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See: Math connections

Coincidences 巧合

I'm collecting various coincidences that may, at some point, prove meaningful.

1+3+3+1

In the standard model, one generation has an electron (with charge {$-1$}, red, blue, green up-type quarks with charge {$+\frac{2}{3}$}, red, blue, green down-type quarks with charge {$-\frac{1}{3}$}, and a neutrino with charge {$0$}. Does this imply a binomial expansion of the form {$(1-\frac{1}{3})^3$}? where the multiplication of the {$-\frac{1}{3}$} operates as addition?

Simplexes

Sūnaus požiūrį 1 + 3 + 3 + 1 = 8 nusako simpleksas. Dvasios požiūrį 4 + 6 nusako dalis simplekso, kurio kita dalis 1 + ... + 4 + 1. Kartu šie du simpleksai apima 24 = 4! Kaip suprasti tą likusią dalį? Tai yra 6 = 3! tad galime pamanyti, kad 3! pridėjus Sūnaus (8) ir Dvasios (10) požiūrius išaugo į 4!. O 3! galima suprasti, kaip 4 + 2!, kaip kad su atvaizdais: 6 = 4+2. Taigi klausimas ir atsakymas išaugo visko atvaizdais - asmenimis. O klausimas ir atsakymas (turinys ir raiška) yra Dievo vidinė įtampa. O 4 yra bene simpleksas 1 + 2 + 1. Ir 2! galima suprasti, kaip 1 + 1! Ir 1! galime suprasti, kaip 0 + 0! Tai ką reiškia, kad prisideda narys? Ir kodėl nėra 4! + 96 = 5! ? Ir kokie simpleksai dalyvauja?

Dievo šokis susideda iš simpleksų:

• Trikampio: 1 + 3 + 3 + 1 tai aštuongubas kelias.
• Trikampainis: 1 + 4 + 6 + 4 + 1 Jame 4 + 6 yra Dešimt Dievo įsakymų. O 1 + 4 + 1 turėtų išsakyti 1 + 1 + 1 + 3 bet kaip suprasti?
• Taip pat, ką reiškia briauna 1 + 2 + 1 ? ir taškas 1 + 1 ? ir židinys ?
• Ir kaip permainos sieja 3 + 3 ir (4 2) ? ir 4 + 2 ?

4+6

• Poincare grupė (space/time) has 6+4=10 generators - {$\mathbf{R}^{1,3} \rtimes \mathrm{O}(1,3)$}
• Catalan skaičių formulė yra: Cn-1 = (4n-6)!!!! / n!

The number 8

• Bott periodicity ir Clifford Algebra periodicity
• John Baez about the number 8
• The 7-sphere {$S_{7}$} represents the octonions of unit norm. So this models the 7/8 of the primary structures.
• Your mind is 8-dimensional. Your brain as math.
• 3+3+2=8 Trimatei erdvei pralįsti už trimatės erdvės reikia aštuonių matų. Tad šešerybės vidaus ir išorės trejybės papildytos nulybe ir septynerybe gali apeiti bet kokius trikdžius.
• Pin group There are as many as eight different double covers of O(p, q), for p, q ≠ 0, which correspond to the extensions of the center (which is either C2 × C2 or C4) by C2. Only two of them are pin groups—those that admit the Clifford algebra as a representation. They are called Pin(p, q) and Pin(q, p) respectively.

The number 24

24=8x3

• Triality of {$\frak{so}(8)$}. Its three representations are related by a three-cycle of Pauli matrices. Division algebraic multiplication is just intertwining Pauli matrices with the three representations {$V,S_+,S_-$}. Normed division algebra gives a trilinear form, "normed triality". There is one for each of the four normed division algebras. Talk 4: A Magic Pyramid of Supergravity Theories (Mia Hughes)
• Dievo šokis gali išreikšti 24 veiksmų lygtis. Aštuongubas kelias išsako padalinimų ratą, tad veiksmą +1. Dievo trejybė gali išsakyti veiksmą +2, sąmonės langus: 0+2=2, 2+2=4, 4+2=6. Dešimt Dievo įsakymų gali išsakyti 4 lygtis +2 ir 6 lygtis +3.
• Veiksmų (+1, +2, +3) lygtys. Jų iš viso yra 3 x 8 = 24.
• Atvaizduoti padalinimai: 4x4 + 2x4 = 24.
• 3 (Dievo trejybė) + 3x3-1 (aštuongubas kelias: 1+3+3+1) + 3x3+1 (dešimt Dievo įsakymų 4+6=1+3 + 3+3) + 3 (žmogaus trejybė)
• (4 1) - (4 0), (4 1) + (4 1), (4 1) + (4 2)... ?
• A cube has four diagonals that can be permuted by S4 (eight vertices, six faces, twelve edges, 24 directed edges)
• Kummer's 24 solutions Symmetries given by Coxeter group D3 with 3!*2^2 elements. Related to the hypergeometric function. The confluent hypergeometric function (or Kummer's function) can be given as a limit of the hypergeometric function

{$M(a,c,z)=\lim_{b\to \infty }$}{$_{2}F_{1}(a,b;c;b^{-1}z)$}

• so all functions that are essentially special cases of it, such as Bessel functions, can be expressed as limits of hypergeometric functions. These include most of the commonly used functions of mathematical physics.
• the 24-cell is the unique convex self-dual regular Euclidean polytope which is neither a polygon nor a simplex. Its symmetry group is F4.
• 24 knygos hebrajų Tanake, mat Samuelio, Karalių ir Kronikų knygos buvo tik vėliau padalintos į dvi dalis, išvertus jas į graikų kalbą, nes graikų kalba raštant su balsiais knygos tampa 50% ilgesnės, o štai ilgiausios knygos
• Classification of abelian spin Chern-Simons theories Z24
• String group stable homotopy groups of spheres
• John Baez about the number 24
• Ramanujan tau function
• Dedekind eta function
• Leech lattice
• Griess algebra
• Binary Golay code
• Monster vertex algebra
• 4 nondesires yield 6x4=24 and if we add the 0-th and 7-th perspectives then we have 24+2=26 as with the Monster group.
• John Baez talk: 24 = 6 (trikampių laukas) x 4 (kvadrato laukas).
• 24 + 2 = 26. Dievo šokis (žmogaus trejybės naryje) veikia ant žmogaus (už šokio) tad žmogus papildo šokį dviem matais. Ir gaunasi "group action". Susiję su Monster group.
• The size of the Monster group is comparable to the number of particles in the universe?

Simetrinė grupė S4 kažkuo išsiskiria iš kitų simetrinių grupių. Pasiskaityti. Ir panagrinėti conjugacy classes:

• () 1
• (ab) 6 lygmenų poros
• (abc) 8 trejybės ratai - aštuongubas kelias
• (abcd) 6 išrašyti šeši keitiniai: S3 - toliau skaidant 1 + 3 + 2
• (ab)(cd) 3 - trejybės ratas

Garrett Lisi: There's an unusual description of spacetime called Cartan geometry that's very interesting. You start with a single ten-dimensional Lie group (a rigid geometric surface) and let it deform along four directions. The resulting structure is our four-dimensional spacetime with the six-dimensional gravitational Lie group twisting over it. It is a very efficient model. A year ago I worked out a generalization of Cartan geometry, allowing spacetime to embed in larger Lie groups. When I do this for E8, there's a symmetry called “triality” linking three different sheets of spacetime; with respect to each different sheet, each of the three different generations of fermions comes out right. If this all works, it would mean the reason we see Lie groups everywhere in physics is because we're inside of one, looking out. Our universe and everything in it might be excitations of a single Lie group. 2014.10.20, Scientific American

• Nobody know what E8 is the symmetry group of. (Going beyond oneself?)

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 {$2^{\frac{n}{2} - 1}$}. 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 {$2^{\frac{n}{2} - 1} = n$}, 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)!

The number 5

• John Baez about the number 5
• Exceptional polytopes based on the pentagon are E6, E7, E8.

The number 3

• Quaternions
• The Jacobi identity for Lie algebras.
• Tėvo požiūris, tai 3=3.
• Sūnaus požiūris: 3x3=1 mod 8
• Dvasios požiūris: 3x3x3=27=3 mod 8
• Toliau: 3x3x3x3=1 mod 8.
• Axiom of Maria One becomes two, two becomes three, and out of the third comes the one as the fourth.

The number 42

John Baez about the number 42

Walks on trees

• Infinite ternary tree. The universal covering space (with no loops) for a wedge B of two circles. All of the coverings of B are given by the 2-oriented graphs. Consider a (finite) graph with 4 edges coming in to any vertex.
• Infinite binary tree.

Infinity

• Begalybė gali kilti iš String group iškilimo Postnikov bokšte.

The cube

Lie algebra dualities

The Acopalypse by St.John contains references to {$3\frac{1}{2}=1+2+\frac{1}{2}$}: a time, two times, and a half-time. This may perhaps be a reference to the four classical root systems - quaternions, complexes, reals: the complexes are a time, the two real systems are two times, and the quaternions are a half-time. But perhaps not.

More coincidences

Number 5

• Golden mean is the "most" irrational of numbers (based on its continued fraction). Consider series of continued fractions... as sequence patterns...
• Relate "most irrational" with "randomness". For what you find out does not tell you anything more about what remains.

More numbers

• http://math.ucr.edu/home/baez/42.html
• http://math.ucr.edu/home/baez/numbers/

Notes

• 6=4+2 representations. Similar to 6 edges of simplex = 4 edges of square + 2 diagonals
• Einstein field equations - energy stress tensor - is 4+6 equations.
• SU(3)xSU(2)xSU(1)xSU(0) is reminiscent of the omniscope.
• John Baez: 24 = 6 x 4 = An x Bn
• Dedekind eta function is based on 24.
• Dimensions needed for monster group: 196883 = (12*4 - 1)(12*5 - 1)(12*6 - 1)
• Thomas: The three-cycle makes sense of three dimensionional space.
• The three-cycle can be thought of spatially in terms of a rotation in three-dimensions, but also temporally as periodicity in 1-dimension.
• There is a Leech lattice representation based on the fact that 24 is the only nontrivial integer N such that {$\sum_{i=1}^{N}n^2=M^2$} for some M. Namely, {$\sum_{i=1}^{24}n^2=70^2$}
• Matthew 12:38–42 The sign of Jonah, who was in the big fish for three days and three nights, just as the Son of Man will be in the heart of the earth three days and three nights. The Son of Man is real Bott periodicity and the Son of God is complex Bott periodicity. Thus between O and 4 there are three levels 1, 2, 3. This also brings to mind consciousness.
• {$\frak{sl}$}{$(2,A)=\frak{so}$}{$(1,\dim A + 1)$} Thus Minkowski space in 3,4,6,10 dimensions is the same as {$\frak{sl}$}{$(2,\mathbb{R})$},{$\frak{sl}$}{$(2,\mathbb{C})$}, {$\frak{sl}$}{$(2,\mathbb{H})$}, {$\frak{sl}$}{$(2,\mathbb{O})$}