The regular icositetragon has Dih24 symmetry, order 48. There are 7 subgroup dihedral symmetries:, and, and 8 cyclic group symmetries:, and. These 16 symmetries can be seen in 22 distinct symmetries on the icositetragon. John Conway labels these by a letter and group order. The full symmetry of the regular form is r48 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices or edges, and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g24 subgroup has no degrees of freedom but can seen as directed edges.
Dissection
states that every zonogon can be dissected into m/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular icositetragon, m=12, and it can be divided into 66: 6 squares and 5 sets of 12 rhombs. This decomposition is based on a Petrie polygon projection of a 12-cube.
A regular triangle, octagon, and icositetragon can completely fill a plane vertex. An icositetragram is a 24-sided star polygon. There are 3 regular forms given by Schläfli symbols:,, and. There are also 7 regular star figures using the same vertex arrangement: 2, 3, 4, 6, 8, 3, and 2. There are also isogonal icositetragrams constructed as deeper truncations of the regular dodecagon and dodecagram. These also generate two quasitruncations: t=, and t=.
Skew icositetragon
A skew icositetragon is a skew polygon with 24 vertices and edges but not existing on the same plane. The interior of such an icositetragon is not generally defined. A skew zig-zag icositetragon has vertices alternating between two parallel planes. A regular skew icositetragon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew icositetragon and can be seen in the vertices and side edges of a dodecagonal antiprism with the same D12d, symmetry, order 48. The dodecagrammic antiprism, s and dodecagrammic crossed-antiprism, s also have regular skew dodecagons.
Petrie polygons
The regular icositetragon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes, including:
