Topologically 4-polytopes are closely related to the [[Convex uniform honeycomb|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 [[polyhedral net|nets]] in 3-space.
Topologically 4-polytopes are closely related to the [[Convex uniform honeycomb|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 [[polyhedral net|nets]] in 3-space.
==Definition==
== Definition ==
A 4-polytope is a closed [[Four-dimensional space|four-dimension]]al figure. It comprises [[vertex (geometry)|vertices]] (corner points), [[edge (geometry)|edges]], [[face (geometry)|faces]] and [[cell (mathematics)|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.
A 4-polytope is a closed [[Four-dimensional space|four-dimension]]al figure. It comprises [[vertex (geometry)|vertices]] (corner points), [[edge (geometry)|edges]], [[face (geometry)|faces]] and [[cell (mathematics)|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.
==Geometry==
== Geometry ==
The convex [[regular 4-polytopes]] are the four-dimensional analogues of the [[Platonic solids]]. The most familiar 4-polytope is the [[tesseract]] or hypercube, the 4D analogue of the cube.
The convex [[regular 4-polytopes]] are the four-dimensional analogues of the [[Platonic solids]]. The most familiar 4-polytope is the [[tesseract]] or hypercube, the 4D analogue of the cube.
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.<ref name="richeson"/>
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.<ref name="richeson"/>
==Classification==
== Classification ==
=== Criteria ===
===Criteria===
Like all polytopes, 4-polytopes may be classified based on properties like "[[convex set|convexity]]" and "[[symmetry]]".
Like all polytopes, 4-polytopes may be classified based on properties like "[[convex set|convexity]]" and "[[symmetry]]".
Line 97:
Line 96:
=== Classes ===
=== Classes ===
The following lists the various categories of 4-polytopes classified according to the criteria above:
The following lists the various categories of 4-polytopes classified according to the criteria above:
[[File:Schlegel half-solid truncated 120-cell.png|150px|thumb|The [[truncated 120-cell]] is one of 47 convex non-prismatic uniform 4-polytopes]]
[[File:Schlegel half-solid truncated 120-cell.png|150px|thumb|The [[truncated 120-cell]] is one of 47 convex non-prismatic uniform 4-polytopes]]
Line 144:
Line 142:
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.
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==
== See also ==
*[[Regular 4-polytope]]
*[[Regular 4-polytope]]
*[[3-sphere]] – analogue of a sphere in 4-dimensional space. This is not a 4-polytope, since it is not bounded by polyhedral cells.
*[[3-sphere]] – analogue of a sphere in 4-dimensional space. This is not a 4-polytope, since it is not bounded by polyhedral cells.
** 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, M.S. Longuet-Higgins and [[J.C.P. Miller]]: ''Uniform Polyhedra'', Philosophical Transactions of the Royal Society of London, Londne, 1954
** '''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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
** '''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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{isbn|978-0-471-01003-6}}
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.
Geometry
The convex regular 4-polytopes are the four-dimensional analogues of the Platonic solids. The most familiar 4-polytope is the tesseract or hypercube, the 4D analogue of the cube.
The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more contentTemplate:Sfn within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering.
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.
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 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 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:
Unknown total number of nonconvex uniform 4-polytopes: Norman Johnson and other collaborators have identified 2191 forms (convex and star, excluding the infinite families), all constructed by vertex figures by Stella4D software.[5]
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.
3-sphere – analogue of a sphere 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.
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
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, wiley.com, Template:Isbn
(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
