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Book: Coincidences

See: Math connections

Coincidences 巧合

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


How to understand God's dance as made up of simplexes?


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:

4+6

The number 8

The number 24

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

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

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

The number 3

The number 42

John Baez about the number 42

Walks on trees

Infinity

The cube

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Puslapis paskutinį kartą pakeistas 2019 balandžio 03 d., 17:29