## Wednesday, March 29, 2006

### M3

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;

12/8=3/2=E3/H3={,{},{,{}}}/{,{,{,{}}}}

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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