List of regular polytopes#Tessellations of Euclidean 4-space

This table shows a summary of regular polytope counts by rank.

{{notelist}}

There are no Euclidean regular star tessellations in any number of dimensions.

There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol { },{{sfnp|Coxeter|1973|p=129}}{{sfnp|McMullen|Schulte|2002|p=30}} or a Coxeter diagram with a single ringed node, {{CDD|node_1}}. Norman Johnson calls it a dion{{cite book |author-link=Norman Johnson (mathematician) |first=N.W. |last=Johnson |title=Geometries and Transformations |year=2018 |isbn=978-1-107-10340-5 |chapter=Chapter 11: Finite symmetry groups |publisher=Cambridge University Press |at=11.1 Polytopes and Honeycombs, p. 224}} and gives it the Schläfli symbol { }.

Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes.{{sfnp|Coxeter|1973|p=120}} It is used in the definition of uniform prisms like Schläfli symbol { }×{p}, or Coxeter diagram {{CDD|node_1|2|node_1|p|node}} as a Cartesian product of a line segment and a regular polygon.{{sfnp|Coxeter|1973|p=124}}

The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A {{mvar|p}}-gonal regular polygon is represented by Schläfli symbol {p}.

Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to be completed.

The Schläfli symbol {p} represents a Regular polygon.

The regular digon {2} can be considered to be a degenerate regular polygon. It can be realized non-degenerately in some non-Euclidean spaces, such as on the surface of a sphere or torus. For example, digon can be realised non-degenerately as a spherical lune. A monogon {1} could also be realised on the sphere as a single point with a great circle through it.Coxeter, Regular Complex Polytopes, p. 9 However, a monogon is not a valid abstract polytope because its single edge is incident to only one vertex rather than two.

There exist infinitely many regular star polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {{math|{{mset|n/m}}}}. They are called star polygons and share the same vertex arrangements of the convex regular polygons.

In general, for any natural number {{mvar|n}}, there are regular {{mvar|n}}-pointed stars with Schläfli symbols {{math|{{mset|n/m}}}} for all {{mvar|m}} such that {{math|m < n/2}} (strictly speaking {{math|{{mset|n/m}} {{=}} {{mset|n/(n − m)}}}}) and {{mvar|m}} and {{mvar|n}} are coprime (as such, all stellations of a polygon with a prime number of sides will be regular stars). Symbols where {{mvar|m}} and {{mvar|n}} are not coprime may be used to represent compound polygons.

Star polygons that can only exist as spherical tilings, similarly to the monogon and digon, may exist (for example: {3/2}, {5/3}, {5/4}, {7/4}, {9/5}), however these have not been studied in detail.

There also exist failed star polygons, such as the piangle, which do not cover the surface of a circle finitely many times.{{Cite web |url=http://chalkdustmagazine.com/blog/fractional-polygons/ |title=Between a square rock and a hard pentagon: Fractional polygons |last=Duncan |first=Hugh |date=28 September 2017 |work=chalkdust |access-date=2018-12-24 |archive-date=2018-12-23 |archive-url=https://web.archive.org/web/20181223030125/http://chalkdustmagazine.com/blog/fractional-polygons/ |url-status=live }}

In addition to the planar regular polygons there are infinitely many regular skew polygons. Skew polygons can be created via the blending operation.

The blend of two polygons {{mvar|P}} and {{mvar|Q}}, written {{math|P#Q}}, can be constructed as follows:

1. take the cartesian product of their vertices {{math|V{{Sub|P}} × V{{sub|Q}}}}.
2. add edges {{math|(p{{sub|0}} × q{{sub|0}}, p{{sub|1}} × q{{sub|1}})}} where {{math|(p{{sub|0}}, p{{sub|1}})}} is an edge of {{mvar|P}} and {{math|(q{{sub|0}}, q{{sub|1}})}} is an edge of {{mvar|Q}}.
3. select an arbitrary connected component of the result.

Alternatively, the blend is the polygon {{math|{{angbr|ρ{{sub|0}}σ{{sub|0}}, ρ{{sub|1}}σ{{sub|1}}}}}} where {{mvar|ρ}} and {{mvar|σ}} are the generating mirrors of {{mvar|P}} and {{mvar|Q}} placed in orthogonal subspaces.{{sfn|McMullen|Schulte|2002}}

The blending operation is commutative, associative and idempotent.

Every regular skew polygon can be expressed as the blend of a unique{{efn|group=lower-roman|(up to identity and idempotency)}} set of planar polygons.{{sfn|McMullen|Schulte|2002}} If {{mvar|P}} and {{mvar|Q}} share no factors then {{math|Dim(P#Q) {{=}} Dim(P) + Dim(Q)}}.

The regular finite polygons in 3 dimensions are exactly the blends of the planar polygons (dimension 2) with the digon (dimension 1). They have vertices corresponding to a prism ({{math|{{mset|n/m}}#{{(}}{{)}}}} where {{mvar|n}} is odd) or an antiprism ({{math|{{mset|n/m}}#{{(}}{{)}}}} where {{mvar|n}} is even). All polygons in 3 space have an even number of vertices and edges.

Several of these appear as the Petrie polygons of regular polyhedra.

The regular finite polygons in 4 dimensions are exactly the polygons formed as a blend of two distinct planar polygons. They have vertices lying on a Clifford torus and related by a Clifford displacement. Unlike 3-dimensional polygons, skew polygons on double rotations can include an odd-number of sides.

Polytopes of rank 3 are called polyhedra:

A regular polyhedron with Schläfli symbol {{math|{{mset|p, q}}}}, Coxeter diagrams {{CDD|node_1|p|node|q|node}}, has a regular face type {{math|{{mset|p}}}}, and regular vertex figure {{math|{{mset|q}}}}.

A vertex figure (of a polyhedron) is a polygon, seen by connecting those vertices which are one edge away from a given vertex. For regular polyhedra, this vertex figure is always a regular (and planar) polygon.

Existence of a regular polyhedron {{math|{{mset|p, q}}}} is constrained by an inequality, related to the vertex figure's angle defect:

\begin{align}

& \frac{1}{p} + \frac{1}{q} > \frac{1}{2} : \text{Polyhedron (existing in Euclidean 3-space)} \\[6pt]

& \frac{1}{p} + \frac{1}{q} = \frac{1}{2} : \text{Euclidean plane tiling} \\[6pt]

& \frac{1}{p} + \frac{1}{q} < \frac{1}{2} : \text{Hyperbolic plane tiling}

\end{align}

By enumerating the permutations, we find five convex forms, four star forms and three plane tilings, all with polygons {{math|{{mset|p}}}} and {{math|{{mset|q}}}} limited to: {3}, {4}, {5}, {5/2}, and {6}.

Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings.

The five convex regular polyhedra are called the Platonic solids. The vertex figure is given with each vertex count. All these polyhedra have an Euler characteristic (χ) of 2.

In spherical geometry, regular spherical polyhedra (tilings of the sphere) exist that would otherwise be degenerate as polytopes. These are the hosohedra {2,n} and their dual dihedra {n,2}. Coxeter calls these cases "improper" tessellations.{{sfnp|Coxeter|1973|pp=66-67}}

The first few cases (n from 2 to 6) are listed below.

Star-dihedra and hosohedra {{math|{{mset|p/q, 2}}}} and {{math|{{mset|2, p/q}}}} also exist for any star polygon {{math|{{mset|p/q}}}}.

The regular star polyhedra are called the Kepler–Poinsot polyhedra and there are four of them, based on the vertex arrangements of the dodecahedron {5,3} and icosahedron {3,5}:

As spherical tilings, these star forms overlap the sphere multiple times, called its density, being 3 or 7 for these forms. The tiling images show a single spherical polygon face in yellow.

There are infinitely many failed star polyhedra. These are also spherical tilings with star polygons in their Schläfli symbols, but they do not cover a sphere finitely many times. Some examples are {5/2,4}, {5/2,9}, {7/2,3}, {5/2,5/2}, {7/2,7/3}, {4,5/2}, and {3,7/3}.

{{Expand section|reason=This section is 75 years behind on mathematical liturature on this topic. It should include Grünbaumian skews.|date=January 2024}}

Regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar vertex figures.

For 4-dimensional skew polyhedra, Coxeter offered a modified Schläfli symbol {l,m|n} for these figures, with {l,m} implying the vertex figure, m l-gons around a vertex, and {{mvar|n}}-gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes.

The regular skew polyhedra, represented by {l,m|n}, follow this equation:

$$2\; \backslash sin\backslash left(\backslash frac\{\backslash pi\}\{l\}\backslash right)\; \backslash sin\backslash left(\backslash frac\{\backslash pi\}\{m\}\backslash right)\; =\; \backslash cos\backslash left(\backslash frac\{\backslash pi\}\{n\}\backslash right)$$

Four of them can be seen in 4-dimensions as a subset of faces of four regular 4-polytopes, sharing the same vertex arrangement and edge arrangement:

Regular 4-polytopes with Schläfli symbol $\backslash \{p,q,r\backslash \}$ have cells of type $\backslash \{p,q\backslash \}$, faces of type $\backslash \{p\backslash \}$, edge figures

$\backslash \{r\backslash \}$, and vertex figures $\backslash \{q,r\backslash \}$.

• A vertex figure (of a 4-polytope) is a polyhedron, seen by the arrangement of neighboring vertices around a given vertex. For regular 4-polytopes, this vertex figure is a regular polyhedron.
• An edge figure is a polygon, seen by the arrangement of faces around an edge. For regular 4-polytopes, this edge figure will always be a regular polygon.

The existence of a regular 4-polytope $\backslash \{p,q,r\backslash \}$ is constrained by the existence of the regular polyhedra $\backslash \{p,q\backslash \},\; \backslash \{q,r\backslash \}$. A suggested name for 4-polytopes is "polychoron".{{cite conference |conference=Convex and Abstract Polytopes (May 19–21, 2005) and Polytopes Day in Calgary (May 22, 2005) |title=Abstracts |url=http://www.mit.edu/~hlb/Associahedron/program.pdf |access-date=2015-02-02 |archive-date=2014-11-29 |archive-url=https://web.archive.org/web/20141129150529/http://www.mit.edu/~hlb/Associahedron/program.pdf |url-status=live }}

Each will exist in a space dependent upon this expression:

: $\backslash sin\; \backslash left\; (\; \backslash frac\{\backslash pi\}\{p\}\; \backslash right\; )\; \backslash sin\; \backslash left(\backslash frac\{\backslash pi\}\{r\}\backslash right)\; -\; \backslash cos\backslash left(\backslash frac\{\backslash pi\}\{q\}\backslash right)$

:: $>\; 0$ : Hyperspherical 3-space honeycomb or 4-polytope

:: $=\; 0$ : Euclidean 3-space honeycomb

:: : Hyperbolic 3-space honeycomb

These constraints allow for 21 forms: 6 are convex, 10 are nonconvex, one is a Euclidean 3-space honeycomb, and 4 are hyperbolic honeycombs.

The Euler characteristic $\backslash chi$ for convex 4-polytopes is zero:

$\backslash chi\; =\; V+F-E-C\; =\; 0$

The 6 convex regular 4-polytopes are shown in the table below. All these 4-polytopes have an Euler characteristic (χ) of 0.

Di-4-topes and hoso-4-topes exist as regular tessellations of the 3-sphere.

Regular di-4-topes (2 facets) include: {3,3,2}, {3,4,2}, {4,3,2}, {5,3,2}, {3,5,2}, {p,2,2}, and their hoso-4-tope duals (2 vertices): {2,3,3}, {2,4,3}, {2,3,4}, {2,3,5}, {2,5,3}, {2,2,p}. 4-polytopes of the form {2,p,2} are the same as {2,2,p}. There are also the cases {p,2,q} which have dihedral cells and hosohedral vertex figures.

There are ten regular star 4-polytopes, which are called the Schläfli–Hess 4-polytopes. Their vertices are based on the convex 120-cell {5,3,3} and 600-cell {3,3,5}.

Ludwig Schläfli found four of them and skipped the last six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F+V−E=2). Edmund Hess (1843–1903) completed the full list of ten in his German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder (1883)[http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8623.0001.001].

There are 4 unique edge arrangements and 7 unique face arrangements from these 10 regular star 4-polytopes, shown as orthogonal projections:

There are 4 failed potential regular star 4-polytopes permutations: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}. Their cells and vertex figures exist, but they do not cover a hypersphere with a finite number of repetitions.

{{Expand section|date=January 2024}}

In addition to the 16 planar 4-polytopes above there are 18 finite skew polytopes.{{sfnp|McMullen|2004}} One of these is obtained as the Petrial of the tesseract, and the other 17 can be formed by applying the kappa operation to the planar polytopes and the Petrial of the tesseract.

5-polytopes can be given the symbol $\backslash \{p,q,r,s\backslash \}$ where $\backslash \{p,q,r\backslash \}$ is the 4-face type, $\backslash \{p,q\backslash \}$ is the cell type, $\backslash \{p\backslash \}$ is the face type, and $\backslash \{s\backslash \}$ is the face figure, $\backslash \{r,s\backslash \}$ is the edge figure, and $\backslash \{q,r,s\backslash \}$ is the vertex figure.

: A vertex figure (of a 5-polytope) is a 4-polytope, seen by the arrangement of neighboring vertices to each vertex.

: An edge figure (of a 5-polytope) is a polyhedron, seen by the arrangement of faces around each edge.

: A face figure (of a 5-polytope) is a polygon, seen by the arrangement of cells around each face.

A regular 5-polytope $\backslash \{p,q,r,s\backslash \}$ exists only if $\backslash \{p,q,r\backslash \}$ and $\backslash \{q,r,s\backslash \}$ are regular 4-polytopes.

The space it fits in is based on the expression:

: $\backslash frac\{\backslash cos^2\backslash left(\backslash frac\{\backslash pi\}\{q\}\backslash right)\}\{\backslash sin^2\backslash left(\backslash frac\{\backslash pi\}\{p\}\backslash right)\}\; +\; \backslash frac\{\backslash cos^2\backslash left(\backslash frac\{\backslash pi\}\{r\}\backslash right)\}\{\backslash sin^2\backslash left(\backslash frac\{\backslash pi\}\{s\}\backslash right)\}$

:: : Spherical 4-space tessellation or 5-space polytope

:: $=\; 1$ : Euclidean 4-space tessellation

:: $>\; 1$ : hyperbolic 4-space tessellation

Enumeration of these constraints produce 3 convex polytopes, no star polytopes, 3 tessellations of Euclidean 4-space, and 5 tessellations of paracompact hyperbolic 4-space. The only non-convex regular polytopes for ranks 5 and higher are skews.

In dimensions 5 and higher, there are only three kinds of convex regular polytopes.{{sfnp|Coxeter|1973|loc=Table I: Regular polytopes, (iii) The three regular polytopes in {{mvar|n}} dimensions (n>=5), pp. 294–295}}

There are also improper cases where some numbers in the Schläfli symbol are 2. For example, {p,q,r,...2} is an improper regular spherical polytope whenever {p,q,r...} is a regular spherical polytope, and {2,...p,q,r} is an improper regular spherical polytope whenever {...p,q,r} is a regular spherical polytope. Such polytopes may also be used as facets, yielding forms such as {p,q,...2...y,z}.

There are no regular star polytopes of rank 5 or higher, with the exception of degenerate polytopes created by the star product of lower rank star polytopes. {{abbr|e.g.}} hosotopes and ditopes.

A projective regular {{math|({{mvar|n}}+1)}}-polytope exists when an original regular {{mvar|n}}-spherical tessellation, {p,q,...}, is centrally symmetric. Such a polytope is named hemi-{p,q,...}, and contain half as many elements. Coxeter gives a symbol {p,q,...}/2, while McMullen writes {p,q,...}_h/2 with h as the coxeter number.{{sfnp|McMullen|Schulte|2002|loc=[https://books.google.com/books?id=JfmlMYe6MJgC&pg=PA162 "6C Projective Regular Polytopes" pp. 162–165]}}

Even-sided regular polygons have hemi-2n-gon projective polygons, {2p}/2.

There are 4 regular projective polyhedra related to 4 of 5 Platonic solids.

The hemi-cube and hemi-octahedron generalize as hemi-hypercube and hemi-{{mvar|n}}-orthoplexes to any rank.

5 of 6 convex regular 4-polytopes are centrally symmetric generating projective 4-polytopes. The 3 special cases are hemi-24-cell, hemi-600-cell, and hemi-120-cell.

Only 2 of 3 regular spherical polytopes are centrally symmetric for ranks 5 or higher. The corresponding regular projective polytopes are the hemi versions of the regular hypercube and orthoplex. They are tabulated below for rank 5, for example:

An apeirotope or infinite polytope is a polytope which has infinitely many facets. An {{mvar|n}}-apeirotope is an infinite {{mvar|n}}-polytope: a 2-apeirotope or apeirogon is an infinite polygon, a 3-apeirotope or apeirohedron is an infinite polyhedron, etc.

There are two main geometric classes of apeirotope:{{cite journal |last=Grünbaum |first=B. |title=Regular Polyhedra—Old and New |journal=Aequationes Mathematicae |volume=16 |date=1977 |issue=1–2 |pages=1–20 |doi=10.1007/BF01836414|s2cid=125049930 }}

• Regular honeycombs in {{mvar|n}} dimensions, which completely fill an {{mvar|n}}-dimensional space.
• Regular skew apeirotopes, comprising an {{mvar|n}}-dimensional manifold in a higher space.

The straight apeirogon is a regular tessellation of the line, subdividing it into infinitely many equal segments. It has infinitely many vertices and edges. Its Schläfli symbol is {∞}, and Coxeter diagram {{CDD|node_1|infin|node}}.

...320px...

It exists as the limit of the {{mvar|p}}-gon as {{mvar|p}} tends to infinity, as follows:

Apeirogons in the hyperbolic plane, most notably the regular apeirogon, {∞}, can have a curvature just like finite polygons of the Euclidean plane, with the vertices circumscribed by horocycles or hypercycles rather than circles.

Regular apeirogons that are scaled to converge at infinity have the symbol {∞} and exist on horocycles, while more generally they can exist on hypercycles.

Above are two regular hyperbolic apeirogons in the Poincaré disk model, the right one shows perpendicular reflection lines of divergent fundamental domains, separated by length λ.

A skew apeirogon in two dimensions forms a zig-zag line in the plane. If the zig-zag is even and symmetrical, then the apeirogon is regular.

Skew apeirogons can be constructed in any number of dimensions. In three dimensions, a regular skew apeirogon traces out a helical spiral and may be either left- or right-handed.

There are six regular tessellations of the plane: the three listed below, and their corresponding Petrials.

There are two improper regular tilings: {∞,2}, an apeirogonal dihedron, made from two apeirogons, each filling half the plane; and secondly, its dual, {2,∞}, an apeirogonal hosohedron, seen as an infinite set of parallel lines.

There are no regular plane tilings of star polygons. There are many enumerations that fit in the plane (1/p + 1/q = 1/2), like {8/3,8}, {10/3,5}, {5/2,10}, {12/5,12}, etc., but none repeat periodically.

Tessellations of hyperbolic 2-space are hyperbolic tilings. There are infinitely many regular tilings in H². As stated above, every positive integer pair {p,q} such that 1/p + 1/q < 1/2 gives a hyperbolic tiling. In fact, for the general Schwarz triangle (p, q, r) the same holds true for 1/p + 1/q + 1/r < 1.

There are a number of different ways to display the hyperbolic plane, including the Poincaré disk model which maps the plane into a circle, as shown below. It should be recognized that all of the polygon faces in the tilings below are equal-sized and only appear to get smaller near the edges due to the projection applied, very similar to the effect of a camera fisheye lens.

There are infinitely many flat regular 3-apeirotopes (apeirohedra) as regular tilings of the hyperbolic plane, of the form {p,q}, with p+q

• {3,7}, {3,8}, {3,9} ... {3,∞}
• {4,5}, {4,6}, {4,7} ... {4,∞}
• {5,4}, {5,5}, {5,6} ... {5,∞}
• {6,4}, {6,5}, {6,6} ... {6,∞}
• {7,3}, {7,4}, {7,5} ... {7,∞}
• {8,3}, {8,4}, {8,5} ... {8,∞}
• {9,3}, {9,4}, {9,5} ... {9,∞}
• ...
• {∞,3}, {∞,4}, {∞,5} ... {∞,∞}

A sampling:

{{Regular hyperbolic tiling table}}

The tilings {p, ∞} have ideal vertices, on the edge of the Poincaré disk model. Their duals {∞, p} have ideal apeirogonal faces, meaning that they are inscribed in horocycles. One could go further (as is done in the table above) and find tilings with ultra-ideal vertices, outside the Poincaré disk, which are dual to tiles inscribed in hypercycles; in what is symbolised {p, iπ/λ} above, infinitely many tiles still fit around each ultra-ideal vertex.Roice Nelson and Henry Segerman, [https://arxiv.org/abs/1511.02851 Visualizing Hyperbolic Honeycombs] {{Webarchive|url=https://web.archive.org/web/20201130191755/https://arxiv.org/abs/1511.02851 |date=2020-11-30 }} (Parallel lines in extended hyperbolic space meet at an ideal point; ultraparallel lines meet at an ultra-ideal point.)Irving Adler, A New Look at Geometry (2012 Dover edition), [https://books.google.com/books?id=EbIAAQAAQBAJ&dq=ultra-ideal+hyperbolic+geometry&pg=PA233 p.233]

There are 2 infinite forms of hyperbolic tilings whose faces or vertex figures are star polygons: {m/2, m} and their duals {m, m/2} with m = 7, 9, 11, ....{{sfnp|Coxeter|1999|loc="Chapter 10"}} The {m/2, m} tilings are stellations of the {m, 3} tilings while the {m, m/2} dual tilings are facetings of the {3, m} tilings and greatenings{{efn|group=lower-roman|In a classification advanced by Conway & adopted by Coxeter,{{refn|group=lower-alpha| {{cite book |last=Coxeter |first=H. M. S. |title=Regular Complex Polytopes |publisher=Cambridge University Press |year=1975 |isbn=9780521201254 |edition=1st |pages=46–7 |language=en}} }} stellation refers to extension of edges, and greatening to extension of faces; the term aggrandizement is given for extension of cells (of polychora), though it appears to be less-commonly used.{{refn|group=lower-alpha|See: {{cite web |last=Inchbald |first=Guy |date=2024-09-09 |title=Stellating and Facetting – A Brief History |url=https://www.steelpillow.com/polyhedra/StelFacet/history.html |url-status=live |archive-url=https://web.archive.org/web/20240520221732/https://www.steelpillow.com/polyhedra/StelFacet/history.html |archive-date=2024-05-20 |website=Guy's Polyhedra Page}} }} }} of the {m, 3} tilings.

The patterns {m/2, m} and {m, m/2} continue for odd m < 7 as polyhedra: when m = 5, we obtain the small stellated dodecahedron and great dodecahedron,{{sfnp|Coxeter|1999|loc="Chapter 10"}} and when m = 3, the case degenerates to a tetrahedron. The other two Kepler–Poinsot polyhedra (the great stellated dodecahedron and great icosahedron) do not have regular hyperbolic tiling analogues. If m is even, depending on how we choose to define {m/2}, we can either obtain degenerate double covers of other tilings or compound tilings.

{{Expand section|reason=Complete the enumeration with details from the sources and explain the operations used.|date=January 2024}}

There are three regular skew apeirohedra in Euclidean 3-space, with planar faces.{{cite journal |last=Coxeter |first=H.S.M. |title=Regular Skew Polyhedra in Three and Four Dimensions |journal=Proc. London Math. Soc. |series=2 |volume=43 |pages=33–62 |date=1938 |doi=10.1112/plms/s2-43.1.33}}{{cite journal |last=Coxeter |first=H.S.M. |date=1985 |title=Regular and semi-regular polytopes II |journal=Mathematische Zeitschrift |volume=188 |issue=4 |pages=559–591 |doi=10.1007/BF01161657|s2cid=120429557 }}{{cite book |last1=Conway |first1=John H. |last2=Burgiel |first2=Heidi |last3=Goodman-Strauss |first3=Chaim |title=The Symmetries of Things |year=2008 |chapter=Chapter 23: Objects with Primary Symmetry, Infinite Platonic Polyhedra |pages=333–335 |publisher=Taylor & Francis |isbn=978-1-568-81220-5}} They share the same vertex arrangement and edge arrangement of 3 convex uniform honeycombs.

• 6 squares around each vertex: {4,6|4}
• 4 hexagons around each vertex: {6,4|4}
• 6 hexagons around each vertex: {6,6|3}

File:Pure 3-dimensional apeirohedra chart.png, {4,3,4}.{{sfnp|McMullen|Schulte|2002|p=224}} A π petrie dual operator replaces faces with petrie polygons; δ is a dual operator reverses vertices and faces; φ_k is a kth facetting operator; η is a halving operator, and σ skewing halving operator.]]

Allowing for skew faces, there are 30 regular apeirohedra in Euclidean 3-space.{{sfnp|McMullen|Schulte|2002|loc=Section 7E}} These include the 12 blended apeirohedra created by blends with the Euclidean planar apeirohedra, and 18 pure apeirohedra, which cannot be expressed as a non-trivial blend including the planar apeirohedra and the three 3-dimensional apeirohedra above.

The 3-dimensional pure apeirohedra are:

• {{math|{{mset|4,6{{!}}4}}}}, the mucube
• {{math|{∞,6}{{sub|4,4}}}}, the Petrial of the mucube
• {{math|{{mset|6,6{{!}}3}}}}, the mutetrahedron
• {{math|{∞,6}{{sub|6,3}}}}, the Petrial of the mutetrahedron
• {{math|{{mset|6,4{{!}}4}}}}, the muoctahedron
• {{math|{∞,4}{{sub|6,4}}}}, the Petrial of the muoctahedron
• {{math|{6,6}{{sub|4}}}}, the halving of the mucube
• {{math|{4,6}{{sub|6}}}}, the Petrial of {{math|{6,6}{{sub|4}}}}
• {{math|{∞,4}{{sub|·,*3}}}}, the skewing of the muoctahedron
• {{math|{6,4}{{sub|6}}}}, the Petrial of {{math|{∞,4}{{sub|·,*3}}}}
• {{math|{∞,3}{{sup|({{mvar|a}})}}}}
• {{math|{∞,3}{{sup|({{mvar|b}})}}}}

There are 31 regular skew apeirohedra with convex faces in hyperbolic 3-space with compact or paracompact symmetry:{{cite journal |last=Garner |first=C.W.L. |title=Regular Skew Polyhedra in Hyperbolic Three-Space |journal=Can. J. Math. |volume=19 |pages=1179–1186 |date=1967 |doi=10.4153/CJM-1967-106-9 |s2cid=124086497 |doi-access=free }} Note: His paper says there are 32, but one is self-dual, leaving 31.

• 14 are compact: {8,10|3}, {10,8|3}, {10,4|3}, {4,10|3}, {6,4|5}, {4,6|5}, {10,6|3}, {6,10|3}, {8,8|3}, {6,6|4}, {10,10|3},{6,6|5}, {8,6|3}, and {6,8|3}.
• 17 are paracompact: {12,10|3}, {10,12|3}, {12,4|3}, {4,12|3}, {6,4|6}, {4,6|6}, {8,4|4}, {4,8|4}, {12,6|3}, {6,12|3}, {12,12|3}, {6,6|6}, {8,6|4}, {6,8|4}, {12,8|3}, {8,12|3}, and {8,8|4}.

Image:Cubic honeycomb.png

There is only one non-degenerate regular tessellation of 3-space (honeycombs), {4, 3, 4}:{{sfnp|Coxeter|1973|loc=Table II: Regular honeycombs, p. 296}}

File:Order-4 square hosohedral honeycomb-sphere.png

There are six improper regular tessellations, pairs based on the three regular Euclidean tilings. Their cells and vertex figures are all regular hosohedra {2,n}, dihedra, {n,2}, and Euclidean tilings. These improper regular tilings are constructionally related to prismatic uniform honeycombs by truncation operations. They are higher-dimensional analogues of the order-2 apeirogonal tiling and apeirogonal hosohedron.

There are 15 flat regular honeycombs of hyperbolic 3-space:

• 4 are compact: {3,5,3}, {4,3,5}, {5,3,4}, and {5,3,5}
• while 11 are paracompact: {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}.

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Tessellations of hyperbolic 3-space can be called hyperbolic honeycombs. There are 15 hyperbolic honeycombs in H³, 4 compact and 11 paracompact.

There are also 11 paracompact H³ honeycombs (those with infinite (Euclidean) cells and/or vertex figures): {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}.

Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental tetrahedron having ultra-ideal vertices). All honeycombs with hyperbolic cells or vertex figures and do not have 2 in their Schläfli symbol are noncompact.

{{Regular honeycomb table}}

There are no regular hyperbolic star-honeycombs in H³: all forms with a regular star polyhedron as cell, vertex figure or both end up being spherical.

Ideal vertices now appear when the vertex figure is a Euclidean tiling, becoming inscribable in a horosphere rather than a sphere. They are dual to ideal cells (Euclidean tilings rather than finite polyhedra). As the last number in the Schläfli symbol rises further, the vertex figure becomes hyperbolic, and vertices become ultra-ideal (so the edges do not meet within hyperbolic space). In honeycombs {p, q, ∞} the edges intersect the Poincaré ball only in one ideal point; the rest of the edge has become ultra-ideal. Continuing further would lead to edges that are completely ultra-ideal, both for the honeycomb and for the fundamental simplex (though still infinitely many {p, q} would meet at such edges). In general, when the last number of the Schläfli symbol becomes ∞, faces of codimension two intersect the Poincaré hyperball only in one ideal point.

There are three kinds of infinite regular tessellations (honeycombs) that can tessellate Euclidean four-dimensional space:

There are also the two improper cases {4,3,4,2} and {2,4,3,4}.

There are three flat regular honeycombs of Euclidean 4-space:{{sfnp|Coxeter|1973|loc=Table II: Regular honeycombs, p. 296}}

• {4,3,3,4}, {3,3,4,3}, and {3,4,3,3}.

There are seven flat regular convex honeycombs of hyperbolic 4-space:{{sfnp|Coxeter|1999|loc="Chapter 10"}}

• 5 are compact: {3,3,3,5}, {5,3,3,3}, {4,3,3,5}, {5,3,3,4}, {5,3,3,5}
• 2 are paracompact: {3,4,3,4}, and {4,3,4,3}.

There are four flat regular star honeycombs of hyperbolic 4-space:{{sfnp|Coxeter|1999|loc="Chapter 10"}}

• {5/2,5,3,3}, {3,3,5,5/2}, {3,5,5/2,5}, and {5,5/2,5,3}.

There are seven convex regular honeycombs and four star-honeycombs in H⁴ space.{{sfnp|Coxeter|1999|loc="Chapter 10" Table IV, p. 213}} Five convex ones are compact, and two are paracompact.

Five compact regular honeycombs in H⁴:

The two paracompact regular H⁴ honeycombs are: {3,4,3,4}, {4,3,4,3}.

Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental 5-cell having some parts inaccessible beyond infinity). All honeycombs which are not shown in the set of tables below and do not have 2 in their Schläfli symbol are noncompact.

{{Regular tetracomb table}}

There are four regular star-honeycombs in H⁴ space, all compact:

There is only one flat regular honeycomb of Euclidean 5-space: (previously listed above as tessellations){{sfnp|Coxeter|1973|loc=Table II: Regular honeycombs, p. 296}}

• {4,3,3,3,4}

There are five flat regular regular honeycombs of hyperbolic 5-space, all paracompact: (previously listed above as tessellations){{sfnp|Coxeter|1999|loc="Chapter 10"}}

• {3,3,3,4,3}, {3,4,3,3,3}, {3,3,4,3,3}, {3,4,3,3,4}, and {4,3,3,4,3}

The hypercubic honeycomb is the only family of regular honeycombs that can tessellate each dimension, five or higher, formed by hypercube facets, four around every ridge.

In E⁵, there are also the improper cases {4,3,3,4,2}, {2,4,3,3,4}, {3,3,4,3,2}, {2,3,3,4,3}, {3,4,3,3,2}, and {2,3,4,3,3}. In Eⁿ, {4,3ⁿ⁻³,4,2} and {2,4,3ⁿ⁻³,4} are always improper Euclidean tessellations.

There are 5 regular honeycombs in H⁵, all paracompact, which include infinite (Euclidean) facets or vertex figures: {3,4,3,3,3}, {3,3,4,3,3}, {3,3,3,4,3}, {3,4,3,3,4}, and {4,3,3,4,3}.

There are no compact regular tessellations of hyperbolic space of dimension 5 or higher and no paracompact regular tessellations in hyperbolic space of dimension 6 or higher.

Since there are no regular star n-polytopes for n ≥ 5, that could be potential cells or vertex figures, there are no more hyperbolic star honeycombs in Hⁿ for n ≥ 5.

There are no regular compact or paracompact tessellations of hyperbolic space of dimension 6 or higher. However, any Schläfli symbol of the form {p,q,r,s,...} not covered above (p,q,r,s,... natural numbers above 2, or infinity) will form a noncompact tessellation of hyperbolic n-space.

The abstract polytopes arose out of an attempt to study polytopes apart from the geometrical space they are embedded in. They include the tessellations of spherical, Euclidean and hyperbolic space, and of other manifolds. There are infinitely many of every rank greater than 1. See [http://www.abstract-polytopes.com/atlas/ this atlas] for a sample. Some notable examples of abstract regular polytopes that do not appear elsewhere in this list are the 11-cell, {3,5,3}, and the 57-cell, {5,3,5}, which have regular projective polyhedra as cells and vertex figures.

The elements of an abstract polyhedron are its body (the maximal element), its faces, edges, vertices and the null polytope or empty set. These abstract elements can be mapped into ordinary space or realised as geometrical figures. Some abstract polyhedra have well-formed or faithful realisations, others do not. A flag is a connected set of elements of each rank - for a polyhedron that is the body, a face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to be regular if its combinatorial symmetries are transitive on its flags - that is to say, that any flag can be mapped onto any other under a symmetry of the polyhedron. Abstract regular polytopes remain an active area of research.

Five such regular abstract polyhedra, which can not be realised faithfully and symmetrically, were identified by H. S. M. Coxeter in his book Regular Polytopes (1977) and again by J. M. Wills in his paper "The combinatorially regular polyhedra of index 2" (1987).{{cite web |url=http://homepages.wmich.edu/~drichter/regularpolyhedra.htm |title=The Regular Polyhedra (of index two) |author=David A. Richter |access-date=2015-03-13 |archive-date=2016-03-04 |archive-url=https://web.archive.org/web/20160304023021/http://homepages.wmich.edu/~drichter/regularpolyhedra.htm |url-status=dead }} They are all topologically equivalent to toroids. Their construction, by arranging n faces around each vertex, can be repeated indefinitely as tilings of the hyperbolic plane. In the diagrams below, the hyperbolic tiling images have colors corresponding to those of the polyhedra images.

:

These occur as dual pairs as follows:

• The medial rhombic triacontahedron and dodecadodecahedron are dual to each other.
• The medial triambic icosahedron and ditrigonal dodecadodecahedron are dual to each other.
• The excavated dodecahedron is self-dual.

• List of regular polytope compounds
• Polygon
• Polyhedron
• Platonic solids
• Kepler–Poinsot solids
• 4-polytope
• Regular 4-polytope (16 regular 4-polytopes, 4 convex and 10 star (Schläfli–Hess))
• Tessellation
• Tilings of regular polygons
• Convex uniform honeycomb
• Regular map (graph theory)

{{notelist-lr}}

{{reflist|group=lower-alpha}}

{{reflist}}

{{refbegin}}

• {{citation

| last = Coxeter | first = H. S. M. | author-link = Harold Scott MacDonald Coxeter

| chapter = Chapter 10: Regular Honeycombs in Hyperbolic Space |pages = 199–214

| isbn = 0-486-40919-8

| mr = 1717154

| lccn = 99035678

| publisher = Dover Publications, Inc. | location = Mineola, NY

| title = The Beauty of Geometry: Twelve Essays

| year = 1999}}. See in particular Summary Tables II, III, IV, V, pp. 212–213.

• Originally published in {{citation

|ref = none

|last = Coxeter

|first = H. S. M.

|author-link = Harold Scott MacDonald Coxeter

|contribution = Regular honeycombs in hyperbolic space

|mr = 0087114

|pages = 155–169

|publisher = North-Holland Publishing Co.

|location = Amsterdam

|title = Proceedings of the International Congress of Mathematicians, 1954, Amsterdam

|volume = III

|contribution-url = http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf

|archive-url = https://web.archive.org/web/20150402180536/http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf

|archive-date = 2015-04-02

|url-status = dead

|year = 1956

}}.

• {{cite book |last=Coxeter |first=H. S. M. |author-link=Harold Scott MacDonald Coxeter |date=1973 |orig-year=1948 |title=Regular Polytopes |location=New York |publisher=Dover Publications |edition=Third |oclc=798003 |isbn=0-486-61480-8 |mr=0370327}} See in particular Tables I and II: Regular polytopes and honeycombs, pp. 294–296.
• {{citation

| last = Johnson

| first = Norman W.

| contribution = Regular inversive polytopes

| pages = 85–95 [http://www.foibg.com/conf/ITA2012/MDA2012_Abstracts_Book.pdf Paper 27]

| title = International Conference on Mathematics of Distances and Applications (July 2–5, 2012, Varna, Bulgaria)

| contribution-url = http://www.foibg.com/ibs_isc/ibs-25/ibs-25-p07.pdf

| year = 2012

| access-date = 2015-01-12

| archive-date = 2017-08-08

| archive-url = https://web.archive.org/web/20170808204226/http://foibg.com/ibs_isc/ibs-25/ibs-25-p07.pdf

| url-status = live

}}

• {{citation

| last1 = McMullen

| first1 = Peter

| author1-link = Peter McMullen

| last2 = Schulte

| first2 = Egon

| author2-link = Egon Schulte

| doi = 10.1017/CBO9780511546686

| isbn = 0-521-81496-0

| mr = 1965665

| publisher = Cambridge University Press

| location = Cambridge

| series = Encyclopedia of Mathematics and its Applications

| title = Abstract Regular Polytopes

| volume = 92

| year = 2002

| s2cid = 115688843

| url-access = registration

| url = https://archive.org/details/abstractregularp0000mcmu

}}

• {{citation|first=Peter|last=McMullen|title=New Trends in Intuitive Geometry |chapter=New Regular Compounds of 4-Polytopes |series=Bolyai Society Mathematical Studies|volume=27|pages=307–320|year=2018|doi=10.1007/978-3-662-57413-3_12|isbn=978-3-662-57412-6}}.
• {{cite arXiv |eprint=1511.02851 |title=Visualizing Hyperbolic Honeycombs |first1=Roice |author2-link=Henry Segerman |last1=Nelson |first2=Henry |last2=Segerman |date=2015|class=math.HO }} [http://hyperbolichoneycombs.org hyperbolichoneycombs.org/] {{Webarchive|url=https://web.archive.org/web/20160304215803/http://hyperbolichoneycombs.org/ |date=2016-03-04 }}
• {{citation

| last = Sommerville | first = D. M. Y. | author-link = Duncan MacLaren Young Sommerville

| mr = 0100239

| publisher = Dover Publications, Inc. | location = New York

| title = An Introduction to the Geometry of {{mvar|n}} Dimensions

| year = 1958}}. Reprint of 1930 ed., published by E. P. Dutton. See in particular Chapter X: The Regular Polytopes.

{{refend}}

• [http://www.math.utah.edu/~alfeld/math/polyhedra/polyhedra.html The Platonic Solids]
• [http://www.georgehart.com/virtual-polyhedra/kepler-poinsot-info.html Kepler-Poinsot Polyhedra]
• [https://web.archive.org/web/20110717184828/http://weimholt.com/andrew/polytope.shtml Regular 4d Polytope Foldouts]
• [https://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html Multidimensional Glossary (Look up Hexacosichoron and Hecatonicosachoron)]
• [http://www.stat.berkeley.edu/~evans/shapiro/TesseractApplet.html Polytope Viewer]
• [https://web.archive.org/web/20060615201915/http://presh.com/hovinga/ Polytopes and optimal packing of p points in n dimensional spheres]
• [http://www.abstract-polytopes.com/atlas/ An atlas of small regular polytopes]
• [https://www.fields.utoronto.ca/programs/scientific/11-12/discretegeom/gradcourses/Course_notes.pdf Regular polyhedra through time] I. Hubard, Polytopes, Maps and their Symmetries
• [http://nan.ma/star/#/ Regular Star Polytopes], Nan Ma

{{Polytopes}}

{{Honeycombs}}

{{Tessellation}}

{{DEFAULTSORT:Polytopes, Regular}}

Regular

Category:Multi-dimensional geometry

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