# 4-polytope

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.

{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} | {5,3,3} | {3,3,5} |

Octaplex 24-cell |
Dodecaplex 120-cell |
Tetraplex 600-cell |

The two-dimensional analogue of a 4-polytope is a polygon, and the three-dimensional analogue is a polyhedron.

Topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space; similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be *cut and unfolded* as nets in 3-space.

## 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.

The most familiar 4-polytope is the tesseract or hypercube, the 4D analogue of the cube.

## Visualisation

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

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.[4]

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.[4]

## 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:
- 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)- 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 1849 known cases (convex and star), all constructed by vertex figures by Stella4D software.[5]

**Other convex 4-polytopes**:

**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:

**Others:**

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

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.

## See also

- Regular 4-polytope
- The 3-sphere (or glome) is another commonly discussed figure that resides in 4-dimensional space. This is not a 4-polytope, since it is not bounded by polyhedral cells.
- The duocylinder is a figure in 4-dimensional space related to the duoprisms. It is also not a 4-polytope because its bounding volumes are not polyhedral.

## 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. - 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:
- 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 - H.S.M. Coxeter,
*Regular Polytopes*, 3rd Edition, Dover New York, 1973

- H. S. M. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller:
**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]

- (Paper 22) H.S.M. Coxeter,
- 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

## External links

Wikimedia Commons has media related to .Polychora |

- Weisstein, Eric W. "Polychoron".
*MathWorld*. - Weisstein, Eric W. "Polyhedral formula".
*MathWorld*. - Weisstein, Eric W. "Regular polychoron Euler characteristics".
*MathWorld*. - Four dimensional figures page, George Olshevsky.
- Olshevsky, George. "Polychoron".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007. - Uniform Polychora, Jonathan Bowers
- Uniform polychoron Viewer - Java3D Applet with sources
- Dr. R. Klitzing, polychora