtle but important ways!'' Physics of fermions.

[[http://math.ucr.edu/home/baez/week61.html | Introduction to rotation groups]] '''Triality of octonions.''' ''More generally, it turns out that the representation theory of Spin(n) depends strongly on whether n is even or odd. When n is even (and bigger than 2), it turns out that Spin(n) has left-handed and right-handed spinor representations, each of dimension 2^{n/2 - 1}. When n is odd there is just one spinor representation. Of course, there is always the representation of Spin(n) coming from the vector representation of SO(n), which is n-dimensional. This leads to something very curious. If you are an ordinary 4-dimensional physicist you undoubtedly tend to think of spinors as "smaller" than vectors, since the spinor representations are 2-dimensional, while the vector representation is 3-dimensional. However, in general, when the dimension n of space (or spacetime) is even, the dimension of the spinor representations is 2^(n/2 - 1), while that of the vector representation is n, so after a while the spinor representation catches up with the vector representation and becomes bigger! This is a little bit curious, or at least it may seem so at first, but what's really curious is what happens exactly when the spinor representation catches up with the vector representation. That's when 2^(n/2 - 1) = n, or n = 8. The group Spin(8) has three 8-dimensional irreducible representations: the vector, left-handed spinor, and right-handed spinor representation. While they are not equivalent to each other, they are darn close; they are related by a symmetry of Spin(8) called "triality". And, to top it off, the octonions can be seen as a kind of spin-off of this triality symmetry... as one might have guessed, from all this 8-dimensional stuff. One can, in fact, describe the product of octonions in these terms. So now let's dig in a bit deeper and describe how this triality business works. For this, unfortunately, I will need to assume some vague familiarity with exterior algebras, Clifford algebras, and their relation to the spin group. But we will have a fair amount of fun getting our hands on a 24-dimensional beast called the Chevalley algebra, which contains the vector and spinor representations of Spin(8)!''
į:
2016 birželio 23 d., 15:34 atliko AndriusKulikauskas -
Pakeistos 5-6 eilutės iš
God is a key concept for me. I am ever trying to imagine everything from God's point of view. I think that God gets expressed in math as the Center which generates the [[simplex | simplexes]]. Everything is then the dual of God, the Totality consisting of all of the vertices and all of the simplexes.
į:
God is a key concept for me. I am ever trying to imagine everything from God's point of view. I think that God gets expressed in math as the [[Center]] which generates the [[simplex | simplexes]]. Everything is then the dual of God, the Totality consisting of all of the vertices and all of the simplexes.
Pakeistos 13-15 eilutės iš
į:
I think that the [[field with one element]] is a model of God's trinity. The sole element of the field can be interpreted as 0, ∞ and 1. 0 makes way for ∞ and 1 is their point of balance.
Pakeistos 18-19 eilutės iš
Dievas
į:
God
* God may also be given by trivial tensor: T00. It is a zero-dimensional array, thus a scalar. An array is a perspective, and so it is having no perspective. A scalar is "spirit". Thus it is spirit with no perspective.
Ištrintos 22-24 eilutės:
* The trivial tensor: T00. It is a zero-dimensional array, thus a scalar. An array is a perspective, and so it is having no perspective. A scalar is "spirit". Thus it is spirit with no perspective.
* Simplex, interpret the -1 face. Sometimes considered the empty set. Attention which is free. It is the center of the simplex. You can always add a new center to imagine the next simplex but in the current dimensions. Interpret (a+x)**n. It is (implicit + explicit)**n dimensions.
** [[http://www.geometrictools.com/Documentation/CentersOfSimplex.pdf | Centers of Simplex]]
2016 birželio 23 d., 15:28 atliko AndriusKulikauskas -
2016 birželio 23 d., 15:28 atliko AndriusKulikauskas -
Pakeistos 3-13 eilutės iš
į:
My philosophy is an exploration of everything, and especially, the conceptual limits of the mind. On this page I am overviewing how the various structures and concepts appear in the [[implicit math]] which we use in our mind to interpret [[explicit math]] that gets expressed in a language of mathematical symbols. That implicit math is especially evident in our mind's system of [[Math Discovery | ways of figuring things out in mathematics]].

God is a key concept for me. I am ever trying to imagine everything from God's point of view. I think that God gets expressed in math as the Center which generates the [[simplex | simplexes]]. Everything is then the dual of God, the Totality consisting of all of the vertices and all of the simplexes.

Four combinations of God and Everything generate four infinite families of polytopes and associated geometries and metalogics. I think these are the for representations of God:
* The simplexes An have a Center and a Totality. They are the basis for affine geometry where directions are preserved.
* The cross-polytopes Cn have a Center but no Totality. They are the basis for projective geometry where lines are preserved.
* The cubes Bn have no Center but have a Totality. They are the basis for conformal geometry where angles are preserved.
* The coordinate systems Dn have no Center and no Totality. They are the basis for symplectic geometry where areas are preserved.
Pakeista 17 eilutė iš:
* Field of one element. Roots of unity = divisions of everything?
į:
* Field of one element.
Pridėtos 23-24 eilutės:

Roots of unity = divisions of everything?
2016 birželio 23 d., 15:00 atliko AndriusKulikauskas -
Pridėtos 1-91 eilutės:
'''Connections between math and my philosophy'''

>>bgcolor=#EEEEEE<<

Dievas
* Field of one element. Roots of unity = divisions of everything?
* [[https://www.youtube.com/watch?v=1XRna0vUYdo | field of one element video]]
* Dievas išeina už savęs: 3 matai -> 2 matai -> (flip to dual) 1 matas -> 0 matas (taškas: gera širdis).
* The trivial tensor: T00. It is a zero-dimensional array, thus a scalar. An array is a perspective, and so it is having no perspective. A scalar is "spirit". Thus it is spirit with no perspective.
* Simplex, interpret the -1 face. Sometimes considered the empty set. Attention which is free. It is the center of the simplex. You can always add a new center to imagine the next simplex but in the current dimensions. Interpret (a+x)**n. It is (implicit + explicit)**n dimensions.
** [[http://www.geometrictools.com/Documentation/CentersOfSimplex.pdf | Centers of Simplex]]

Septynerybė aštuonerybė
* triangle 1 unknown 3 vertices +3 edges +1 whole

Unmarked opposite
* turinys = raiška. "Those things are which show themselves to be." buvimo pagrindas
* inner 2-cycle, kurio paprastai nebūna.
* complex numbers i=j iš kurio atsiveria 1<>-1, i<>-i. Paprastai i -> j -> i ... banguoja, o šitą sustabdžius gaunasi +1 +1 +1 +1 amžinai ir atitinkamai -1-1-1-1 amžinai.

* Bott periodicity [[http://math.ucr.edu/home/baez/week105.html | John Baez]]
* [[http://www.math.illinois.edu/K-theory/handbook/1-111-138.pdf | Max Karoubi vadovėlis apie Bott periodicity]]
* palyginti susijusias Lie grupes (ir jų ryšį su gaubliu) su požiūrių permainomis
* Bott periodicity turėtų būti susijęs su aštuonerybės sugriuvimu prieštaravimu
* susipažinti su Clifford algebra ir clock shift veiksmais
* Max Karoubi savo video paskaitoje paminėjo loop lygtį žiedams kurioje R,C,H,H' ir epsilon = +/-1 gaunasi 10 homotopy equivalences. Kodėl 10? 8+2=10? ar 6+4=10, dešimt Dievo įsakymų?
* Žiūrėk taip pat [[https://en.wikipedia.org/wiki/Hopf_fibration Hopf fibration]], [[https://en.wikipedia.org/wiki/Hopf_invariant | Hopf invariant]] ir Adam's theorem. [[https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres | Homotopy group of spheres]]. [[https://en.wikipedia.org/wiki/Clifford_parallel | Clifford paralells]] ir quaternions.
* [[https://golem.ph.utexas.edu/category/2007/10/higher_clifford_algebras.html | Higher Clifford Algebras]]

[[http://math.ucr.edu/home/baez/week82.html | Clifford algebra periodicity]]
* C0 R
* C1 C
* C2 H
* C3 H + H
* C4 H(2)
* C5 C(4)
* C6 R(8)
* C7 R(8) + R(8)
* C8 R(16)
''C_{n+8} consists of 16 x 16 matrices with entries in Cn ! For a proof you might try

2) H. Blaine Lawson, Jr. and Marie-Louise Michelson, "Spin Geometry", Princeton U. Press, Princeton, 1989. or 3) Dale Husemoller, "Fibre Bundles", Springer-Verlag, Berlin, 1994. These books also describe some of the amazing consequences of this periodicity phenomenon. The topology of n-dimensional manifolds is very similar to the topology of (n+8)-dimensional manifolds in some subtle but important ways!'' Physics of fermions.

[[http://math.ucr.edu/home/baez/week61.html | Introduction to rotation groups]] '''Triality of octonions.''' ''More generally, it turns out that the representation theory of Spin(n) depends strongly on whether n is even or odd. When n is even (and bigger than 2), it turns out that Spin(n) has left-handed and right-handed spinor representations, each of dimension 2^{n/2 - 1}. When n is odd there is just one spinor representation. Of course, there is always the representation of Spin(n) coming from the vector representation of SO(n), which is n-dimensional. This leads to something very curious. If you are an ordinary 4-dimensional physicist you undoubtedly tend to think of spinors as "smaller" than vectors, since the spinor representations are 2-dimensional, while the vector representation is 3-dimensional. However, in general, when the dimension n of space (or spacetime) is even, the dimension of the spinor representations is 2^(n/2 - 1), while that of the vector representation is n, so after a while the spinor representation catches up with the vector representation and becomes bigger! This is a little bit curious, or at least it may seem so at first, but what's really curious is what happens exactly when the spinor representation catches up with the vector representation. That's when 2^(n/2 - 1) = n, or n = 8. The group Spin(8) has three 8-dimensional irreducible representations: the vector, left-handed spinor, and right-handed spinor representation. While they are not equivalent to each other, they are darn close; they are related by a symmetry of Spin(8) called "triality". And, to top it off, the octonions can be seen as a kind of spin-off of this triality symmetry... as one might have guessed, from all this 8-dimensional stuff. One can, in fact, describe the product of octonions in these terms. So now let's dig in a bit deeper and describe how this triality business works. For this, unfortunately, I will need to assume some vague familiarity with exterior algebras, Clifford algebras, and their relation to the spin group. But we will have a fair amount of fun getting our hands on a 24-dimensional beast called the Chevalley algebra, which contains the vector and spinor representations of Spin(8)!''

Gyvenimo lygtis:
* Dvasia ir sandara susieti "duality", veiksmu +2.

Požiūriai:
* Complex numbers: dvimačiai: širdies tiesa. Real numbers: pasaulio tiesa.
* Kategorijų teorija.
* Kvantoriai ir septynerybė.
* Nėra quantum frequency. Fotonai yra "požiūriai". Bosonai yra "jėgų nešėjai", "santykiai", jie "neegzistuoja".
* Ar požiūriai yra neasociatyvūs?
* Perspectives are (multidimensional) arrays. The number of array dimensions is the number of divisions of everything.

Ketverybė:
* Recursive functions - There is a jump hierarchy of recursive functions that (by the Yates index theorem) has one level be "conscious" of the level that is three levels below it, which is thus relevant for the foursome's role in consciousness.
* Reikėtų išmokti Yates Index Theorem, jinai pakankamai trumpa ir turbūt ne tokia sudėtinga. Ir paaiškinti ką jinai turi bendro su sąmoningumu.

Penkerybė:
* Analysis allows for work with limits.
* Eccentricity of conic sections - there are five eccentricities (for the circle, parabola, ellipse, hyperbola, line).

Septynerybė:
* Logic is the end result of structure, see the sevensome and Greimas' semiotic square.
* Reikėtų gerai išmokti aritmetinę hierarchiją ir bandyti ją taikyti kitur. Kaip jinai rūšiuoja sąvokas? Kaip ji siejasi su pirmos eilės, antros eilės ir kitokiomis logikomis? Kaip ji siejasi su tikrųjų skaičių ir kitokių skaičių tvėrimu? Kaip kvantoriai išsako septynerybę? Ar septynerybė išsako kvantorių ir neigimo derinius? Kaip jie susiję su požiūriais, požiūrių sudūrimu ir požiūrių grandinėmis, tad su kategorijų teorija ir požiūrių algebra?

Pertvarkymai:
* Aibės struktūra primena medį nes gali būti aibės aibėse. Tačiau svarbu, kad nėra ratų.

Niekas
* Taškas yra niekas.

Aštuongubas kelias
* SetTheory - The Zermelo Frankel axioms of set theory seem to be structured by what I call the EightfoldWay.

* Topologies - Systems of constraints that may be thought of as defining worlds. Topology is the study of topologies.

Matematikos įrodymų būdai - laipsnynas

24
* 4 netroškimai: 6x4=24 ir dar 0-inis požiūris ir 7-as požiūris. Iš viso 24+2=26 kaip Monster grupėje.

Walks on trees
* Julia sets

>><<

#### MathConnections

Naujausi pakeitimai

 Puslapis paskutinį kartą pakeistas 2018 vasario 21 d., 13:58