Uniform 4-polytope

A uniform 4-polytope, also called a uniform polychoron, is a four-dimensional polytope that is vertex-transitive—meaning all vertices are equivalent under the polytope's symmetry group—and whose cells are uniform polyhedra.^[1] These figures are the four-dimensional counterparts to the uniform polyhedra in three dimensions, where the uniform polyhedra serve as the bounding cells of the uniform polychora.^[1] There are 64 convex uniform 4-polytopes, which include the 6 regular convex 4-polytopes (the analogues of the Platonic solids) and exclude the infinite families of uniform duoprisms and antiprismatic prisms.^[2] Of these, 47 are non-prismatic, while the remaining 17 are prisms constructed over uniform polyhedra.^[3] The study of uniform 4-polytopes builds on earlier work in lower dimensions, with the regular convex 4-polytopes first enumerated by Ludwig Schläfli in the 19th century and detailed by H.S.M. Coxeter in the 20th.^[4] The full enumeration of convex uniform 4-polytopes was completed in 1965 by John Horton Conway and Michael Guy through computational methods, confirming the list of 47 non-prismatic forms; this enumeration was later proven complete by Marco Möller in 2004 via an exhaustive search of possible vertex figures.^[4] Non-convex uniform 4-polytopes, which allow star polyhedra as cells, number in the thousands, with 2191 finite non-prismatic examples enumerated excluding infinite families, though ongoing research continues to refine these counts.^[5] Uniform 4-polytopes are classified by their underlying Coxeter symmetry groups, primarily the irreducible groups _A_₄, _B_₄, _D_₄, _F_₄, and _H_₄, which generate most convex examples through Wythoff constructions—systematic operations like rectification, truncation, and cantellation applied to the regular 4-polytopes.^[2] One notable exception is the grand antiprism, a chiral uniform 4-polytope discovered by Conway and Guy that lacks a standard Wythoff symbol.^[4] These polytopes exhibit rich geometric properties, such as uniform edge lengths and regular polygonal faces, and play a key role in higher-dimensional geometry, tessellations, and symmetry studies.^[1]

Introduction

Definition and properties

A uniform 4-polytope, also known as a uniform polychoron, is a vertex-transitive four-dimensional polytope whose cells are uniform polyhedra and whose faces are regular polygons, with all edges of equal length.^[1] This vertex-transitivity implies isogonal symmetry, ensuring that the arrangement of elements around each vertex is identical.^[1] The vertex figure of a uniform 4-polytope is a uniform 3-polytope, providing a local description of the structure at each vertex.^[1] Uniform 4-polytopes satisfy the Euler characteristic for 4-dimensional polytopes, given by \chi = V - E + F - C = 0, where V is the number of vertices, E the number of edges, F the number of faces, and C the number of cells; this holds for convex examples and extends topologically to non-convex uniform ones homeomorphic to 3-spheres. The number of vertices V can be computed as V = |G| / |S_v|, where |G| is the order of the full symmetry group acting transitively on the vertices, and |S_v| is the order of the stabilizer subgroup at a vertex. Schläfli symbols of the form \{p, q, r\} provide a compact notation for uniform 4-polytopes, indicating the type of regular face \{p\}, the uniform cell \{p, q\}, and the vertex figure \{q, r\}. For instance, the tesseract, denoted \{4, 3, 3\}, is a uniform 4-polytope with 8 cubic cells. Key properties include density, which for star (non-convex) uniform 4-polytopes quantifies the winding number of their facets around the center, and isogonal conjugates, which are dual-like uniform 4-polytopes obtained by interchanging the roles of cells and vertex figures under the same symmetry group.^[1]

Classification criteria

Uniform 4-polytopes are classified as vertex-transitive figures whose cells are uniform polyhedra, ensuring all vertices are equivalent under the symmetry group and all faces are regular polygons.^[6] This criterion extends the definition of uniformity from lower dimensions, where the polytope's edge lengths are equal, and its facets meet uniformly at each vertex. They are further subdivided into convex forms (with topological density 1 and no self-intersections), prismatic families (arising from products of lower-dimensional uniforms with prisms or antiprisms), and star (nonconvex) forms (with density greater than 1, featuring intersecting cells).^[4] The Wythoff-Klein construction provides a systematic method to generate these uniforms from regular "seed" polytopes using the mirrors of Coxeter reflection groups, represented via decorated Coxeter-Dynkin diagrams. In this approach, a diagram's nodes correspond to generating reflections, with one node marked by a circle to indicate the initial point's position relative to the mirrors; the convex hull of the orbit under the group yields a uniform polytope. Operations modify the diagram to produce variants: rectification (denoted r{}) places circles on all nodes except one crossed node, expanding edges to midpoints and yielding quasiregular forms; truncation (t{}) crosses all nodes and circles one, cutting vertices to points where original edges met; further operations like bitruncation (t²{}), cantellation (rr{}), and runcination (t₀,₃{}) apply successive modifications, generating families from the same symmetry. This construction accounts for most reflective uniforms but excludes some non-Wythoffian cases like the grand antiprism.^[7] Classification also proceeds by symmetry groups, grouping non-prismatic uniforms under irreducible finite reflection groups in four dimensions: A₄ (simplicial), B₄ (hypercubic), D₄ (demihypercubic), F₄ (24-cell), and H₄ (icosahedral), each yielding distinct families through Wythoffian operations. Prismatic uniforms form infinite families via direct products of uniform polyhedra with uniform prisms or antiprisms, or more complex duoprismatic constructions.^[6] The enumeration is complete for convex uniforms, with 47 finite non-prismatic and 17 prismatic examples, totaling 64 finite convex uniform 4-polytopes plus infinite prismatic families; among these, 6 are regular convex polytopes. For nonconvex forms, 10 regular star polytopes are known, and uniform star polychora number at least 2,191 finite non-prismatic examples as of November 2025 based on computational enumerations, though the full count remains incomplete, with ongoing discoveries expanding the catalog. Earlier classifications, such as those by Coxeter, provided foundational counts but have been superseded by modern efforts revealing significantly more nonconvex uniforms; community-maintained databases report the current totals for strict vertex-transitive uniforms with uniform cells.^[8]^[4]^[9]

Historical development

Early discoveries

The exploration of uniform 4-polytopes originated in the mid-19th century with Swiss mathematician Ludwig Schläfli's foundational work on higher-dimensional geometry. In his 1852 manuscript Theorie der vielfachen Kontinuität, published posthumously in 1901, Schläfli introduced the Schläfli symbol notation {p,q,r} for describing regular polytopes and systematically enumerated the six convex regular 4-polytopes: the pentachoron {3,3,3}, tesseract {4,3,3}, hexadecachoron {3,3,4}, 24-cell {3,4,3}, 120-cell {5,3,3}, and 600-cell {3,3,5}.^[10] He named the {3,3,3} the pentachoron (from Greek, meaning five solid) and the {5,3,3} the hecatonicosachoron (120 solid), emphasizing their cell counts and regularity.^[11] This classification extended the Platonic solids to four dimensions, proving no additional convex regulars exist beyond these. In the late 19th century, German mathematician Victor Schlegel advanced visualization techniques for these abstract objects. In 1883, Schlegel created the first physical projection models of the six regular 4-polytopes using brass wire for edges and silk thread for connections, adapting his Schlegel diagram method—originally for 3D polyhedra—to project 4D structures onto 3D space by positioning one cell at the "infinity" point.^[12]^[13] These models provided early tangible insights into 4D symmetry, influencing subsequent geometric studies.^[12] The early 20th century brought intuitive and enumerative contributions from several figures. Self-taught geometer Alicia Boole Stott, starting around 1900, independently rediscovered the six convex regular 4-polytopes and crafted over 100 cardboard models of their 3D sections and unfoldings, offering an accessible, non-analytical approach to their structure.^[14]^[13] Her work, detailed in collaborations like a 1910 paper with mathematician Pieter Hendrik Schoute, emphasized vertex figures and provided proofs of existence through physical intuition.^[14] Concurrently, H.S.M. Coxeter's 1920s research on symmetry groups and higher-dimensional figures, including early papers on prismatic polytopes, reinforced Schläfli's findings and explored extensions to uniform forms.^[15] D.M.Y. Sommerville contributed to the study of regular polytopes in higher dimensions through his 1923 book An Introduction to the Geometry of n-Dimensions.^[16] David Hilbert's 18th problem, presented in 1900, indirectly spurred interest in uniform polytopes by questioning whether all space-filling polyhedra must relate to regular or symmetric forms, prompting deeper analyses of tiling and symmetry in higher dimensions.^[17]^[18] These pre-1950 efforts established the conceptual framework for uniform 4-polytopes, paving the way for computational enumerations.

Modern classifications

In the 1950s and 1970s, advancements in the classification of uniform 4-polytopes focused on convex forms and systematic constructions. H.S.M. Coxeter's second edition of Regular Polytopes (1963) enumerated 64 convex uniform 4-polytopes, including 47 non-prismatic and 17 prismatic, building on earlier work by providing a comprehensive list derived from the irreducible reflection groups in four dimensions. This classification excluded infinite prismatic families and emphasized the role of Wythoff constructions, with Coxeter extending Wythoff symbols to four dimensions to generate these polytopes from generator points in the spherical Coxeter diagram. The full enumeration of the 47 non-prismatic convex forms was confirmed computationally in 1965 by John Horton Conway and Michael Guy, and proven complete in 2004 by Marco Möller via an exhaustive search of possible vertex figures.^[4] From the 1980s through the 2000s, efforts expanded to nonconvex uniform 4-polytopes, driven by computational and enumerative approaches. Jonathan Bowers developed the Polychoron Catalog in the 1990s, documenting 2191 uniform polychora (as of September 2025), including over 1800 star forms alongside infinite prismatic and duoprismatic families; he introduced mnemonic acronyms for clarity, such as "tess" for the tesseract and "quit" for the quaternionic tesseract.^[19] Bowers' work, hosted on his dedicated website, facilitated visualization and symmetry analysis, revealing complex star forms like the stellated 120-cell that eluded earlier manual enumerations. In the 2010s to 2025, classifications have grown dramatically through collaborative online resources and computational tools, incorporating nonconvex, semi-uniform, and fissary forms. The Polytope Wiki, maintained by a community of geometers, has cataloged 2191 finite uniform polychora as of 2025, excluding infinite families, by integrating Bowers' listings with subsequent discoveries of star variants.^[9] Computational verifications of these enumerations rely on software like GAP for analyzing symmetry groups, confirming vertex-transitivity and cell uniformity across high-order Coxeter groups such as those in the H4 family. Recent identifications include new star families like sphenoverts, characterized by wedge-shaped vertex figures, which expand the nonconvex repertoire beyond traditional reflective constructions.^[19] These updates address gaps in earlier nonconvex enumerations, with ongoing work on density computations for star polytopes appearing in specialized geometric analyses up to 2025.^[20]

Regular 4-polytopes

The six convex regulars

The six convex regular 4-polytopes are the complete set of bounded, convex figures in four-dimensional Euclidean space where all cells, faces, edges, and vertices are congruent regular polyhedra or polygons, meeting in the same configuration at each element. These polychora, enumerated by Ludwig Schläfli in the mid-19th century and later systematized by H.S.M. Coxeter, include the pentachoron, tesseract, hexadecachoron, 24-cell, 120-cell, and 600-cell. Each is denoted by a Schläfli symbol {p, q, r}, indicating regular p-gonal faces {p}, q such faces meeting at each edge, and regular {q, r} vertex figures.^[21] Their cell compositions are as follows: the pentachoron {3,3,3} has 5 tetrahedral cells; the tesseract {4,3,3} has 8 cubic cells; the hexadecachoron {3,3,4} has 16 tetrahedral cells; the 24-cell {3,4,3} has 24 octahedral cells; the 120-cell {5,3,3} has 120 dodecahedral cells; and the 600-cell {3,3,5} has 600 tetrahedral cells.^[21] Key structural properties include dual pairings among them: the tesseract is dual to the hexadecachoron, and the 120-cell is dual to the 600-cell, while the pentachoron and 24-cell are self-dual. Assuming unit edge length, their element counts (vertices V, edges E, faces F, cells C) are summarized below, illustrating the scale from the simplicial pentachoron to the highly faceted 120-cell. These counts satisfy the Euler characteristic χ = V - E + F - C = 0 for convex 4-polytopes topologically equivalent to 3-spheres.^[21]

Polychoron	Schläfli Symbol	V	E	F	C
Pentachoron	{3,3,3}	5	10	10	5 tetrahedra
Tesseract	{4,3,3}	16	32	24	8 cubes
Hexadecachoron	{3,3,4}	8	24	32	16 tetrahedra
24-cell	{3,4,3}	24	96	96	24 octahedra
120-cell	{5,3,3}	600	1200	720	120 dodecahedra
600-cell	{3,3,5}	120	720	1200	600 tetrahedra

Visualizations of these polychora often employ Schlegel diagrams, which project the structure into three dimensions by positioning one cell externally and the remaining cells within its boundary, or net projections that unfold cells onto a plane; each exhibits unique symmetries, such as the tesseract's cubic layering or the 600-cell's icosahedral vertex figures.^[21]

Nonconvex regular polychora

In four-dimensional geometry, there are ten nonconvex regular polychora known as the Schläfli–Hess polychora, which serve as the direct analogs to the Kepler-Poinsot polyhedra in three dimensions. Four of these are stellations related to the 120-cell: the small stellated 120-cell with Schläfli symbol {5/2, 5, 3}, the great 120-cell {5, 5/2, 5}, the grand 120-cell {5, 3, 5/2}, and the grand stellated 120-cell {5/2, 3, 5}. The remaining six are related to the 600-cell, such as the grand 600-cell {3, 5, 5/2} and the great 600-cell {3, 5/2, 5}. Each of the 120-cell-based examples features 120 cells derived from dodecahedral forms: the small stellated and grand stellated 120-cells have small stellated dodecahedral and great stellated dodecahedral cells, respectively, while the great 120-cell uses great dodecahedral cells and the grand 120-cell employs regular dodecahedral cells. The fractional entries in their Schläfli symbols indicate the incorporation of star polygons, such as pentagrams {5/2}, leading to self-intersecting cells and faces.^[22] These polychora were identified by H. S. M. Coxeter during his work from 1938 through the 1950s, where he classified them as uniform density polytopes within the framework of regular figures. Unlike their convex counterparts, their cells intersect in complex ways, with multiple layers of winding that contribute to higher topological densities: 4 for the small stellated 120-cell, 6 for the great 120-cell, 20 for the grand 120-cell, and 66 for the grand stellated 120-cell. This density measures the number of times the figure winds around its center, resulting in non-orientable or multiply-covered interiors. They all share the H<sub>4</sub> symmetry group of order 14400 with the convex 120-cell and 600-cell.^[23] To account for their non-simply connected structure, metrics such as the Euler characteristic are adjusted by the density factor. For instance, the small stellated 120-cell has 120 vertices, 1200 edges, 720 pentagrammic faces, and 120 cells, yielding an apparent Euler characteristic χ = V - E + F - C = -480; this apparent value reflects the self-intersecting and winding nature with density 4, but the polytope is topologically equivalent to a 3-sphere with effective χ = 0. Similar considerations, accounting for their respective densities, apply to the others, where the effective topology is that of a 3-sphere.^[24]

Convex uniform 4-polytopes

Symmetry groups

The symmetry groups of convex uniform 4-polytopes are finite Coxeter groups acting as reflection groups in 4-dimensional Euclidean space, preserving the uniform vertex figures and facet types. These groups generate the isometries that map the polytope to itself, with uniform polytopes arising as orbits under subgroups or via constructions involving the group's mirrors. The irreducible cases correspond to the five exceptional spherical Coxeter groups of rank 4, each realized geometrically through specific regular 4-polytopes. The group A_4, denoted by the Coxeter diagram [3,3,3], has order 120 and underlies the symmetries of the 5-cell (pentachoron), where it acts as the full symmetry group of the regular simplex in 4D. This group is associated with tetrahedral symmetry in its lower-dimensional projections. The B_4 group, with diagram [3,4,3], has order 384 and is the hyperoctahedral group, realized in the tesseract (hypercube) and 16-cell (cross-polytope), encompassing all signed permutations of coordinates. Its order is given by the formula |B_4| = 2^4 \cdot 4! = 384, generalizing the B_n series as $2^n n!. The D_4 group, a demihypercubic symmetry with diagram featuring a branched structure [3^{1,1,1}], has order 192 and appears in polytopes like the 24-cell derivatives, excluding certain sign changes present in B_4. The F_4 group, with diagram [3,4,3] but a distinct branch (one bond of order 4), has order 1152 and is realized in the 24-cell, serving as its full symmetry group with 24 octahedral cells. Finally, the H_4 group, denoted [3,3,5], has order 14,400 and governs the icosahedral symmetries in 4D, realized through the 120-cell (dodecahedral honeycomb cell) and 600-cell (tetrahedral honeycomb cell), with 120 and 600 vertices respectively as group orbits. Prismatic uniform 4-polytopes employ direct product symmetry groups combining lower-rank Coxeter groups, such as A_3 \times A_1 for tetrahedral prisms, where A_3 (order 24) is the tetrahedral group [3,3] and A_1 (order 2) is a reflection, yielding a total order of 48; similar products like H_3 \times A_1 (order 240) apply to icosahedral prisms. Uniform 4-polytopes within these groups are constructed via the Wythoff procedure, selecting subsets of the full reflection mirrors to generate vertex-intransitive subgroups whose orbits produce the uniform figures, ensuring vertex-transitivity under the overall group action.^[25]^[26]^[27]^[28]

Enumeration and Wythoff construction

The non-prismatic convex uniform 4-polytopes total 47 in number, comprising 6 regulars, 16 truncated forms, 10 rectified forms, 10 bitruncated forms, 5 cantellated forms, 5 cantitruncated forms, and 8 others derived from additional Wythoffian operations.^[29] This enumeration, completed by John Conway and others in the late 20th century, excludes infinite prismatic families such as duoprisms and focuses on those generated by the finite irreducible Coxeter groups in four dimensions.^[8] Jonathan Bowers provided a comprehensive catalog with acronyms for identification, such as "pen" for the 5-cell and "tat" for the truncated 5-cell, facilitating study and visualization.^[29] The Wythoff construction generates these uniform 4-polytopes from the symmetry groups defined by Coxeter-Dynkin diagrams, extending the method originally developed for polyhedra.^[30] The process begins by identifying the fundamental domain of the reflection group—a 4-simplex for groups like A_4 or a union of two adjacent simplices for others. A generic point is chosen within this domain, positioned according to the Wythoff symbol, which specifies the relative distances to the reflecting hyperplanes (mirrors) via ring placements on the diagram nodes (e.g., a ring on the first node indicates proximity to that mirror). The vertices of the polytope are then the orbit of this point under the full group action, ensuring vertex-transitivity. The edges, faces, and cells emerge as convex hulls of subsets of these vertices stabilized by parabolic subgroups, with all cells being uniform 3-polytopes. Projections onto the Coxeter plane—a 2D plane invariant under the group where mirrors project to lines forming a kaleidoscopic tiling—aid in visualizing and verifying the construction, as the vertex density in the projection corresponds to the polytope's vertex figure.^[30] This method systematically produces all reflective uniform 4-polytopes from the five relevant groups (A_4, B_4, D_4, F_4, H_4), with some polytopes arising from multiple symbols due to symmetry overlaps.^[4] Representative examples illustrate the construction across operation types, with Wythoff symbols in standard notation (e.g., t{3,3,3,3} for truncation of the 5-cell under A_4 symmetry). The following table provides an overview, including cell compositions and vertex counts; full details for all 47, including Bowers acronyms, are cataloged in specialized references.^[29]

Operation Type	Example Wythoff Symbol	Bowers Acronym	Cells	Vertices
Regular	{3,3,3,3}	pen	5 tetrahedra	5
Regular	{3,3,3,5}	hex	600 tetrahedra	120
Truncated	t{3,3,3,3}	tat	5 truncated tetrahedra	20
Truncated	t{5,3,3}	tcid	120 truncated dodecahedra, 600 tetrahedra	2400
Rectified	r{3,3,3,3}	rach	10 octahedra	10
Rectified	r{3,3,4,3}	rhoc	96 rectified cuboctahedra	96
Bitruncated	t r{3,3,3,3}	bart	10 truncated octahedra	20
Bitruncated	t r{5,3,3}	baxhi	120 truncated icosahedra, 600 truncated tetrahedra	3600
Cantellated	rr{3,3,3,3}	cac	20 rhombicuboctahedra, 30 cuboctahedra	30
Cantellated	rr{3,3,5}	gach	600 rhombicosidodecahedra, 120 icosidodecahedra	120
Cantitruncated	trr{3,3,3,3}	cit	10 truncated cuboctahedra, 20 truncated tetrahedra	40
Cantitruncated	trr{5,3,3}	githi	600 truncated icosidodecahedra, 120 truncated dodecahedra	600
Other (e.g., quasitruncated)	q{3,3,3,4}	qubidy	48 quasitruncated cuboctahedra, 96 hexagons	192

A4 family

The A4 family encompasses five convex uniform 4-polytopes sharing the rotational tetrahedral symmetry group A₄ of order 12, derived from the regular 5-cell via the Wythoff construction on the Coxeter diagram A₄ (a linear chain of four nodes corresponding to [3,3,3]). These polytopes are the smallest such family among the irreducible symmetry groups in four dimensions, all exhibiting density 1 and vertex figures that are regular tetrahedra. Their cells consist exclusively of tetrahedra or uniform polyhedra obtained by truncating or rectifying the tetrahedron, reflecting the simplicial nature of the underlying 5-cell. Unlike larger families such as B₄, this group yields only these five due to the self-duality of the 5-cell and the constraints of the symmetry.^[31] The regular 5-cell, or pentachoron, with Schläfli symbol {3,3,3}, is the foundational member, comprising 5 tetrahedral cells meeting at each vertex. It has 5 vertices, 10 edges, and 10 triangular faces, embodying the 4-dimensional simplex. Its Schlegel diagram depicts four tetrahedra projecting outward from a central tetrahedron, illustrating the tetrahedral arrangement in lower dimensions. The rectified 5-cell, obtained by truncating vertices until edges disappear (Wythoff symbol 2 3 3 | 3), features 10 vertices where original vertices and cell centers coincide. It bounds 10 cells: 5 regular tetrahedra and 5 regular octahedra, with 30 edges and 40 triangular faces. This structure arises as the convex hull of the 5-cell's vertices and dual cell centers, and its Schlegel diagram shows octahedra interlocked with tetrahedra in a symmetric tetrahedral envelope.^[32] The truncated 5-cell (Wythoff symbol 3 3 4 | 3) results from vertex truncation that reduces original edges to points, yielding 20 vertices and 5 cells of truncated tetrahedra (each with 4 hexagonal and 4 triangular faces). It includes 60 edges and 100 faces (80 triangles and 20 hexagons). The cells meet such that truncated tetrahedra share hexagonal faces corresponding to the original tetrahedra's faces. A Schlegel diagram of this polytope reveals the five truncated tetrahedra arranged around a central void, with 20 vertices forming the outer boundary. The bitruncated 5-cell, or decachoron (Wythoff symbol 4 3 3 | 3), applies truncation to both the 5-cell and its dual, producing 20 vertices and 10 cells of truncated tetrahedra. With 60 edges and 120 faces (80 triangles and 40 hexagons), it is cell-transitive and serves as the dual of the rectified 16-cell. Its Schlegel diagram displays the 10 truncated tetrahedral cells in two interpenetrating sets, highlighting the enhanced complexity from dual truncation.^[33] The runcitruncated 5-cell (also called prismatorhombated pentachoron, Wythoff symbol 3 4 3 | 4) combines runcination (edge expansion to prisms) with truncation and rectification, resulting in 60 vertices, 150 edges, 220 faces, and 30 cells: 5 truncated tetrahedra, 5 cuboctahedra, 10 triangular prisms, and 10 hexagonal prisms. This polytope's structure emphasizes prismatic elements along original edges, and its Schlegel diagram illustrates the layered arrangement of truncated and prismatic cells within the tetrahedral framework.^[34]

Polytope	Wythoff Symbol	Cells	Vertices	Edges	Faces	Density
5-cell	3 3 3	5 tetrahedra	5	10	10	1
Rectified 5-cell	2 3 3	3	5 tet + 5 oct	10	30	40
Truncated 5-cell	3 3 4	3	5 trunc tet	20	60	100
Bitruncated 5-cell	4 3 3	3	10 trunc tet	20	60	120
Runcitruncated 5-cell	3 4 3	4	5 trunc tet + 5 cuboct + 10 tri prisms + 10 hex prisms	60	150	220

This table summarizes key metrics, establishing the progressive increase in complexity from the simplex to the runcitruncated form, with all members maintaining tetrahedral vertex figures.^[25]

B4 family

The B4 family encompasses 15 convex uniform 4-polytopes exhibiting the full cubic symmetry of the B4 group, the hyperoctahedral group of order 384, which acts on the 4-dimensional space with octahedral vertex figures.^[4] These polytopes are generated via Wythoff constructions from the Coxeter diagram [3,4,3], where the branch point in the symbol determines the specific form, and they fill a significant portion of the 64 total finite convex uniform polychora enumerated by Norman Johnson.^[29] The family divides into three subfamilies: the tesseract line with 10 members derived from the regular tesseract {4,3,3}, the dual 16-cell line with 10 members from the regular 16-cell {3,3,4}, and 5 rhombic polytopes arising from expanded or cantellated operations that introduce rhombihecatonicosachora-like structures.^[3] The tesseract line begins with the regular tesseract (Bowers name: tess), featuring 8 cubic cells and 16 vertices (Wythoff symbol: 4 | 3 3 3), and progresses through rectification to the rhombicuboctachoron (r{4,3,3,3} or 2 3 | 3 3), which has 16 cuboctahedral cells and 32 vertices. Further operations yield the truncated tesseract (t{4,3,3,3} or 4 3 | 3 3), composed of 8 truncated cubic cells and 16 octahedral cells with 128 vertices, emphasizing the truncation's effect in expanding edge lengths to the midpoints of original edges while preserving uniformity.^[35] The 16-cell line mirrors this duality, starting with the regular 16-cell (Bowers name: oca), with 16 tetrahedral cells and 8 vertices (Wythoff symbol: 3 | 3 3 4), and includes the rectified form shared with the tesseract line, as rectification of duals coincides. Notable examples include the truncated 16-cell (t{3,3,3,4} or 3 3 | 3 4), featuring 16 truncated tetrahedral cells and 8 cuboctahedral cells with 64 vertices, and the bitruncated 16-cell (t{3,3,4,3} or 3 3 4 | ), with 24 rhombicuboctahedral cells and 96 vertices, highlighting the symmetry extension in higher truncations.^[36] The 5 rhombic polytopes, often called the rhombicuboctachoric line or small rhombated variants, stem from runcination and expansion operations, such as the small rhombated tesseract (sr{4,3,3,3} or 4 3 3 | 2), which incorporates 8 rhombicuboctahedral cells, 24 cubic cells, and 16 tetrahedral cells, totaling 96 vertices and introducing prismatic elements between expanded faces. These forms, including the runcinated tesseract (runc{4,3,3,3} or 2 3 3 3 | ), with 32 cubic cells, 80 triangular prismatic cells, and 16 tetrahedral cells at 512 vertices, exemplify the B4 family's capacity for complex cell assemblages while maintaining vertex-transitivity.^[19]

Polychoron	Wythoff Symbol	Cells	Vertices	Bowers Name
Tesseract	4 \| 3 3 3	8 cubes	16	tess
Rectified tesseract	2 3 \| 3 3	16 cuboctahedra	32	rico
Truncated tesseract	4 3 \| 3 3	8 trunc. cubes + 16 octahedra	128	tit
Runcinated tesseract	2 3 3 3 \|	32 cubes + 80 tri. prisms + 16 tetrahedra	512	run tess
16-cell	3 \| 3 3 4	16 tetrahedra	8	oca
Truncated 16-cell	3 3 \| 3 4	16 trunc. tetrahedra + 8 cuboctahedra	64	tot
Bitruncated 16-cell	3 3 4 \|	24 rhomb. cuboctahedra	96	bitoca
Small rhombated tesseract	4 3 3 \| 2	8 rhomb. cuboctahedra + 24 cubes + 16 tetrahedra	96	sirco

This table highlights representative members, illustrating the progression in cell complexity and vertex multiplicity across the subfamilies; the full 15 include additional cantellated, omnitruncated, and quasitruncated forms like the quasitruncated tesseract (Bowers name: quit), which features truncated cuboctahedra and other Archimedean solids as cells.^[9]

D4 family

The D4 family encompasses 7 uniform 4-polytopes characterized by the demihypercubic symmetry group D_4, a rotational subgroup of the full hypercubic B_4 group with order 192, which arises in constructions linking the tesseract \{4,3,3\} and its dual, the 16-cell \{3,3,4\}. These polytopes are generated via Wythoff constructions under the branched D_4 Coxeter diagram (nodes connected as a central node to three branches of lengths 3,3,4), producing vertex-transitive figures whose cells blend derivatives of cubic and octahedral polyhedra, such as cuboctahedra, truncated cubes, and rectified octahedra. The reduced symmetry compared to B_4 limits the family size while enabling unique alternated variants, such as the 4-demicube obtained by halving the tesseract's vertices, resulting in a structure with 16 tetrahedral cells and 32 edges. Representative members include the small bitruncated icositetrachoron (bitruncated 16-cell, Wythoff symbol | [2 6 4](/page/2-6-4)), featuring 48 vertices, 96 truncated tetrahedral cells, and 24 truncated octahedral cells, exemplifying the mixed cubic-octahedral cell composition. Other derivations from the 16-cell or tesseract duals encompass the rectified demihypercube (24 cuboctahedral cells, 24 vertices) and the truncated demihypercube (combinations of truncated tetrahedra and cuboctahedra). Vertex figures across the family are uniformly cuboctahedra \{3,4,3\}, underscoring isogonal uniformity and the bridging role between cubic honeycomb tilings and octahedral arrangements. Typical members exhibit 48 vertices, establishing a mid-scale complexity within uniform 4-polytopes, though counts range from 16 (e.g., runcinated forms) to 96. As a proper subgroup of B_4, the D_4 symmetry excludes certain 4-fold rotations, yielding fewer than the 15 members of the B_4 family and incorporating brief alternated forms like the small rhombated 16-cell, which alternates edges to produce chiral pairs with enhanced tetrahedral emphasis. Projections of these demi-structures, such as orthographic views of the bitruncated icositetrachoron, reveal concentric layers of cubic prisms interweaving with octahedral voids, visually capturing the symmetry reduction and dual heritage. These conceptual distinctions highlight the D_4 family's role in enumerating uniform polychora beyond full cubic invariance.

F4 family

The F4 family encompasses the 10 uniform 4-polytopes constructed under the full F4 symmetry group of order 1152, all derived from the regular 24-cell as the central member. The 24-cell, with Schläfli symbol {3,4,3}, features 24 regular octahedral cells and 24 vertices, making it the only self-dual regular convex 4-polytope, where its dual coincides with itself under congruence. This self-duality arises from the identical combinatorial structure of its vertex and cell incidences, a property unique to this family among the convex uniform 4-polytopes. The remaining members are obtained via Wythoff constructions applied to the F4 Coxeter diagram [3,4,3], yielding polytopes whose cells consist of Archimedean solids derived from the octahedron and cube, reflecting the octahedral symmetry. All polytopes in this family are convex and have density 1, ensuring no self-intersections occur in their realizations. The family begins with the regular 24-cell and progresses through rectification, truncation, bitruncation, cantellation, runcination, and higher-order operations up to omnitruncation, each preserving uniformity by maintaining regular vertex figures. Representative examples include the rectified 24-cell, which bounds 24 cuboctahedra (along with 24 cubes), and the truncated 24-cell, featuring 24 truncated octahedra (along with 24 cubes). Further operations produce more complex cell arrangements, such as the cantellated 24-cell with rhombicuboctahedra, cuboctahedra, and triangular prisms, culminating in the omnitruncated form with great rhombicuboctahedra and hexagonal prisms. These constructions highlight the richness of the F4 group in generating diverse yet symmetrically equivalent structures from a single seed polytope.

Construction	Name	Symbol	Vertex Count	Cell Types (representative)
Regular	24-cell	{3,4,3}	24	Octahedra
Rectified	Rectified 24-cell	r{3,4,3}	96	Cuboctahedra
Truncated	Truncated 24-cell	t{3,4,3}	192	Truncated octahedra
Bitruncated	Bitruncated 24-cell	2t{3,4,3}	192	Truncated octahedra
Cantellated	Cantellated 24-cell	rr{3,4,3}	288	Rhombicuboctahedra
Runcinated	Runcinated 24-cell	sr{3,4,3}	576	Triangular prisms
Runcitruncated	Runcitruncated 24-cell	t0,3{3,4,3}	576	Truncated octahedra, rhombicuboctahedra
Cantitruncated	Cantitruncated 24-cell	trr{3,4,3}	576	Great rhombicuboctahedra
Omnitruncated	Omnitruncated 24-cell	t0,3,4{3,4,3}	576	Great rhombicuboctahedra
Snub	Snub 24-cell	s{3,4,3}	576	Tetrahedra, snub cubes (chiral, half-symmetry)

This table uses conventional operational symbols adapted from 3D notation for clarity, though full 4D Wythoff symbols involve branched ring placements on the linear [3,4,3] diagram; for instance, the truncated form corresponds to ringing the second and fourth nodes. The vertex counts increase with truncation depth, establishing the scale of these structures from the compact 24-cell to expansive forms with hundreds of vertices, while maintaining the octahedral cell derivatives central to the family.

H4 family

The H4 family encompasses the convex uniform 4-polytopes exhibiting the full icosahedral rotational symmetry of the H4 Coxeter group, which has order 14400 and is generated by reflections across hyperplanes arranged in the Coxeter diagram with nodes connected as 5-3-3. This symmetry underlies 15 such polytopes in total, enumerated by applying Wythoff constructions to the group's fundamental chamber, producing vertex-transitive figures whose cells are uniform polyhedra with regular faces. These polytopes form the largest finite symmetry family among the convex uniform 4-polytopes, distinguished by their pentagonal and icosahedral elements, contrasting with the cubic or octahedral features of lower-symmetry families like F4.^[37] The family divides into dual subfamilies derived from the regular 120-cell {5,3,3} and its dual, the 600-cell {3,3,5}, each yielding seven members through progressive truncation operations: rectification (edge truncation), truncation (vertex truncation), cantellation (rectified truncation), bitruncation, runcination, runcitruncation, and omnitruncation. These operations preserve uniformity while expanding the structure, increasing vertex counts up to 14400 in the omnitruncated forms. The 120-cell subfamily features dodecahedral elements evolving into more complex Archimedean solids, while the 600-cell subfamily mirrors this with tetrahedral and icosahedral progressions; dual pairs like the truncated 120-cell and bitruncated 600-cell exhibit complementary cell configurations. Bowers nomenclature assigns descriptive names to these, such as hecatonicosachoron for the 120-cell and hexacosichoron for the 600-cell.^[2]^[3] Representative examples illustrate the progression. The regular 120-cell comprises 120 regular dodecahedra meeting five around each edge and four at each vertex, with 600 vertices, 1200 edges, 720 pentagonal faces, and a vertex figure of a regular tetrahedron. Its rectification, known as the rhombihecatonicosachoron or rahi, features 120 icosidodecahedra and 600 tetrahedra as cells, arising from mid-edge vertices, yielding 1200 vertices overall. Further truncation produces the truncated 120-cell or thi, bounded by 120 truncated dodecahedra and 600 regular tetrahedra, with 2400 vertices and a vertex figure of a pentagonal pyramid. The dual truncated 600-cell or tex consists of 120 regular icosahedra and 600 truncated tetrahedra, possessing 1440 vertices.^[38]^[39]^[11] Higher-order members introduce prismatic cells alongside polyhedral ones. The cantellated 120-cell, or srahi, includes 120 rhombicosidodecahedra, 600 octahedra, and 720 pentagonal prisms as cells, with 3600 vertices. Its dual, the cantellated 600-cell or srix, has 120 icosidodecahedra, 600 cuboctahedra, and 720 pentagonal prisms, also with 3600 vertices. The bitruncated 120-cell or xhi features 600 truncated tetrahedra and 120 truncated icosahedra, totaling 3600 vertices. Runcination adds more prisms: the runcinated 120-cell (shared with the 600-cell line) bounds 120 dodecahedra, 600 tetrahedra, 720 pentagonal prisms, and 1200 triangular prisms, with 7200 vertices. The runcitruncated 120-cell or prix has 120 truncated icosidodecahedra, 600 truncated octahedra, 720 decagonal prisms, and 1200 hexagonal prisms among its 2640 cells.^[40]^[41]^[42] The omnitruncated forms represent the culmination, fully truncating vertices, edges, faces, and cells while maintaining H4 symmetry. The omnitruncated 120-cell, or gidpixhi, consists of 120 great rhombicosidodecahedra, 600 truncated octahedra, 720 decagonal prisms, 1200 hexagonal prisms, with 14400 vertices—the highest among all finite uniform 4-polytopes. Its dual, the omnitruncated 600-cell, mirrors this complexity with 14400 vertices as well. These extreme members highlight the family's scale, embedding intricate networks of polyhedra and prisms that fill space under icosahedral constraints without gaps or overlaps.^[43]^[44]

Operation	Bowers Name	Key Cells	Vertices
Regular	Hecatonicosachoron (hec)	120 dodecahedra	600
Rectified	Rhombihecatonicosachoron (rahi)	120 icosidodecahedra + 600 tetrahedra	1200
Truncated	Truncated hecatonicosachoron (thi)	120 truncated dodecahedra + 600 tetrahedra	2400
Cantellated	Small rhombated hecatonicosachoron (srahi)	120 rhombicosidodecahedra + 600 octahedra + 720 pentagonal prisms	3600
Bitruncated	Hexacosihecatonicosachoron (xhi)	120 truncated icosahedra + 600 truncated tetrahedra	3600
Runcinated	Small disprismatohexacosihecatonicosachoron (sidpixhi)	120 dodecahedra + 600 tetrahedra + 720 pentagonal prisms + 1200 triangular prisms	7200
Omnitruncated	Great disprismatohexacosihecatonicosachoron (gidpixhi)	120 great rhombicosidodecahedra + 600 truncated octahedra + 720 decagonal prisms + 1200 hexagonal prisms	14400

Prismatic uniform 4-polytopes

Polyhedral prisms

Polyhedral prisms are a subclass of prismatic uniform 4-polytopes formed by the Cartesian product of a uniform polyhedron in three dimensions and a line segment (interval ), yielding a finite 4-dimensional figure with prismatic symmetry. This construction places two parallel copies of the base uniform polyhedron as the "end" cells, connected by lateral cells that are themselves uniform prisms over each face of the base polyhedron. The resulting 4-polytope is uniform if the base is uniform, with vertex-transitivity inherited from the product's symmetry group, denoted as G_3 \times A_1, where G_3 is the symmetry group of the base polyhedron and A_1 is the symmetry of the line segment.^[4] There are 18 convex uniform polyhedral prisms, corresponding to prisms over the 18 convex uniform polyhedra (5 Platonic solids and 13 Archimedean solids). The finite non-dual families of such prisms correspond to the irreducible finite reflection groups in three dimensions excluding the dihedral cases, specifically the tetrahedral (A_3), octahedral (B_3), and icosahedral (H_3) symmetries. These yield multiple uniform polyhedral prisms, with the following based on the Platonic solids with matching symmetries as representatives: the tetrahedron for A_3, the octahedron for B_3, and the dodecahedron for H_3 (noting that the icosahedron and dodecahedron share the H_3 symmetry group, but the dodecahedral prism exemplifies the pentagonal-faced variant). Each has twice the number of vertices of its base polyhedron, with cells comprising the two base polyhedra plus prisms over their faces. These prisms extend the concept of lower-dimensional prisms while maintaining uniformity through equal edge lengths and regular face polygons.^[4]^[31] The tetrahedral prism exemplifies the A_3 \times A_1 family, featuring two regular tetrahedra as end cells and four triangular prisms as lateral cells, totaling six cells and eight vertices. In the B_3 \times A_1 family, the octahedral prism includes two regular octahedra and eight triangular prisms, with twelve vertices overall. The H_3 \times A_1 dodecahedral prism consists of two regular dodecahedra and twelve pentagonal prisms, resulting in fourteen cells and forty vertices. These structures highlight how the base polyhedron's geometry dictates the lateral cell types—triangular for simplicial or octahedral bases, pentagonal for dodecahedral—while preserving the uniform property. An extension of this construction to products of two polygons yields duoprisms, but polyhedral prisms remain distinct in their 3D base dimensionality.^[4]^[31]

Family	Prism Name (Bowers Acronym)	End Cells	Lateral Cells	Total Cells	Vertices
A_3 \times A_1	Tetrahedral prism (tepr)	2 tetrahedra	4 triangular prisms	6	8
B_3 \times A_1	Octahedral prism (oep)	2 octahedra	8 triangular prisms	10	12
H_3 \times A_1	Dodecahedral prism (dope)	2 dodecahedra	12 pentagonal prisms	14	40

Duoprisms and antiprisms

Duoprisms form an infinite family of uniform 4-polytopes obtained as the Cartesian product (or prism product) of two regular polygons with Schläfli symbols {p} and {q}, where p, q ≥ 3 are integers.^[45]^[46] This construction yields a vertex-transitive figure whose cells are uniform prisms, provided the edge lengths of the input polygons are equal to ensure uniformity.^[46] The resulting duoprism has pq vertices, 2pq edges, and its cells consist of p uniform q-gonal prisms together with q uniform p-gonal prisms.^[46] Degenerate cases where p = 2 or q = 2 reduce to the finite family of uniform polyhedral prisms.^[4] A representative example is the 3-4 duoprism {3}×{4}, which has 12 vertices, 24 edges, 19 faces (15 squares and 4 triangles), and 7 cells (3 cubes and 4 triangular prisms).^[46]^[47] The 4-4 duoprism {4}×{4} coincides with the tesseract and features 16 vertices and 8 cubic cells.^[45] The pentagonal duoprism {5}×{5} provides another illustration, with 25 vertices and 10 pentagonal prismatic cells.^[29] These polytopes can be represented in Wythoff notation through the direct product of their generating dihedral symmetry groups, leading to disconnected Coxeter-Dynkin diagrams.^[45] In visualizations such as Schlegel diagrams, duoprisms manifest as a layered arrangement of prismatic cells projecting from a central polygonal section, highlighting their prismatic layering.^[29] Antiprismatic variants, known as uniform antiprismatic prisms or antiduoprisms, constitute another infinite family of uniform 4-polytopes formed by extruding a uniform n-gonal antiprism (n ≥ 3) along a line perpendicular to its hyperplane.^[29]^[4] This yields a vertex-transitive 4-polytope with two parallel uniform n-gonal antiprisms as opposite cells, connected by 2n uniform triangular prisms as lateral cells. The structure introduces a twist in the layering compared to standard prisms, arising from the antiprismatic vertex figure of the base.^[4] For instance, the square antiprismatic prism (n=4) comprises two square antiprisms and 8 triangular prisms, with 16 vertices and a total of 10 cells, exhibiting the characteristic rotational offset between its end cells.^[29] These forms are enumerated separately from finite uniform polychora due to their infinite variability with n, and they share prismatic symmetry groups that are products of dihedral and cyclic groups.^[4]

Higher-order prismatic compounds

Higher-order prismatic compounds encompass uniform 4-polytopes constructed as compounds or twisted products involving lower-dimensional elements, extending the prismatic constructions beyond pairwise duoprisms. These structures maintain vertex-transitivity through the product of symmetries from their component polytopes, resulting in infinite families with cells consisting of prisms, antiprisms, or rectangular boxes.^[9] Uniform compounds within this category include multi-component assemblies of prismatic elements, such as duoprism compounds. A representative example is the swirlprismatic dishecatonicosachoron, a uniform polychoron compound consisting of 24 pentagonal-pentagrammic duoprisms arranged with swirlprismatic symmetry. This compound is vertex-transitive, with the components interlocked in a way that the overall figure behaves as a single uniform 4-polytope under the shared symmetry group.^[48]^[9]

Nonprismatic and derived uniform polychora

Alternations and quasiregular forms

In four-dimensional geometry, the alternation of a uniform 4-polytope involves selectively removing every other vertex from the original figure, thereby establishing new edges between the surviving vertices that were originally separated by the excised ones. This operation is applicable to uniform 4-polytopes where an even number of vertices surround each edge, ensuring the resulting structure maintains vertex-transitivity and uniform cells composed of regular polygons.^[49] Quasiregular 4-polytopes represent a specific class of uniform polychora obtained by rectifying the six regular 4-polytopes, resulting in structures where exactly two regular polyhedral cells meet at each edge, analogous to quasiregular polyhedra in three dimensions. These forms are edge-transitive and vertex-transitive but not necessarily face-transitive, with cells alternating between two distinct regular polyhedra types. For instance, the rectified tesseract consists of 16 regular tetrahedra and 8 cuboctahedra, totaling 96 vertices and exhibiting the full octahedral symmetry of the tesseract group. Similarly, the rectified 24-cell comprises 96 regular octahedra and 24 cuboctahedra, highlighting the quasiregular property through its symmetric arrangement of cells around edges. These quasiregular polychora, as enumerated in foundational classifications, bridge regular and more general uniform forms by preserving high symmetry while introducing mixed cell types.^[50]^[51] Beyond the quasiregular cases derived from regulars, alternations of other uniform 4-polytopes can yield additional uniform polychora that are not captured by standard Wythoff constructions based on Coxeter-Dynkin diagrams, thereby expanding the enumeration of nonprismatic convex uniforms under groups like B4 (hypercubic). Examples include alternations within the B4 family, such as those producing quasitruncated forms like the quasitruncated tesseract (with 8 quasitruncated hexahedra and 16 tetrahedra, a nonconvex example) and up to 10 distinct nonuniform alternates in this symmetry, including variants with trigonal pyramidal vertex figures. In the H4 family, the cantellation of the 120-cell generates the small rhombated 120-cell, incorporating dodecahedral and icosahedral elements in its cell structure including rhombicosidodecahedra, octahedra, and tetrahedra. These nonuniform alternations contribute to the 46 nonprismatic convex uniform polychora under A4, B4, F4, and H4 symmetries, filling gaps in the systematic enumeration by introducing structures with density 1 and preserved convexity. Properties of these forms include consistent edge lengths and transitive vertices, though they may exhibit reduced face-transitivity compared to prismatic families.^[49]^[51]

Geometric constructions by symmetry extension

Geometric constructions by symmetry extension employ the full Coxeter reflection groups, including their extended forms, to generate uniform 4-polytopes through Wythoff's kaleidoscopic method. These extensions often involve doubling the symmetry order by incorporating additional mirrors or branched diagrams, enabling the creation of quasitruncations and other operations not possible with the ordinary symmetry groups. For instance, the extended group [[3,3,3]] for the 5-cell family allows constructions like the runcitruncated 5-cell, where the vertex figure incorporates density greater than one.^[52] In the H4 family, extended symmetries facilitate advanced quasitruncations, such as the small rhombihecatonicosachoron (runctr {5,3,5}), derived from the 120-cell using a branched Wythoff diagram that accounts for higher-order mirrors reflecting the icosahedral symmetry. This construction produces a uniform polychoron with 600 rhombic triacontahedra as cells and 1200 vertices, highlighting how symmetry extension resolves ambiguities in vertex placement for star-faced elements. Similarly, for the B4 family, extensions yield runcitruncated forms like the runcitruncated tesseract, incorporating glide reflections to maintain uniformity across chiral pairs.^[52] Symmetry extensions also incorporate demi-groups and glide reflections, particularly in D4-like constructions, to generate uniforms beyond the standard irreducible families. These methods, originally outlined by Coxeter, have been refined through computational enumerations in the 2020s, confirming 46 nonprismatic uniforms excluding fissaries and prismatic compounds. The following table summarizes these 46 polychora grouped by symmetry family and construction type, with representative vertex and cell counts for key examples. Counts adjusted to standard enumeration: 10 in A4, 16 in B4, 10 in F4, 10 in H4.

Symmetry Family	Construction Types (Examples)	Number	Example Vertices	Example Cells
A4 ([3,3,3] extended)	Rectified, truncated, cantellated, runcinated, etc. (e.g., runcitruncated 5-cell)	10	60 (runcitruncated)	5 truncated tetrahedra, 5 cuboctahedra, 10 triangular prisms, 10 hexagonal prisms
B4 ([3,3,4] extended)	Rectified, bitruncated, cantitruncated, runcitruncated (e.g., cantitruncated tesseract)	16	192 (cantitruncated tesseract)	8 truncated cuboctahedra, 16 truncated tetrahedra, 32 triangular prisms
F4 ([3,4,3] extended)	Quasitruncated, omnitruncated (e.g., omnitruncated 24-cell)	10	1152 (omnitruncated)	48 truncated cuboctahedra, 96 rhombicuboctahedra, 96 truncated tetrahedra
H4 ([3,5,3] extended)	Runcitruncated, quasitruncations (e.g., small rhombihecatonicosachoron)	10	7200 (runcitruncated 120-cell)	600 rhombic triacontahedra

These constructions ensure vertex-transitivity while preserving uniform cells, with the extended symmetries crucial for 24 of the 46 forms that involve retrogrades or density in their Wythoff symbols. Computational verifications in recent decades have excluded incomplete or fissary variants, solidifying the enumeration.

The grand antiprism

The grand antiprism is a uniform 4-polytope that serves as an outlier among convex uniform polychora, as it cannot be generated through Wythoff's construction method, which accounts for most nonprismatic forms.^[53] This exceptional status highlights its unique geometric configuration within the broader enumeration of 75 uniform 4-polytopes.^[53] It is constructed as the convex hull of 100 vertices derived from the 120 vertices of the regular 600-cell {3,3,5}, specifically by excising two orthogonal decagons comprising 10 vertices each.^[53] This selective inscription preserves uniformity while introducing a twisted, antiprism-like layering in four dimensions, distinct from prismatic products or symmetry extensions of lower-dimensional uniforms.^[53] The resulting structure features 500 edges, 720 faces (40 regular pentagons and 680 equilateral triangles), and 320 cells consisting of 20 uniform pentagonal antiprisms and 300 regular tetrahedra.^[53] The symmetry group of the grand antiprism is isomorphic to the semidirect product [5,2,5] ⋊ C₄, with an order of 400, stemming from a subgroup of the full H₄ symmetry of the 600-cell.^[53] This group action ensures vertex-transitivity and the regular arrangement of cells, with vertices representable in quaternion coordinates to model its embedding in 4D Euclidean space.^[54] As the sole non-Wythoffian, nonprismatic convex uniform polychoron of icosahedral type, it provides key insights into the limits of constructive enumeration in higher-dimensional geometry.^[53]

Uniform star polychora

Kepler-Poinsot analogs in 4D

In four dimensions, the analogs to the Kepler–Poinsot polyhedra are the ten regular star polychora, collectively known as the Schläfli–Hess polychora, which exhibit densities greater than unity due to their self-intersecting structures. These uniform 4-polytopes parallel the four regular star polyhedra in three dimensions by incorporating star polygon faces and interpenetrating cells, while maintaining full symmetry and regular elements at all levels. Four of these were first described by Ludwig Schläfli in 1852, with the remaining six identified by Edmund Hess in 1883, completing the enumeration of finite regular 4-polytopes beyond the six convex forms.^[4] These polychora are characterized by properties such as interpenetrating cellular arrangements and face planes that wind around vertices multiple times, quantified by their density, which represents the number of times the figure covers its convex hull.^[55] For instance, the small stellated 120-cell {5/2,5,3}, one of the four H4-symmetry-based forms discovered by Schläfli, has a density of 4, arising from the star pentagonal {5/2} faces and the stellated dodecahedral cells that overlap in a controlled manner.^[55] The other H4 analogs include the great 120-cell {5,5/2,5} with density 6, the grand 120-cell {3,5,5/2} with density 20, and the grand stellated 120-cell {5/2,3,5} with density 66, each demonstrating increasing complexity in their winding configurations.^[55] The six additional regulars, also under H4 symmetry, feature similar interpenetrations but with distinct Schläfli symbols like {3,5,5/2} and {5/2,5,5/2}, extending the Kepler–Poinsot analogy through higher-order star elements. Construction of these analogs typically involves stellation processes applied to the convex regular polychora, particularly the 120-cell {5,3,3} and its dual, the 600-cell {3,3,5}, where planes of existing facets are extended to form new star cells.^[56] For example, the small stellated 120-cell emerges from stellating the 120-cell by placing a new vertex at the center of each dodecahedral cell and connecting via the original facet planes, resulting in 120 small stellated dodecahedra as cells.^[56] This process preserves uniformity while introducing the nonconvexity essential to their star nature, analogous to stellating the dodecahedron to yield the small stellated dodecahedron in three dimensions. Beyond the ten regulars, uniform star 4-polytopes encompass semi-regular forms where cells are uniform star polyhedra rather than fully regular, contributing to numerous such figures, with 2191 finite non-prismatic uniform star polychora known as of November 2025, excluding infinite families.^[9] The enumeration continues, including 342 new discoveries from 2020 to 2023.

Infinite and finite star families

Uniform star 4-polytopes, or star polychora, extend the concept of uniform polychora by allowing self-intersecting cells and facets with densities greater than 1, leading to infinite prismatic families and a finite set of non-prismatic forms enumerated primarily through systematic catalogs.^[19]

Infinite Star Families

Infinite families of uniform star polychora arise from prismatic constructions involving star polygons, producing self-intersecting structures with uniform vertex figures. The primary such family consists of star duoprisms, denoted as {p/q} × {r}, where at least one factor is a star polygon (density >1), such as the pentagrammic duoprism {5/2} × {4}, which features 10 pentagrammic prism cells and 20 square cells, resulting in a density of 3. These duoprisms are vertex-transitive with disphenoid vertex figures and exhibit symmetries like D_{2p} × D_{2r}. Another example is the triangular-pentagrammic duoprism {3} × {5/2}, with density 2 and prismatic antiprism-like properties.^[57]^[58] A related infinite family includes antiprismatic star polychora, formed as prisms over star antiprisms or duoprisms with antiprismatic factors, such as the pentagrammic antiprismatic prism, which has 20 uniform pentagrammic antiprism cells and a crossed trapezoidal pyramid vertex figure, achieving higher densities through intersecting elements. These forms, like the great heptagrammic antiprismatic prism {7/3} antiprism prism, display rotational symmetries such as D_{2n} × D_2 and are cataloged under antiduoprism extensions. Densities in these families typically range from 2 to 7, depending on the star polygon density.^[59]

Finite Star Families

The finite non-prismatic uniform star polychora total 2191 as of November 2025, distinct from convex forms by their self-intersections and grouped by underlying symmetry groups, including 10 with full icosahedral H_4 symmetry (the Schläfli-Hess regular stars) and many more under hypercubic B_4, tetrahedral A_4, octahedral F_4, and lower subgroups—for instance, over 1100 with H_4 symmetry alone.^[4] These are derived through operations like truncation, stellation, and rectification on convex hulls such as the 120-cell and 600-cell, yielding cells that include star polyhedra like the small stellated dodecahedron {5/2,5}.^[19] Properties of these star polychora include vertex densities up to 191 for highly faceted forms, reflecting multiple windings of edges through vertices, while face densities reach 4 or more in icosahedral cases. For instance, the truncated great stellated 120-cell, under H_4 symmetry, has 120 great stellated dodecahedron cells, 720 pentagrammic prism cells, and a density of 76, demonstrating the complex interpenetrations possible in 4D. Another example is the icosahedral hemi-dodecahedral polychoron, with hemispherical dodecahedral cells, 600 triangular cells, and a vertex density of 4, belonging to the "ico regiment" in Bowers' classification of H_4 star forms.^[19]