There are only a few types of symmetry a polyhedron can have: 7 infinite families corresponding to the 2D frieze groups, and 7 special groups related to the Platonic solids. Here are representative figures with the frieze groups as symmetries; they're supposed to extend infinitely along one dimension:
- Code: Select all
1:
⌊ - ⌊ - ⌊ - ⌊ - ⌊ -
2:
⌊ - ⌊ - ⌊ - ⌊ - ⌊ -
⌈ - ⌈ - ⌈ - ⌈ - ⌈ -
3:
⌊ - ⌊ - ⌊ - ⌊ - ⌊ -
- ⌈ - ⌈ - ⌈ - ⌈ - ⌈
4:
⌊ - ⌊ - ⌊ - ⌊ - ⌊ -
⌉ - ⌉ - ⌉ - ⌉ - ⌉ -
5:
⌊⌋ -- ⌊⌋ -- ⌊⌋ -- ⌊⌋ -- ⌊⌋ --
6:
⌊⌋ -- ⌊⌋ -- ⌊⌋ -- ⌊⌋ -- ⌊⌋ --
⌈⌉ -- ⌈⌉ -- ⌈⌉ -- ⌈⌉ -- ⌈⌉ --
7:
⌊⌋ -- ⌊⌋ -- ⌊⌋ -- ⌊⌋ -- ⌊⌋ --
-- ⌈⌉ -- ⌈⌉ -- ⌈⌉ -- ⌈⌉ -- ⌈⌉
Now imagine wrapping each figure around a cylinder in 3D (like a label on a can), so that ⌊ appears n times in a circle. Then figure 5 has the symmetry of an n-gon pyramid, figure 6 has the symmetry of an n-gon prism, and figure 7 has the symmetry of an n-gon antiprism. Figures 2 and 3 are like the treads on tyres. The symmetry group of figure 1 contains only n rotations. Figures 2 and 3, n rotations and n rotoreflections. Figure 4, n rotations around the main axis and n 180° rotations around perpendicular axes. Figure 5, n rotations and n reflections. Figures 6 and 7, n rotations around the main axis, n 180° rotations around perpendicular axes, n reflections, and n rotoreflections.
(A rotoreflection is a reflection across a plane, followed by a rotation around the axis perpendicular to the plane.)
The 7 special groups are:
Tetrahedral; 8 120° rotations, 3 180° rotations, 1 identity, 6 reflections, 6 90° rotoreflections; total 24
Chiral tetrahedral; 8 120° rotations, 3 180° rotations, 1 identity; total 12
Cubic; 6 90° rotations, 8 120° rotations, 6+3 180° rotations, 1 identity, 6+3 reflections, 6 90° rotoreflections, 8 60° rotoreflections, 1 inversion; total 48
Chiral cubic (the symmetry of the
snub cube); 6 90° rotations, 8 120° rotations, 6+3 180° rotations, 1 identity; total 24
Pyritohedral (something between chiral tetrahedral and cubic); 8 120° rotations, 3 180° rotations, 1 identity, 3 reflections, 8 60° rotoreflections, 1 inversion; total 24
Dodecahedral; 12 72° rotations, 12 144° rotations, 20 120° rotations, 15 180° rotations, 1 identity, 15 reflections, 20 60° rotoreflections, 12 36° rotoreflections, 12 108° rotoreflections, 1 inversion; total 120
Chiral dodecahedral; 12 72° rotations, 12 144° rotations, 20 120° rotations, 15 180° rotations, 1 identity; total 60
The inversion can be considered a 180° rotoreflection, but the axis of rotation isn't unique. It just negates the coordinates of everything. Similarly the identity can be considered a 0° rotation, and a reflection can be considered a 0° rotoreflection.
Now quaternions can only represent rotations, not reflections. So, in the above, only the symmetries of figures 1 and 4, and the three chiral special groups, can be made into quaternions. We've already considered the cubic and tetrahedral groups. I believe the dodecahedral rotations generate the cells of a 120-cell, or the vertices of a 600-cell. Figure 1 is rather degenerate; it generates 2n quaternions of the form cos(kπ/n)+
isin(kπ/n)+0
j+0
k, which are the vertices of a polygon, not a polychoron. Figure 4 generates the cells of a 2n,2n-
duoprism. This includes the 8-cell (or tesseract, or 4-cube, or hypercube, or 4,4-duoprism) in the case n=2, corresponding to the rotational symmetries of a box with three different edge lengths.
But figure 4 looks nothing like a box!
Well, if you introduce some reflections (and no new rotations) to figure 4, you get figure 6, representing a prism of a regular n-gon, or of an irregular but isogonal 2n-gon. An isogonal 4-gon is just a rectangle. Hence, figure 6 in the case n=2 is a box.
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