Uniform 4-polytope


In geometry, a uniform 4-polytope 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.

History of discovery

Regular 4-polytopes are a subset of the uniform 4-polytopes, which satisfy additional requirements. Regular 4-polytopes can be expressed with Schläfli symbol have cells of type, faces of type, edge figures, and vertex figures.
The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra which becomes cells, and which becomes the vertex figure.
Existence as a finite 4-polytope is dependent upon an inequality:
The 16 regular 4-polytopes, with the property that all cells, faces, edges, and vertices are congruent:

Symmetry of uniform 4-polytopes in four dimensions

The 16 mirrors of B4 can be decomposed into 2 orthogonal groups, 4A1 and D4:
  1. =
  2. =
The 24 mirrors of F4 can be decomposed into 2 orthogonal D4 groups:
  • =
  • =
    • The 10 mirrors of B3×A1 can be decomposed into orthogonal groups, 4A1 and D3:
  • =
  • =
    • There are 5 fundamental mirror symmetry point group families in 4-dimensions: A4 =, B4 =, D4 =, F4 =, H4 =. There are also 3 prismatic groups A3A1 =, B3A1 =, H3A1 =, and duoprismatic groups: I2×I2 =. 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 , have an extended symmetry, a,b,a, doubling the symmetry order. This includes , , and . Uniform polytopes in these group with symmetric rings contain this extended symmetry.
    If all mirrors of a given color are unringed 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, an alternation operation can generate a new 4-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is [|not generally adjustable to create uniform solutions].

    Enumeration

    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 Pentachoric symmetry|diploid pentachoric symmetry, of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way.
    Facets 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, 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 +, order 60, or its doubling +, order 120, defining an omnisnub 5-cell which is listed for completeness, but is not uniform.

    The B4 family

    This family has diploid hexadecachoric symmetry, , 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, , , and +, all order 192.

    Tesseract truncations

    16-cell truncations

    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 . The truncated octahedral cells become icosahedra. The cubes becomes tetrahedra, and 96 new tetrahedra are created in the gaps from the removed vertices.

    The F4 family

    This family has diploid icositetrachoric symmetry, , 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, , , and +, 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.

    The H4 family

    This family has diploid hexacosichoric symmetry, , 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 +, all order 7200.

    120-cell truncations

    600-cell truncations

    The D4 family

    This demitesseract family, , 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, +, order 96.
    When the 3 bifurcated branch nodes are identically ringed, the symmetry can be increased by 6, as , and thus these polytopes are repeated from the 24-cell family.
    Here again the snub 24-cell, with the symmetry group + 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.

    The grand antiprism

    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.

    Prismatic uniform 4-polytopes

    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 and a layer of prisms joining them. This family includes prisms for the 75 nonprismatic uniform polyhedra.
    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. The symmetry number of a polyhedral prism is twice that of the base polyhedron.

    Tetrahedral prisms: A3 × A1

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

    Octahedral prisms: B3 × A1

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

    Icosahedral prisms: H3 × A1

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

    Duoprisms: p × q

    The second is the infinite family of uniform duoprisms, products of two regular polygons. A duoprism's Coxeter-Dynkin diagram is. Its vertex figure is a disphenoid tetrahedron,.
    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 is 4pq if p≠q; if the factors are both p-gons, the symmetry number is 8p2. The tesseract can also be considered a 4,4-duoprism.
    The elements of a p,q-duoprism are:
    There is no uniform analogue in four dimensions to the infinite family of three-dimensional antiprisms.
    Infinite set of p-q duoprism - - p q-gonal prisms, q p-gonal prisms:
    NameCoxeter graphCellsImagesNet
    3-3 duoprism3+3 triangular prisms
    3-4 duoprism3 cubes
    4 triangular prisms
    4-4 duoprism
    		4+4 cubes
    3-5 duoprism3 pentagonal prisms
    5 triangular prisms
    4-5 duoprism4 pentagonal prisms
    5 cubes
    5-5 duoprism5+5 pentagonal prisms
    3-6 duoprism3 hexagonal prisms
    6 triangular prisms
    4-6 duoprism4 hexagonal prisms
    6 cubes
    5-6 duoprism5 hexagonal prisms
    6 pentagonal prisms
    6-6 duoprism6+6 hexagonal prisms


    3-3
    3-4
    3-5
    3-6
    3-7
    3-8

    4-3
    4-4
    4-5
    4-6
    4-7
    4-8

    5-3
    5-4
    5-5
    5-6
    5-7
    5-8

    6-3
    6-4
    6-5
    6-6
    6-7
    6-8

    7-3
    7-4
    7-5
    7-6
    7-7
    7-8

    8-3
    8-4
    8-5
    8-6
    8-7
    8-8

    Polygonal prismatic prisms: p × ×

    The infinite set of uniform prismatic prisms overlaps with the 4-p duoprisms: - - p cubes and 4 p-gonal prisms - The second polytope in the series is a lower symmetry of the regular tesseract, ×.

    Polygonal antiprismatic prisms: p × ×

    The infinite sets of uniform antiprismatic prisms are constructed from two parallel uniform antiprisms): - - 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

    Coxeter showed only two uniform solutions for rank 4 Coxeter groups with all rings alternated. The first is, s which represented an index 24 subgroup form of the demitesseract,, h. The second is, s, which is an index 6 subgroup form of the snub 24-cell,, s,.
    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. 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 +, 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 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 pramid 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 two convex cases are given below, along with a list of cells around each vertex.
    Coxeter
    diagram    		s3, s3,
    Relation24 of 48 vertices of
    rhombicuboctahedral prism    		288 of 576 vertices of
    runcitruncated 24-cell
    Net
    runcic snub cubic hosochoron
    runcic snub 24-cell
    Cells
    Vertex
    figure
    3.4.3.4: triangular cupola
    3.4.6: triangular cupola
    3.3.3: tetrahedron
    3.6.6: truncated tetrahedron
    3.4.3.4: triangular cupola
    3.4.6: triangular cupola
    3.4.4: triangular prism
    3.6.6: truncated tetrahedron
    3.3.3.3.3: icosahedron

    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.

    Summary chart of truncation operations
    Example locations of kaleidoscopic generator point on fundamental domain.

    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. 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.
    OperationSchläfli symbolSymmetryCoxeter diagramDescription
    Parentt0Original regular form
    Rectificationt1Truncation operation applied until the original edges are degenerated into points.
    Birectification
    		t2Face are fully truncated to points. Same as rectified dual.
    Trirectification
    		t3Cells are truncated to points. Regular dual
    Truncationt0,1Each vertex is cut off so that the middle of each original edge remains. Where the vertex was, there appears a new cell, the parent's vertex figure. Each original cell is likewise truncated.
    Bitruncationt1,2A truncation between a rectified form and the dual rectified form.
    Tritruncationt2,3Truncated dual.
    Cantellationt0,2A truncation applied to edges and vertices and defines a progression between the regular and dual rectified form.
    Bicantellationt1,3Cantellated dual.
    Runcination
    		t0,3A truncation applied to the cells, faces and edges; defines a progression between a regular form and the dual.
    Cantitruncationt0,1,2Both the cantellation and truncation operations applied together.
    Bicantitruncationt1,2,3Cantitruncated dual.
    Runcitruncationt0,1,3Both the runcination and truncation operations applied together.
    Runcicantellationt0,1,3Runcitruncated dual.
    Omnitruncation
    		t0,1,2,3Application of all three operators.
    Halfh
    =    		Alternation of, same as
    Cantich2
    =    		Same as
    Runcich3
    =    		Same as
    Runcicantich2,3
    =    		Same as
    QuarterqSame as
    SnubsAlternated truncation
    Cantic snubs2Cantellated alternated truncation
    Runcic snubs3Runcinated alternated truncation
    Runcicantic snubs2,3Runcicantellated alternated truncation
    Snub rectifiedsrAlternated truncated rectification
    ht0,3Alternated runcination
    Bisnub2sAlternated bitruncation
    Omnisnubht0,1,2,3+Alternated omnitruncation

    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, 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. 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.