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Graphs of six convex regular 4-polytopes
{3,3,3} {3,3,4} {4,3,3}
4-simplex t0.svg
4-cube t3.svg
4-cube t0.svg
{3,4,3} {5,3,3} {3,3,5}
24-cell t0 F4.svg
120-cell graph H4.svg
600-cell graph H4.svg

In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope.[1][2] 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.


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.


Example presentations of a 24-cell
Sectioning Net
24cell section anim.gif Polychoron 24-cell net.png
Schlegel 2D orthogonal 3D orthogonal
Schlegel wireframe 24-cell.png 24-cell t0 F4.svg Orthogonal projection envelopes 24-cell.png

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.


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.


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[edit]

The topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients.[3]

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.[3]

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.[3]



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


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

Uniform 4-polytope (vertex-transitive):

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)

Dual uniform 4-polytope (cell-transitive):


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.

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.

See also[edit]

  • 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.



  1. ^ Vialar, T. (2009). Complex and Chaotic Nonlinear Dynamics: Advances in Economics and Finance. Springer. p. 674. ISBN 978-3-540-85977-2. 
  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. 
  3. ^ a b c Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
  4. ^ Uniform Polychora, Norman W. Johnson (Wheaton College), 1845 cases in 2005


  • 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]

External links[edit]