Symmetry of uniform 4-polytopes in four dimensions There are 5 fundamental mirror symmetry
point group families in 4-dimensions:
A4 = ,
B4 = ,
D4 = ,
F4 = ,
H4 = .
Polygonal prismatic prisms: [p] × [ ] × [ ] The infinite set of uniform prismatic prisms overlaps with the 4-p duoprisms: (p≥3) - -
p cubes and 4
p-gonal prisms - (All are the same as
4-p duoprism) The second polytope in the series is a lower symmetry of the regular
tesseract, {4}×{4}.
Polygonal antiprismatic prisms: [p] × [ ] × [ ] The infinite sets of
uniform antiprismatic prisms are constructed from two parallel uniform
antiprisms): (p≥2) - - 2
p-gonal antiprisms, connected by 2
p-gonal prisms and
2p triangular prisms. A
p-gonal antiprismatic prism has
4p triangle,
4p square and
4 p-gon faces. It has
10p edges, and
4p vertices.
Nonuniform alternations , , an
alternation removes half the vertices, in two chiral sets of vertices from the ringed form , however the
uniform solution requires the vertex positions be adjusted for equal lengths. In four dimensions, this adjustment is only possible for 2 alternated figures, while the rest only exist as nonequilateral alternated figures. Coxeter showed only two uniform solutions for rank 4 Coxeter groups with all rings
alternated (shown with empty circle nodes). The first is , s{21,1,1} which represented an index 24 subgroup (
symmetry [2,2,2]+, order 8) form of the
demitesseract, , h{4,3,3} (symmetry [1+,4,3,3] = [31,1,1], order 192). The second is , s{31,1,1}, which is an index 6 subgroup (symmetry [31,1,1]+, order 96) form of the
snub 24-cell, , s{3,4,3}, (symmetry [3+,4,3], order 576). Other alternations, such as , as an alternation from the
omnitruncated tesseract , can not be made uniform as solving for equal edge lengths are in general
overdetermined (there are six equations but only four variables). Such nonuniform alternated figures can be constructed as
vertex-transitive 4-polytopes by the removal of one of two half sets of the vertices of the full ringed figure, but will have unequal edge lengths. Just like uniform alternations, they will have half of the symmetry of uniform figure, like [4,3,3]+, order 192, is the symmetry of the
alternated omnitruncated tesseract. Wythoff constructions with alternations produce
vertex-transitive figures that can be made equilateral, but not uniform because the alternated gaps (around the removed vertices) create cells that are not regular or semiregular. A proposed name for such figures is
scaliform polytopes. This category allows a subset of
Johnson solids as cells, for example
triangular cupola. Each
vertex configuration within a Johnson solid must exist within the vertex figure. For example, a square pyramid has two vertex configurations: 3.3.4 around the base, and 3.3.3.3 at the apex. The nets and vertex figures of the four convex equilateral cases are given below, along with a list of cells around each vertex.
Geometric derivations for 46 nonprismatic Wythoffian uniform polychora The 46 Wythoffian 4-polytopes include the six
convex regular 4-polytopes. The other forty can be derived from the regular polychora by geometric operations which preserve most or all of their
symmetries, and therefore may be classified by the
symmetry groups that they have in common. The geometric operations that derive the 40 uniform 4-polytopes from the regular 4-polytopes are
truncating operations. A 4-polytope may be truncated at the vertices, edges or faces, leading to addition of cells corresponding to those elements, as shown in the columns of the tables below. The
Coxeter-Dynkin diagram shows the four mirrors of the Wythoffian kaleidoscope as nodes, and the edges between the nodes are labeled by an integer showing the angle between the mirrors (
π/
n radians or 180/
n degrees). Circled nodes show which mirrors are active for each form; a mirror is active with respect to a vertex that does not lie on it. See also
convex uniform honeycombs, some of which illustrate these operations as applied to the regular
cubic honeycomb. If two polytopes are
duals of each other (such as the tesseract and 16-cell, or the 120-cell and 600-cell), then
bitruncating,
runcinating or
omnitruncating either produces the same figure as the same operation to the other. Thus where only the participle appears in the table it should be understood to apply to either parent.
Summary of constructions by extended symmetry The 46 uniform polychora constructed from the A4, B4, F4, H4 symmetry are given in this table by their full extended symmetry and Coxeter diagrams. The D4 symmetry is also included, though it only creates duplicates. Alternations are grouped by their chiral symmetry. All alternations are given, although the
snub 24-cell, with its 3 constructions from different families is the only one that is uniform. Counts in parentheses are either repeats or nonuniform. The Coxeter diagrams are given with subscript indices 1 through 46. The 3-3 and 4-4 duoprismatic family is included, the second for its relation to the B4 family. == Uniform star polychora ==