Uniform 4-polytope

In geometry, a uniform 4-polytope (or uniform polychoron)[1] is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

Forty-seven non-prismatic convex uniform 4-polytopes, one finite set of convex prismatic forms, and two infinite sets of convex prismatic forms have been described. There are also an unknown number of non-convex star forms.

The 16 regular 4-polytopes, with the property that all cells, faces, edges, and vertices are congruent:

There are 5 fundamental mirror symmetry point group families in 4-dimensions: A4 = CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, B4 = CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, D4 = CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, F4 = CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, H4 = CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png.[7] There are also 3 prismatic groups A3A1 = CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png, B3A1 = CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png, H3A1 = CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png, and duoprismatic groups: I2(p)×I2(q) = CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.png. Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes.

Each reflective uniform 4-polytope can be constructed in one or more reflective point group in 4 dimensions by a Wythoff construction, represented by rings around permutations of nodes in a Coxeter diagram. Mirror hyperplanes can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form [a,b,a], have an extended symmetry, [[a,b,a]], doubling the symmetry order. This includes [3,3,3], [3,4,3], and [p,2,p]. Uniform polytopes in these group with symmetric rings contain this extended symmetry.

If all mirrors of a given color are unringed (inactive) in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed (active), an alternation operation can generate a new 4-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is .

There are 64 convex uniform 4-polytopes, including the 6 regular convex 4-polytopes, and excluding the infinite sets of the duoprisms and the antiprismatic prisms.

These 64 uniform 4-polytopes are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets.

In addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms:

The 5-cell has diploid pentachoric [3,3,3] symmetry,[7] of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way.

Facets (cells) are given, grouped in their Coxeter diagram locations by removing specified nodes.

The three uniform 4-polytopes forms marked with an asterisk, *, have the higher extended pentachoric symmetry, of order 240, [[3,3,3]] because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual. There is one small index subgroup [3,3,3]+, order 60, or its doubling [[3,3,3]]+, order 120, defining an omnisnub 5-cell which is listed for completeness, but is not uniform.

This family has diploid hexadecachoric symmetry,[7] [4,3,3], of order 24×16=384: 4!=24 permutations of the four axes, 24=16 for reflection in each axis. There are 3 small index subgroups, with the first two generate uniform 4-polytopes which are also repeated in other families, [1+,4,3,3], [4,(3,3)+], and [4,3,3]+, all order 192.

The snub 24-cell is repeat to this family for completeness. It is an alternation of the cantitruncated 16-cell or truncated 24-cell, with the half symmetry group [(3,3)+,4]. The truncated octahedral cells become icosahedra. The cubes becomes tetrahedra, and 96 new tetrahedra are created in the gaps from the removed vertices.

This family has diploid icositetrachoric symmetry,[7] [3,4,3], of order 24×48=1152: the 48 symmetries of the octahedron for each of the 24 cells. There are 3 small index subgroups, with the first two isomorphic pairs generating uniform 4-polytopes which are also repeated in other families, [3+,4,3], [3,4,3+], and [3,4,3]+, all order 576.

Like the 5-cell, the 24-cell is self-dual, and so the following three forms have twice as many symmetries, bringing their total to 2304 (extended icositetrachoric symmetry [[3,4,3]]).

This family has diploid hexacosichoric symmetry,[7] [5,3,3], of order 120×120=24×600=14400: 120 for each of the 120 dodecahedra, or 24 for each of the 600 tetrahedra. There is one small index subgroups [5,3,3]+, all order 7200.

This demitesseract family, [31,1,1], introduces no new uniform 4-polytopes, but it is worthy to repeat these alternative constructions. This family has order 12×16=192: 4!/2=12 permutations of the four axes, half as alternated, 24=16 for reflection in each axis. There is one small index subgroups that generating uniform 4-polytopes, [31,1,1]+, order 96.

When the 3 bifurcated branch nodes are identically ringed, the symmetry can be increased by 6, as [3[31,1,1]] = [3,4,3], and thus these polytopes are repeated from the 24-cell family.

Here again the snub 24-cell, with the symmetry group [31,1,1]+ this time, represents an alternated truncation of the truncated 24-cell creating 96 new tetrahedra at the position of the deleted vertices. In contrast to its appearance within former groups as partly snubbed 4-polytope, only within this symmetry group it has the full analogy to the Kepler snubs, i.e. the snub cube and the snub dodecahedron.

There is one non-Wythoffian uniform convex 4-polytope, known as the grand antiprism, consisting of 20 pentagonal antiprisms forming two perpendicular rings joined by 300 tetrahedra. It is loosely analogous to the three-dimensional antiprisms, which consist of two parallel polygons joined by a band of triangles. Unlike them, however, the grand antiprism is not a member of an infinite family of uniform polytopes.

Its symmetry is the ionic diminished Coxeter group, [[10,2+,10]], order 400.

A prismatic polytope is a Cartesian product of two polytopes of lower dimension; familiar examples are the 3-dimensional prisms, which are products of a polygon and a line segment. The prismatic uniform 4-polytopes consist of two infinite families:

The most obvious family of prismatic 4-polytopes is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a 4-polytopes are two identical uniform polyhedra lying in parallel hyperplanes (the base cells) and a layer of prisms joining them (the lateral cells). This family includes prisms for the 75 nonprismatic uniform polyhedra (of which 18 are convex; one of these, the cube-prism, is listed above as the tesseract).[citation needed]

There are 18 convex polyhedral prisms created from 5 Platonic solids and 13 Archimedean solids as well as for the infinite families of three-dimensional prisms and antiprisms.[citation needed] The symmetry number of a polyhedral prism is twice that of the base polyhedron.

This prismatic tetrahedral symmetry is [3,3,2], order 48. There are two index 2 subgroups, [(3,3)+,2] and [3,3,2]+, but the second doesn't generate a uniform 4-polytope.

This prismatic octahedral family symmetry is [4,3,2], order 96. There are 6 subgroups of index 2, order 48 that are expressed in alternated 4-polytopes below. Symmetries are [(4,3)+,2], [1+,4,3,2], [4,3,2+], [4,3+,2], [4,(3,2)+], and [4,3,2]+.

This prismatic icosahedral symmetry is [5,3,2], order 240. There are two index 2 subgroups, [(5,3)+,2] and [5,3,2]+, but the second doesn't generate a uniform polychoron.

The second is the infinite family of uniform duoprisms, products of two regular polygons. A duoprism's Coxeter-Dynkin diagram is CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel q.pngCDel node.png. Its vertex figure is a disphenoid tetrahedron, Pq-duoprism verf.png.

This family overlaps with the first: when one of the two "factor" polygons is a square, the product is equivalent to a hyperprism whose base is a three-dimensional prism. The symmetry number of a duoprism whose factors are a p-gon and a q-gon (a "p,q-duoprism") is 4pq if pq; if the factors are both p-gons, the symmetry number is 8p2. The tesseract can also be considered a 4,4-duoprism.

There is no uniform analogue in four dimensions to the infinite family of three-dimensional antiprisms.

The infinite sets of uniform antiprismatic prisms are constructed from two parallel uniform antiprisms): (p≥2) - CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png - 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.

Other alternations, such as CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png, as an alternation from the omnitruncated tesseract CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png, 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.[18]

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.[19] 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 pramid has two vertex configurations: 3.3.4 around the base, and at the apex.

The nets and vertex figures of the two convex cases are given below, along with a list of cells around each vertex.

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.

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. Alternations are grouped by their chiral symmetry. All alternations are given, although the snub 24-cell, with its 3 family of constructions is the only one that is uniform. Counts in parenthesis 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.