Dividing a sphere into twelve "identical" shapes.

Thread,

It takes a while for me to build each truncated octagon, but finally today I finished the first level buildout. 14 truncated octagons touching the 14 faces of a central truncated octagon.

I have not studied it much yet. Just built it, but I was intrigued at the fact that when you look at each diamond shaped foursome it is arranged exactly like the 12 diamonds of the rhombic dodecahedron.

However, it does not seem to be arranged in the same lattice structure as the rhombic dodecahedron. Although there are four hex planes and 3 square planes. Have not figured it yet.

In any case it is very interesting to me, in terms of the unit volume idea I am working on because it seems to have identical symmetries and angles even though one is 14 faces and the other 12.

Sort a dualness about these two figures, the rhombic dodecahedron and the truncated octagon. As you can tell looking at the toothpicks with the red tips and the plain toothpicks in exactly the same angle positions in each figure, one the one the faces correspond to the apexes and vice a versa.

Excited to build out the next level...but I am figuring it will take 42 to build out the next level so I may just do an octant. and imagine what shape is developing.

The rhombic dodecahedra build out to a cuboctahedron and I was guessing the truncated octagon would build out to an octagon, but now, seeing the first layer looking like a rhombic dodecahedra...i wonder.

The eight hex sides point exactly though the middle of each octant of the cartesian coordinate system, xyz and the square sides point right down each axis.