4-polytope

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Graphs of six convex regular 4-polytopes
{3,3,3} {3,3,4} {4,3,3}
4-simplex t0.svg
5-cell
Pentatope
4-simplex
4-cube t3.svg
16-cell
Orthoplex
4-orthoplex
4-cube t0.svg
8-cell
Tesseract
4-cube
{3,4,3} {3,3,5} {5,3,3}
24-cell t0 F4.svg
Octaplex
24-cell
120-cell graph H4.svg
Dodecaplex
120-cell
600-cell graph H4.svg
Tetraplex
600-cell

In geometry, a 4-polytope (also called a polychoron) 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 term polychoron (plural polychora), from the Greek roots poly ("many") and choros ("room" or "space") and has been advocated by Norman Johnson and George Olshevsky in the context of uniform polychora,[3] Other names for 4-polytopes include polyhedroid and polycell.

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[edit]

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[edit]

Example presentations of a 24-cell
Sectioning Net
24cell section anim.gif Polychoron 24-cell net.png
Projections
Schlegel 2D orthogonal 3D orthogonal
Schlegel half-solid rectified 16-cell.png 24-cell t0 F4.svg Orthogonal projection envelopes 24-cell.png

4-polytopes can not 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[edit]

The Euler characteristic for any 4-polytope of whatever topology, including a topological 3-sphere, is zero:[4]

χ = V − E + F − C = 0.

For example the hypercube or tesseract {4, 3, 3} has 16 vertices, 32 edges, 24 faces, and 8 cells giving

χ = 16 − 32 + 24 − 8 = 0.

χ = 0 for all polytopes in any even number of dimensions, including 2-polytopes (polygons) and 4-polytopes. This lack of ability for χ to generally distinguish between different topologies led to the discovery of more sophisticated topological characteristics including Betti numbers and torsion coefficients.[4]

Classification[edit]

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

Categories[edit]

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-polytopes (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-polytopes (cell-transitive):

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.

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]

  • Convex 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.

Footnotes[edit]

  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. ^ "Convex and abstract polytopes, Thursday, May 19, 2005 to Saturday, May, 21, 2005", Programme and abstracts, MIT, 2005. [1]
  4. ^ a b Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
  5. ^ http://www.mit.edu/~hlb/Associahedron/program.pdf Uniform Polychora], Norman W. Johnson (Wheaton College), 1845 cases in 2005

References[edit]

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
  • 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 [2]
    • (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 [3]

External links[edit]