{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} | {5,3,3} | {3,3,5} |
![]() 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 | |||||||
Graph | |||||||
Vertices | 5 | 8 | 16 | 24 | 120 | 600 | |
Edges | 10 | 24 | 32 | 96 | 720 | 1200 | |
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 | 1 16-cell | 2 16-cells | 3 8-cells | 5 24-cells x 5 | 5 600-cells x 2 | |
Great polygons | 2 𝝅/2 squares x 3 | 4 𝝅/2 rectangles x 3 | 4 𝝅/3 hexagons x 4 | 12 𝝅/5 decagons x 6 | 50 𝝅/15 dodecagons x 4 | ||
Petrie polygons | 1 pentagon | 1 octagon | 2 octagons | 2 dodecagons | 4 30-gons | 20 30-gons | |
Isocline polygons | 1 {8/2}=2{4} x {8/2}=2{4} | 2 {8/2}=2{4} x {8/2}=2{4} | 2 {12/2}=2{6} x {12/6}=6{2} | 4 {30/2}=2{15} x 30{0} | 20 {30/2}=2{15} x 30{0} | ||
Long radius | 1 | 1 | 1 | 1 | 1 | 1 | |
Edge length | √5/√2 ≈ 1.581 | √2 ≈ 1.414 | 1 | 1 | 1/ϕ ≈ 0.618 | 1/√2ϕ2 ≈ 0.270 | |
Short radius | 1/4 | 1/2 | 1/2 | √2/2 ≈ 0.707 | 1 - (√2/2√3φ)2 ≈ 0.936 | 1 - (1/2√3φ)2 ≈ 0.968 | |
Area | 10•√8/3 ≈ 9.428 | 32•√3/4 ≈ 13.856 | 24 | 96•√3/4 ≈ 41.569 | 1200•√3/8φ2 ≈ 99.238 | 720•25+10√5/8φ4 ≈ 621.9 | |
Volume | 5•5√5/24 ≈ 2.329 | 16•1/3 ≈ 5.333 | 8 | 24•√2/3 ≈ 11.314 | 600•1/3√8φ3 ≈ 16.693 | 120•2 + φ/2√8φ3 ≈ 18.118 | |
4-Content | √5/24•(√5/2)4 ≈ 0.146 | 2/3 ≈ 0.667 | 1 | 2 | Short∙Vol/4 ≈ 3.907 | Short∙Vol/4 ≈ 4.385 |
Sectioning | Net | |
---|---|---|
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:
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.
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