Four-dimensional polyhedron

A 4-dimensional polyhedron is a closed four-dimensional figure. It consists of vertices (corner points), edges , faces, and . A cell is a three-dimensional analogue of a face and is a (3-dimensional) polyhedron . Each (2-dimensional) face must connect exactly two cells, in the same way as the edges of a three-dimensional polyhedron connect exactly two faces. Like other polyhedra, elements of a 4-dimensional polyhedron cannot be divided into two or more sets, which are also 4-polyhedra, that is, it is not composite.

The most famous 4-dimensional polyhedron is a tesseract (hypercube), a four-dimensional analogue of a cube.

4-dimensional polyhedra cannot be represented in three-dimensional space due to excess dimension. A number of techniques are used for visualization.

Orthogonal projection

Orthogonal projections can be used to show different symmetries of a 4-dimensional polyhedron. Projections can be represented as two-dimensional graphs, or can be represented as three-dimensional bodies as .

Perspective projection

Just like three-dimensional shapes can be projected onto a flat sheet, 4-dimensional shapes can be projected into 3-dimensional space, or even onto a plane. A common type of projection is the Schlegel diagram , which uses the stereographic projection of points on the surface of a 3-sphere in three-dimensional space, connected in 3-dimensional space by straight edges, faces and cells.

Slice

Just as a cut of a polyhedron reveals a cut surface, a cut of a 4-dimensional polyhedron gives a “hypersurface” in three-dimensional space. The sequence of such slices can be used to understand the whole figure. Extra dimension can be equated to time for the formation of animation of these sections.

Reamers

The development of a 4-dimensional polyhedron consists of polyhedral connected by faces and located in three-dimensional space, just as the polygonal development faces of a three-dimensional polyhedron are connected by edges and are all in the same plane.

The topology of any given 4-polyhedron is determined by its Betty numbers and ^[3] .

The value of the Euler characteristic used to characterize polyhedra is not properly generalized to higher dimensions and is equal to zero for all 4-dimensional polytopes, whatever the underlying topology. This mismatch of the Euler characteristics for a reliable distinction between different topologies in high dimensions leads to the appearance of more refined Betty numbers ^[3] .

Similarly, the concept of orientability of a polyhedron is not enough to characterize the twisting of surfaces of toroidal polyhedra, which leads to the use of torsion coefficients ^[3] .

Criteria

4-dimensional polyhedra can be classified by properties, such as “ convexity ” and “ symmetry ” ^[3] .

• A 4-dimensional polyhedron is convex if its boundaries (including cells, (3-dimensional) faces and edges) do not intersect themselves (in principle, the faces of a polyhedron can pass inside the shell) and the segments connecting any two points of the four-dimensional polyhedron are completely inside him .. Otherwise, the polyhedron is considered non-convex . Self-intersecting 4-dimensional polyhedra are also known as star-shaped polyhedra by analogy with star- like forms of non - convex Kepler-Poinsot polyhedra .
• A 4-dimensional polyhedron is correct if it is transitive with respect to its flags . This means that all its cells are congruent regular polyhedra , and also all its vertex figures are congruent to another kind of regular polyhedra.
• A convex 4-polyhedron is semiregular if it has a symmetry group such that all vertices are equivalent ( vertex transitive ) and the cells are regular polyhedra . Cells can be two or more types, provided that they have the same kind of faces. There are only 3 such figures found by in 1900 - a , and .
• A 4-polyhedron is if it has a symmetry group in which all vertices are equivalent and the cells are . The faces (2-dimensional) of a homogeneous 4-polyhedron must be regular polygons .
• A 4-polyhedron is an ^[4] if it is vertex transitive and has edges of the same length. That is, inhomogeneous cells, for example, convex Johnson polyhedra , are allowed.
• The regular 4-dimensional polyhedron, which is also convex , is spoken of as the .
• A 4-dimensional polyhedron is prismatic if it is a direct product of two or more polyhedra of smaller dimension. A prismatic 4-dimensional polyhedron is homogeneous if its factors in the direct product are homogeneous. The hypercube is prismatic (the product of two squares or a cube and a segment ), but is considered separately, since it has higher symmetry than the symmetries inherited from the factors.
• a mosaic or honeycomb in three-dimensional space is the decomposition of three - dimensional Euclidean space into a repeating polyhedral cells. Such mosaics or tilings are infinite and not limited by the “4D” volume, so they are examples of infinite 4-polyhedra. A homogeneous mosaic of 3-dimensional space is a mosaic in which the vertices are congruent and connected by a crystallographic group , and the cells are .

Classes

The following list of different categories of 4-dimensional polyhedra is classified according to the criteria set out above:

( vertex-transitive ):

• Convex homogeneous 4-dimensional polyhedra (64, plus two infinite families)
  • 47 non-prismatic convex homogeneous 4-dimensional polyhedra include:
    • 6 regular 4-dimensional polyhedra
  • :
    • {} × {p, q}: 18 (including cubic hyperprisms, regular hypercubes )
    • Prisms built on antiprisms (endless family)
    • {p} × {q}: Duoprisms (infinite family)
• Non-convex homogeneous 4-dimensional polyhedra (10 + unknown)
   

  With its 600 vertices, the is the largest of 10 regular star-shaped 4-dimensional polyhedra
  • 10 (correct)
  • 57 hyperprisms built on non-convex homogeneous polyhedra
  • An unknown number of non-convex homogeneous 4-dimensional polyhedra - and other co-authors found 1849 polyhedra (convex and star-shaped), all built on vertex shapes using the program ^[5]

Other convex 4-dimensional polyhedra :

Infinite homogeneous 4-dimensional polyhedra in Euclidean 3-dimensional space (homogeneous tilings by convex homogeneous cells)

• 28 (homogeneous convex tilings), including:
  • 1 correct tiling, : {4,3,4}

Infinite homogeneous 4-polyhedra of hyperbolic 3-dimensional space (homogeneous tilings by convex homogeneous cells)

• 76 Withhoff , including:
  • 4 regular tilings of a compact hyperbolic 3-dimensional space : {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}

Dual ( ):

• 41 unique dual homogeneous 4-dimensional polyhedra
• 17 unique dual homogeneous multifaceted prisms
• infinite family of dual convex homogeneous duoprisms (with irregular tetrahedral cells)
• 27 unique dual homogeneous cells, including:

Others:

• periodic cell-filling cells with irregular cells

:

These categories include only 4-dimensional polygons with a high degree of symmetry. Many other 4-dimensional polyhedra may exist, but they have not been studied as intensively as the ones listed above.

• The correct four-dimensional polyhedron
• The 3-sphere is another widely discussed figure located in four-dimensional space. But it is not a 4-dimensional polyhedron, since it is not limited to polyhedral cells.
• is a figure in 4-dimensional space associated with duoprisms , although it is also not a polyhedron.

Literature

• T. Vialar. Complex and Chaotic Nonlinear Dynamics: Advances in Economics and Finance. - Springer, 2009 .-- S. 674. - ISBN 978-3-540-85977-2 .
• V. Capecchi, P. Capecchi, M. Buscema, B. D'Amore. Applications of Mathematics in Models, Artificial Neural Networks and Arts. - Springer, 2010 .-- S. 598. - ISBN 978-90-481-8580-1 . - DOI : 10.1007 / 978-90-481-8581-8 .
• HSM Coxeter :
  • HSM Coxeter, MS Longuet-Higgins, JCP Miller: Uniform Polyhedra , Philosophical Transactions of the Royal Society of London, Londne, 1954
  • HSM Coxeter . . - 3rd (1947, 63, 73). - New York: Dover Publications Inc., 1973. - ISBN 0-486-61480-8 .
• HSM Coxeter . Kaleidoscopes: Selected Writings of HSM Coxeter / F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss. - Wiley-Interscience Publication, 1995. - ISBN 978-0-471-01003-6 .
  • (Paper 22) HSM Coxeter, Regular and Semi Regular Polytopes I , [Math. Zeit. 46 (1940) 380-407, MR 2.10]
  • (Paper 23) HSM Coxeter, Regular and Semi-Regular Polytopes II , [Math. Zeit. 188 (1985) 559-591]
  • (Paper 24) HSM Coxeter, Regular and Semi-Regular Polytopes III , [Math. Zeit. 200 (1988) 3-45]
• JH Conway , MJT Guy. Proceedings of the Colloquium on Convexity at Copenhagen. - 1965. - S. 38-39.
• The Theory of Uniform Polytopes and Honeycombs. - Ph.D. Dissertation. - University of Toronto, 1966.
• Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation [1]

• Weisstein, Eric W. Polychoron on Wolfram MathWorld .
• Weisstein, Eric W. Polyhedral formula on the Wolfram MathWorld website.
• Weisstein, Eric W. Regular polychoron Euler characteristics ( Wolfram MathWorld) . *
• Four dimensional figures page
• Polychoron on Glossary for Hyperspace
• Uniform Polychora , Jonathan Bowers
• Uniform polychoron Viewer - Java3D Applet with sources
• Dr. R. Klitzing, polychora