# 4-polytope

(Redirected from Polychora)
Graphs of 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} {5,3,3} {3,3,5}

Octaplex
24-cell

Dodecaplex
120-cell

Tetraplex
600-cell

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 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

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 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".

### Classes

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

The truncated 120-cell is one of 47 convex non-prismatic uniform 4-polytopes

Other convex 4-polytopes:

The regular cubic honeycomb is the only infinite regular 4-polytope in Euclidean 3-dimensional space.

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

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

• 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:

The 11-cell is an abstract regular 4-polytope, existing in the real projective plane, it can be seen by presenting its 11 hemi-icosahedral vertices and cells by index and color.

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.

• 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

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.
4. ^ a b c Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
5. ^ 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
• 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 [1]
• (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 [2]