by **Mercurial, the Spectre** » Fri Dec 29, 2017 4:33 pm

When finding the coordinates of polyhedra with high symmetry, it is best to use the octahedral model (all permutations of x,y,z with all sign changes) since it's relatively easy. For every vertex (x,y,z), there is an octahedral symmetry polyhedron (generally isogonal) that has this coordinate.

Say, you have the coordinate set (1,2,3). You have these 6 permutations:

(1,2,3)

(1,3,2)

(2,1,3)

(2,3,1)

(3,1,2)

(3,2,1)

and, accounting for 8 sign changes (+++, ++-, +-+, -++, +--, -+-, --+, ---), you have 6*8 = 48 possible coordinates, representing a truncated cuboctahedron.

In fact that is the order of octahedral symmetry.

Now, when building icosahedral symmetry figures, the coordinates generally involve the golden ratio (phi) because the regular icosahedron has 3 golden rectangles (1:phi) that are identical in its pyritohedral subsymmetry plus a regular pentagon embedded inside the icosahedron; this is marked by edges but not considered a face.

In fact, you can look at the icosahedron standing on its edge. Two opposite edges form a golden rectangle of coordinates (±1,±phi,0). Then there is another of coordinates (0,±1,±phi), and another of coordinates (±phi,0,±1). These rectangles lie on the xy, xz, and yz planes, so they are mutually perpendicular. This represents the pyritohedral subsymmetry of the icosahedron which has half the order of octahedral symmetry.

One thing is, the icosahedron does not have full octahedral subsymmetry. It doesn't have any 4-fold rotations the cube and octahedron has. This means that the coordinates of the icosahedron need to be in cyclical order, like abc -> cab -> bca where the symbols are interpreted as vertices of a triangle and labeled by moving in only one direction like a circle. So you have to base on pyritohedral symmetry to construct an icosahedral-symmetric object. In fact there are two variants, one from either abc or cba. However, you still have the 8 possible sign permutations.

For tetrahedral-symmetric polyhedra, you have the 6 permutations but only four sign changes (+++, +--, --+, -+-, or ++-, +-+, -++, ---), distinguished by whether the number of minus signs in a given coordinate is even or odd. In fact, the tetrahedron can be inscribed in a cube, which is indeed the cube's own alternation.

Both symmetries have half the order of full octahedral symmetry, but they are not the same. Pyritohedral symmetry has no 4-fold projections but central inversion, while tetrahedral symmetry has 4-fold projections but no central inversion. Central inversion is the property that for a coordinate (+a,+b,+c), you have (-a,-b,-c).

That is all I can explain to you regarding the construction of polyhedra through coordinates. Hope you can learn more about these funky solids!

Ilazhra!