ATON is a hopefully evolving classification theory. It aspires to unify knowledge around numbers and prefers naive methods. Some of the older posts are wrong but I'll keep them for the sake of continuity.

Wednesday, March 29, 2006


Above are 3 visualisations of M3 with no perspective. It is a shape I found while playing with equal length bamboo sticks. Previously I called it Bion. I decided to call it just M3. Now I aso don't divide the outer cubes. It's outer aspect is the 3 interlaced cubes like the Escher's. Inside, it has a cube made of two reflected tetrahedrons. The number of total faces of the outer cubes is 18. The tetrahedrons have 8 triangle faces. Note the resemblance to some ancient Asian yantras, specially Shri Yantra.

I found it when I was admiring the beauty of a tetrahedron made using 6 equal length sticks. Then I rotated it pi/2 and added them to each other making a cube made of two reflected tetrahedrons. This is a cube made by it's diagonals. The diagonals of the cube make two interlaced tetrahedrons. Then I wondered what would happen if I kept this iteration of making a transformation and adding it to the parent shape. I rotated the cube pi/2 and translated it by 1/2 in each of the 6 face directions and added them all to each other. Then I completed the manifested cubes and also divided the cubes to into 8 equal pieces.

There may be more things one can do to Bion with equal length sticks as suggested by M6. There is already a large octahedron inside. I expect all the Plato's polyhedrons to appear in some iteration of Bion. I had enough of playing with sticks, too tedious.
I am thinking about a little program to experiment with Bion. Someday it may happen. It would be wonderful to make a full connection with M6.

In the middle there are two interlaced Tetrahedrons, made by rotating one of them pi/4 and adding. Each T is made of 6 sticks representing the edges. T+T(pi/4) has 2*6=12 sticks. There are 6 crosses making the faces of a cube. There are 8 corners made by the tips of Ts. Thus we got our;
T^2 is a cube made by it's diagonals.

The outer layer of Bion has 3 cubes made by sticks at the edges. There are 3*6=18=SE4/SH4 faces.

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