Symmetry Triangles
There are 4 symmetry classes of reflection on the sphere, and two in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are also listed. (Increasing any of the numbers defining a hyperbolic or Euclidean tiling makes another hyperbolic tiling.)
Point groups:
- (p 2 2) dihedral symmetry, p = 2, 3, 4... (order 4p)
- (3 3 2) tetrahedral symmetry (order 24)
- (4 3 2) octahedral symmetry (order 48)
- (5 3 2) icosahedral symmetry (order 120)
Euclidean (affine) groups:
- (4 4 2) *442 symmetry: 45°-45°-90° triangle
- (6 3 2) *632 symmetry: 30°-60°-90° triangle
- (3 3 3) *333 symmetry (60°-60°-60° plane)
Hyperbolic groups:
- (7 3 2) *732 symmetry
- (8 3 2) *832 symmetry
- (4 3 3) *433 symmetry
- (4 4 3) *443 symmetry
- (4 4 4) *444 symmetry
- (5 4 2) *542 symmetry
- (6 4 2) *642 symmetry
| Dihedral spherical | Spherical | ||||||
|---|---|---|---|---|---|---|---|
| D2h | D3h | D4h | D5h | D6h | Td | Oh | Ih |
| *222 | *322 | *422 | *522 | *622 | *332 | *432 | *532 |
(2 2 2) |
(3 2 2) |
(4 2 2) |
(5 2 2) |
(6 2 2) |
(3 3 2) |
(4 3 2) |
(5 3 2) |
The above symmetry groups only includes the integer solutions on the sphere. The list of Schwarz triangles includes rational numbers, and determine the full set of solutions of nonconvex uniform polyhedra.
| p4m | p3m | p6m |
|---|---|---|
| *442 | *333 | *632 |
(4 4 2) |
(3 3 3) |
(6 3 2) |
| *732 | *542 | *433 |
|---|---|---|
(7 3 2) |
(5 4 2) |
(4 3 3) |
In the tilings above, each triangle is a fundamental domain, colored by even and odd reflections.
Read more about this topic: Wythoff Symbol
Famous quotes containing the words symmetry and/or triangles:
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—George Gordon Noel Byron (1788–1824)
“If triangles had a god, they would give him three sides.”
—Charles Louis de Secondat Montesquieu (1689–1755)