4-polytope

Last updated October 17, 2024

Graphs of the six convex regular 4-polytopes
{3,3,3}	{3,3,4}	{4,3,3}
5-cell Pentatope 4-simplex	16-cell Orthoplex 4-orthoplex	8-cell Tesseract 4-cube
{3,4,3}	{3,3,5}	{5,3,3}
24-cell Octaplex	600-cell Tetraplex	120-cell Dodecaplex

In geometry, a 4-polytope (sometimes also called a polychoron,^[1]polycell, or polyhedroid) is a four-dimensional polytope.^[2]^[3] It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.^[4]

Definition

A 4-polytope is a closed four-dimensional figure. It comprises vertices (corner points), edges, faces and cells. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Each face must join exactly two cells, analogous to the way in which each edge of a polyhedron joins just two faces. Like any polytope, the elements of a 4-polytope cannot be subdivided into two or more sets which are also 4-polytopes, i.e. it is not a compound.

Geometry

The convex regular 4-polytopes are the four-dimensional analogues of the Platonic solids. The most familiar 4-polytope is the tesseract or hypercube, the 4D analogue of the cube.

The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content^[5] within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering.

Regular convex 4-polytopes
Symmetry group	A₄	B₄		F₄	H₄
Name	5-cell Hyper-tetrahedron 5-point	16-cell Hyper-octahedron 8-point	8-cell Hyper-cube 16-point	24-cell 24-point	600-cell Hyper-icosahedron 120-point	120-cell Hyper-dodecahedron 600-point
Schläfli symbol	{3, 3, 3}	{3, 3, 4}	{4, 3, 3}	{3, 4, 3}	{3, 3, 5}	{5, 3, 3}
Coxeter mirrors
Mirror dihedrals	⁠𝝅/3⁠⁠𝝅/3⁠⁠𝝅/3⁠⁠𝝅/2⁠⁠𝝅/2⁠⁠𝝅/2⁠	⁠𝝅/3⁠⁠𝝅/3⁠⁠𝝅/4⁠⁠𝝅/2⁠⁠𝝅/2⁠⁠𝝅/2⁠	⁠𝝅/4⁠⁠𝝅/3⁠⁠𝝅/3⁠⁠𝝅/2⁠⁠𝝅/2⁠⁠𝝅/2⁠	⁠𝝅/3⁠⁠𝝅/4⁠⁠𝝅/3⁠⁠𝝅/2⁠⁠𝝅/2⁠⁠𝝅/2⁠	⁠𝝅/3⁠⁠𝝅/3⁠⁠𝝅/5⁠⁠𝝅/2⁠⁠𝝅/2⁠⁠𝝅/2⁠	⁠𝝅/5⁠⁠𝝅/3⁠⁠𝝅/3⁠⁠𝝅/2⁠⁠𝝅/2⁠⁠𝝅/2⁠
Graph
Vertices	5 tetrahedral	8 octahedral	16 tetrahedral	24 cubical	120 icosahedral	600 tetrahedral
Edges	10 triangular	24 square	32 triangular	96 triangular	720 pentagonal	1200 triangular
Faces	10 triangles	32 triangles	24 squares	96 triangles	1200 triangles	720 pentagons
Cells	5 tetrahedra	16 tetrahedra	8 cubes	24 octahedra	600 tetrahedra	120 dodecahedra
Tori	1 5-tetrahedron	2 8-tetrahedron	2 4-cube	4 6-octahedron	20 30-tetrahedron	12 10-dodecahedron
Inscribed	120 in 120-cell	675 in 120-cell	2 16-cells	3 8-cells	25 24-cells	10 600-cells
Great polygons		2 squares x 3	4 rectangles x 4	4 hexagons x 4	12 decagons x 6	100 irregular hexagons x 4
Petrie polygons	1 pentagon x 2	1 octagon x 3	2 octagons x 4	2 dodecagons x 4	4 30-gons x 6	20 30-gons x 4
Long radius	$1$	$1$	$1$	$1$	$1$	$1$
Edge length	${\sqrt {\tfrac {5}{2}}}\approx 1.581$	${\sqrt {2}}\approx 1.414$	$1$	$1$	${\tfrac {1}{\phi }}\approx 0.618$	${\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270$
Short radius	${\tfrac {1}{4}}$	${\tfrac {1}{2}}$	${\tfrac {1}{2}}$	${\sqrt {\tfrac {1}{2}}}\approx 0.707$	${\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926$	${\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926$
Area	$10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825$	$32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713$	$24$	$96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569$	$1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48$	$720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366$
Volume	$5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329$	$16\left({\tfrac {1}{3}}\right)\approx 5.333$	$8$	$24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314$	$600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693$	$120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118$
4-Content	${\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146$	${\tfrac {2}{3}}\approx 0.667$	$1$	$2$	${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863$	${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193$

Visualisation

Example presentations of a 24-cell
Sectioning		Net

Projections
Schlegel	2D orthogonal	3D orthogonal

4-polytopes cannot be seen in three-dimensional space due to their extra dimension. Several techniques are used to help visualise them.

Orthogonal projection

Orthogonal projections can be used to show various symmetry orientations of a 4-polytope. They can be drawn in 2D as vertex-edge graphs, and can be shown in 3D with solid faces as visible projective envelopes.

Perspective projection

Just as a 3D shape can be projected onto a flat sheet, so a 4-D shape can be projected onto 3-space or even onto a flat sheet. One common projection is a Schlegel diagram which uses stereographic projection of points on the surface of a 3-sphere into three dimensions, connected by straight edges, faces, and cells drawn in 3-space.

Sectioning

Just as a slice through a polyhedron reveals a cut surface, so a slice through a 4-polytope reveals a cut "hypersurface" in three dimensions. A sequence of such sections can be used to build up an understanding of the overall shape. The extra dimension can be equated with time to produce a smooth animation of these cross sections.

Nets

A net of a 4-polytope is composed of polyhedral cells that are connected by their faces and all occupy the same three-dimensional space, just as the polygon faces of a net of a polyhedron are connected by their edges and all occupy the same plane.

Topological characteristics

The tesseract as a Schlegel diagram Hypercube.svg — The tesseract as a Schlegel diagram

The topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients.^[6]

The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 4-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.^[6]

Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal 4-polytopes, and this led to the use of torsion coefficients.^[6]

Classification

Criteria

Like all polytopes, 4-polytopes may be classified based on properties like "convexity" and "symmetry".

A 4-polytope is convex if its boundary (including its cells, faces and edges) does not intersect itself and the line segment joining any two points of the 4-polytope is contained in the 4-polytope or its interior; otherwise, it is non-convex. Self-intersecting 4-polytopes are also known as star 4-polytopes, from analogy with the star-like shapes of the non-convex star polygons and Kepler–Poinsot polyhedra.
A 4-polytope is regular if it is transitive on its flags. This means that its cells are all congruent regular polyhedra, and similarly its vertex figures are congruent and of another kind of regular polyhedron.
A convex 4-polytope is semi-regular if it has a symmetry group under which all vertices are equivalent (vertex-transitive) and its cells are regular polyhedra. The cells may be of two or more kinds, provided that they have the same kind of face. There are only 3 cases identified by Thorold Gosset in 1900: the rectified 5-cell, rectified 600-cell, and snub 24-cell.
A 4-polytope is uniform if it has a symmetry group under which all vertices are equivalent, and its cells are uniform polyhedra. The faces of a uniform 4-polytope must be regular.
A 4-polytope is scaliform if it is vertex-transitive, and has all equal length edges. This allows cells which are not uniform, such as the regular-faced convex Johnson solids.
A regular 4-polytope which is also convex is said to be a convex regular 4-polytope.
A 4-polytope is prismatic if it is the Cartesian product of two or more lower-dimensional polytopes. A prismatic 4-polytope is uniform if its factors are uniform. The hypercube is prismatic (product of two squares, or of a cube and line segment), but is considered separately because it has symmetries other than those inherited from its factors.
A tiling or honeycomb of 3-space is the division of three-dimensional Euclidean space into a repetitive grid of polyhedral cells. Such tilings or tessellations are infinite and do not bound a "4D" volume, and are examples of infinite 4-polytopes. A uniform tiling of 3-space is one whose vertices are congruent and related by a space group and whose cells are uniform polyhedra.

Classes

The following lists the various categories of 4-polytopes classified according to the criteria above:

Uniform 4-polytope (vertex-transitive):

Convex uniform 4-polytopes (64, plus two infinite families)
- 47 non-prismatic convex uniform 4-polytope including:
  - 6 Convex regular 4-polytope
- Prismatic uniform 4-polytopes:
  - {} × {p,q} : 18 polyhedral hyperprisms (including cubic hyperprism, the regular hypercube)
  - Prisms built on antiprisms (infinite family)
  - {p} × {q} : duoprisms (infinite family)
Non-convex uniform 4-polytopes (10 + unknown)

The great grand stellated 120-cell is the largest of 10 regular star 4-polytopes, having 600 vertices.
- 10 (regular) Schläfli-Hess polytopes
- 57 hyperprisms built on nonconvex uniform polyhedra
- Unknown total number of nonconvex uniform 4-polytopes: Norman Johnson and other collaborators have identified 2191 forms (convex and star, excluding the infinite families), all constructed by vertex figures by Stella4D software.^[7]

Other convex 4-polytopes:

Polyhedral pyramid
Polyhedral bipyramid
Polyhedral prism

Infinite uniform 4-polytopes of Euclidean 3-space (uniform tessellations of convex uniform cells)

28 convex uniform honeycombs: uniform convex polyhedral tessellations, including:
- 1 regular tessellation, cubic honeycomb: {4,3,4}

Infinite uniform 4-polytopes of hyperbolic 3-space (uniform tessellations of convex uniform cells)

76 Wythoffian convex uniform honeycombs in hyperbolic space, including:
- 4 regular tessellation of compact hyperbolic 3-space: {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}

Dual uniform 4-polytope (cell-transitive):

41 unique dual convex uniform 4-polytopes
17 unique dual convex uniform polyhedral prisms
infinite family of dual convex uniform duoprisms (irregular tetrahedral cells)
27 unique convex dual uniform honeycombs, including:
- Rhombic dodecahedral honeycomb
- Disphenoid tetrahedral honeycomb

Others:

Weaire–Phelan structure periodic space-filling honeycomb with irregular cells

Abstract regular 4-polytopes :

These categories include only the 4-polytopes that exhibit a high degree of symmetry. Many other 4-polytopes are possible, but they have not been studied as extensively as the ones included in these categories.

Related Research Articles

In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.

<span class="mw-page-title-main">Cuboctahedron</span> Polyhedron with 8 triangular faces and 6 square faces

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron.

<span class="mw-page-title-main">Polyhedron</span> 3D shape with flat faces, straight edges and sharp corners

In geometry, a polyhedron is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices.

In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.

<span class="mw-page-title-main">Schläfli symbol</span> Notation that defines regular polytopes and tessellations

In geometry, the Schläfli symbol is a notation of the form $that defines regular polytopes and tessellations.$

<span class="mw-page-title-main">Vertex figure</span> Shape made by slicing off a corner of a polytope

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

In geometry, a uniform 4-polytope is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.

The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

In geometry, a truncated 5-cell is a uniform 4-polytope formed as the truncation of the regular 5-cell.

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.

In geometry, a five-dimensional polytope is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.

In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the same must also be true within each lower-dimensional face of the polytope. In two dimensions a stronger definition is used: only the regular polygons are considered as uniform, disallowing polygons that alternate between two different lengths of edges.

In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface.

In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.

<span class="mw-page-title-main">Regular 4-polytope</span> Four-dimensional analogues of the regular polyhedra in three dimensions

In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.

In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.

In four-dimensional geometry, a prismatic uniform 4-polytope is a uniform 4-polytope with a nonconnected Coxeter diagram symmetry group. These figures are analogous to the set of prisms and antiprism uniform polyhedra, but add a third category called duoprisms, constructed as a product of two regular polygons.

References

Notes

↑ N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.1 Polytopes and Honeycombs, p.224
↑ Vialar, T. (2009). Complex and Chaotic Nonlinear Dynamics: Advances in Economics and Finance. Springer. p. 674. ISBN 978-3-540-85977-2.
↑ Capecchi, V.; Contucci, P.; Buscema, M.; D'Amore, B. (2010). Applications of Mathematics in Models, Artificial Neural Networks and Arts. Springer. p. 598. doi:10.1007/978-90-481-8581-8. ISBN 978-90-481-8580-1.
↑ Coxeter 1973, p. 141, §7-x. Historical remarks.
↑ Coxeter 1973, pp. 292–293, Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions: [An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.]
1 2 3 Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
↑ Uniform Polychora, Norman W. Johnson (Wheaton College), 1845 cases in 2005

Bibliography

H.S.M. Coxeter:
- Coxeter, H.S.M. (1973) [1948]. Regular Polytopes (3rd ed.). New York: Dover.
- H.S.M. Coxeter, M.S. Longuet-Higgins and J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
  - (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
  - (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
  - (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation Archived 2005-03-22 at the Wayback Machine

External links

Weisstein, Eric W. "Polychoron". MathWorld .
Weisstein, Eric W. "Polyhedral formula". MathWorld .
Weisstein, Eric W. "Regular polychoron Euler characteristics". MathWorld .
Uniform Polychora, Jonathan Bowers
Uniform polychoron Viewer - Java3D Applet with sources
R. Klitzing, polychora

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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[1] N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.1 Polytopes and Honeycombs, p.224

[2] Vialar, T. (2009). Complex and Chaotic Nonlinear Dynamics: Advances in Economics and Finance. Springer. p. 674. ISBN 978-3-540-85977-2.

[3] Capecchi, V.; Contucci, P.; Buscema, M.; D'Amore, B. (2010). Applications of Mathematics in Models, Artificial Neural Networks and Arts. Springer. p. 598. doi:10.1007/978-90-481-8581-8. ISBN 978-90-481-8580-1.

[FOOTNOTECoxeter1973141§7-x._Historical_remarks-4] Coxeter 1973, p. 141, §7-x. Historical remarks.

[FOOTNOTECoxeter1973292–293Table_I(ii):_The_sixteen_regular_polytopes_{''p,q,r''}_in_four_dimensions-5] Coxeter 1973, pp. 292–293, Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions: [An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.]

[richeson-6] 1 2 3 Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.

[7] Uniform Polychora, Norman W. Johnson (Wheaton College), 1845 cases in 2005

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