Arcadian Functor

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Marni D. Sheppeard

Monday, May 14, 2007

Tetractys

An anonymous commenter has mentioned the Pythagorean Tetractys, a triangle on 10 points, divided into 9 triangular pieces. Knutson, Tao and Woodward studied the correspondence between such tetractys diagrams and honeycombs in the plane. The vertices of the honeycomb become small triangles in the tetractys, as shown. This example is a three dimensional tetractys, for which each side has three small faces. In the NxN case, a puzzle diagram is built from a triangle of side length N. Puzzles use three kinds of pieces: small triangles with edge labels 1, small triangles with edge labels 0, and rhombi (joined triangles) with paired edges labelled 0 and 1.

We can view this as a ternary analogue of a top down view of a binary tree, for which a simple Y tree would look like an interval divided into two parts, with the central point marking the vertex. Recall that such viewpoints were used by Devadoss in his study of generalised associahedra. Further levels of the tree correspond to further subdivisions of subintervals. A well known instance of this game, which considers binary subdivisions in a 2D piece of the tetractys, is Sierpinski's triangle, of Hausdorff dimension log(3)/log(2).

2 Comments:

Anonymous Anonymous said...

According to Matti Pitkänen, we perceive infinite primes as geometric shapes. A 3D tetraclys is triangular pyramid better known as the Platonic form Tetrahedron: http://en.wikipedia.org/wiki/Tetrahedron.

Attempts to understand TGD and natural numbers have lead to obsession with a combination of two tetrahedrons into non-platonic hexahedron, a "Heraclitian cube" or a triangurar dipyramid: http://en.wikipedia.org/wiki/Triangular_dipyramid

Numerologically it seems very productive form of 5 points. 2 points of the pyramid connect to the 3 shared point along 3 lines and the 3 shared points connect 4 lines. So this compact hexahedron has 9 edges.

Actually the 6-face hexahedron consists of 7 triangles including the shared one, and when the tips of the pyramids are also connected by a line, there are 10 lines. The sacred number of the Pythagorean Tetractys is preserved!

What seems to be missing is the number 8 or 2^3 of the Platonic hexahedron, the cube with 12 edges. Now, being total ignoramus of math and having absolutely zero technical skills (therefore anonymous), I can't help thinking here the platonic cubeness of Octonions and a fuzzy analogy between the more compact "cubeness" of the "Heraclitian hexahedron" and e.g. Lie groups, especially the exceptional cases. Also the analogy of a "p-adic" point and a "real" point of the two pyramids sharing a triangular partition comes to mind.

May 14, 2007 8:37 PM  
Blogger kneemo said...

Kea

I found a nice little article by Matthew Headrick (MIT) explaining how the Higgs particle is actually a tachyon in the string theory picture.

A mathematical treatment of the Higgs-Tachyon model is given in Bergman and Lifschytz's hep-th/0606216, where the four Hopf maps play an interesting role.

May 15, 2007 5:45 AM  

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