{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} | {3,3,5} | {5,3,3} |
![]() 24-cell Octaplex | ![]() 600-cell Tetraplex | ![]() 120-cell Dodecaplex |
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 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853. [4]
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 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 content [5] 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.
Regular convex 4-polytopes | |||||||
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Symmetry group | A4 | B4 | F4 | H4 | |||
Name | 5-cell Hyper-tetrahedron | 16-cell Hyper-octahedron | 8-cell Hyper-cube | 24-cell
| 600-cell Hyper-icosahedron | 120-cell Hyper-dodecahedron | |
Schläfli symbol | {3, 3, 3} | {3, 3, 4} | {4, 3, 3} | {3, 4, 3} | {3, 3, 5} | {5, 3, 3} | |
Coxeter mirrors | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
Mirror dihedrals | 𝝅/3𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2 | 𝝅/3𝝅/3𝝅/4𝝅/2𝝅/2𝝅/2 | 𝝅/4𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2 | 𝝅/3𝝅/4𝝅/3𝝅/2𝝅/2𝝅/2 | 𝝅/3𝝅/3𝝅/5𝝅/2𝝅/2𝝅/2 | 𝝅/5𝝅/3𝝅/3𝝅/2𝝅/2𝝅/2 | |
Graph | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | |
Vertices | 5 tetrahedral | 8 octahedral | 16 tetrahedral | 24 cubical | 120 icosahedral | 600 tetrahedral | |
Edges | 10 triangular | 24 square | 32 triangular | 96 triangular | 720 pentagonal | 1200 triangular | |
Faces | 10 triangles | 32 triangles | 24 squares | 96 triangles | 1200 triangles | 720 pentagons | |
Cells | 5 tetrahedra | 16 tetrahedra | 8 cubes | 24 octahedra | 600 tetrahedra | 120 dodecahedra | |
Tori | 1 5-tetrahedron | 2 8-tetrahedron | 2 4-cube | 4 6-octahedron | 20 30-tetrahedron | 12 10-dodecahedron | |
Inscribed | 120 in 120-cell | 675 in 120-cell | 2 16-cells | 3 8-cells | 25 24-cells | 10 600-cells | |
Great polygons | 2 squares x 3 | 4 rectangles x 4 | 4 hexagons x 4 | 12 decagons x 6 | 100 irregular hexagons x 4 | ||
Petrie polygons | 1 pentagon x 2 | 1 octagon x 3 | 2 octagons x 4 | 2 dodecagons x 4 | 4 30-gons x 6 | 20 30-gons x 4 | |
Long radius | |||||||
Edge length | |||||||
Short radius | |||||||
Area | |||||||
Volume | |||||||
4-Content |
Sectioning | Net | |
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![]() | ![]() | |
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 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.
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.
The topology of any given 4-polytope is defined by its Betti numbers and torsion coefficients. [6]
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. [6]
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. [6]
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:
Uniform 4-polytope (vertex-transitive):
Other convex 4-polytopes:
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):
Others:
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.
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.
In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
In geometry, a uniform 4-polytope is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
In geometry, a truncated 5-cell is a uniform 4-polytope formed as the truncation of the regular 5-cell.
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.
In geometry, a five-dimensional polytope is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.
The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed entirely of triangular prisms.
In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface.
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.
In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.
In four-dimensional geometry, a prismatic uniform 4-polytope is a uniform 4-polytope with a nonconnected Coxeter diagram symmetry group. These figures are analogous to the set of prisms and antiprism uniform polyhedra, but add a third category called duoprisms, constructed as a product of two regular polygons.