Improper Self-Dual Polyhedra |
The canonical form of a polyhedron (when it exists) can be used to generate the polyhedron's dual using an algorithm like the following:
When this algorithm is applied to a chiral self-dual polyhedron, there are two possibilities: Either the resulting dual polyhedron is an identical copy of the original polyhedron (only rotated), or it is an opposite-handed version of the original polyhedron. In most cases, the dual is identical to the original. In a small percentage of the cases, however, the dual is the opposite-handed version. An appropriate name for these opposite-hand-pairing chiral self-dual polyhedra is "improper" because of the similarity to improper rotations: The opposite-handed dual cannot be brought into identity alignment with the original using rotations alone. A reflection is necessary.
The following table shows the counts of all chiral self-dual polyhedra with up to 16 faces, cross tabulated by number of faces and symmetry type. Each count is shown as a fraction, where the numerator represents the number of improper self duals, and the denominator represents the number of chiral self duals. Many of the counts can be clicked to show representative polyhedra.
FACES | C1 | C2 | C3 | C4 | C5 | D2 | T | O | I | TOTAL | PERCENT |
4 | ...... | ...... | ...... | ...... | ...... | ...... | ...... | ...... | ...... | ...... | ...... |
5 | ...... | ...... | ...... | ...... | ...... | ...... | ...... | ...... | ...... | ...... | ...... |
6 | ...... | 0/1 | ...... | ...... | ...... | ...... | ...... | ...... | ...... | 0/1 | 0.00 % |
7 | 0/2 | 0/1 | ...... | ...... | ...... | ...... | ...... | ...... | ...... | 0/3 | 0.00 % |
8 | 0/10 | 0/3 | ...... | ...... | ...... | ...... | ...... | ...... | ...... | 0/13 | 0.00 % |
9 | 0/39 | 0/4 | ...... | ...... | ...... | ...... | ...... | ...... | ...... | 0/43 | 0.00 % |
10 | 2/137 | 0/10 | 0/3 | ...... | ...... | 0/1 | ...... | ...... | ...... | 2/151 | 1.32 % |
11 | 13/514 | 0/18 | ...... | ...... | ...... | ...... | ...... | ...... | ...... | 13/532 | 2.44 % |
12 | 46/1817 | 0/37 | ...... | ...... | ...... | 0/2 | ...... | ...... | ...... | 46/1856 | 2.48 % |
13 | 187/6490 | 2/68 | 0/12 | 0/3 | ...... | ...... | ...... | ...... | ...... | 189/6573 | 2.88 % |
14 | 660/23199 | 6/155 | ...... | ...... | ...... | 0/3 | ...... | ...... | ...... | 666/23357 | 2.85 % |
15 | 2387/83406 | 8/275 | ...... | ...... | ...... | ...... | ...... | ...... | ...... | 2395/83681 | 2.86 % |
16 | 8379/300928 | 24/587 | 2/52 | ...... | 0/3 | 1/6 | 0/1 | ...... | ...... | 8406/301577 | 2.79 % |
TOTAL | 11674/416542 | 40/1159 | 2/67 | 0/3 | 0/3 | 1/12 | 0/1 | ...... | ...... | 11717/417787 | 2.80 % |