5-cell


In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, hypertetrahedron, pentachoron, pentatope, pentahedroid, tetrahedral pyramid, or 4-simplex, the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.
The regular 5-cell is bounded by five regular tetrahedra, and is one of the six regular convex 4-polytopes. A regular 5-cell can be constructed from a regular tetrahedron by adding a fifth vertex one edge length distant from all the vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: Make 10 equilateral triangles, all of the same size, using 10 matchsticks, where each side of every triangle is exactly one matchstick, and none of the triangles and matchsticks intersect one another. No solution exists in three dimensions.

Properties

The 5-cell is the 4-dimensional simplex, the simplest possible 4-polytope. In other words, the 5-cell is a polychoron analogous to a tetrahedron in high dimension. It is formed by any five points which are not all in the same hyperplane. Any such five points constitute a 5-cell, though not usually a regular 5-cell. The regular 5-cell is not found within any of the other regular convex 4-polytopes except one: the 600-vertex 120-cell is a compound of 120 regular 5-cells.
The 5-cell is self-dual, meaning its dual polytope is 5-cell itself. Its maximal intersection with 3-dimensional space is the triangular prism. Its dichoral angle is.
It is the first in the sequence of 6 convex regular 4-polytopes, in order of volume at a given radius or number of vertexes.
The convex hull of two 5-cells in dual configuration is the disphenoidal 30-cell, dual of the bitruncated 5-cell.

As a configuration

This configuration matrix represents the 5-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 5-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual polytope's matrix is identical to its 180 degree rotation. The k-faces can be read as rows left of the diagonal, while the k-figures are read as rows after the diagonal.
Elementk-facefkf0f1f2f3k-figs
f05464Tetrahedron|
f121033Triangle|
triangle|f233102
Tetrahedron|f34645

All these elements of the 5-cell are enumerated in Branko Grünbaum's Venn diagram of 5 points, which is literally an illustration of the regular 5-cell in [|projection] to the plane.

Geodesics and rotations

The 5-cell has only digon central planes through vertices. It has 10 digon central planes, where each vertex pair is an edge, not an axis, of the 5-cell. Each digon plane is orthogonal to 3 others, but completely orthogonal to none of them. The characteristic isoclinic rotation of the 5-cell has, as pairs of invariant planes, those 10 digon planes and their completely orthogonal central planes, which are 0-gon planes which intersect no vertices of the 5-cell.
Below, a spinning 5-cell is visualized with the fourth dimension squashed and displayed as colour. The Clifford torus is depicted in its rectangular form.

Projections

The A4 Coxeter plane projects the 5-cell into a regular pentagon and pentagram. The A3 Coxeter plane projection of the 5-cell is that of a square pyramid. The A2 Coxeter plane projection of the regular 5-cell is that of a triangular bipyramid with the two opposite vertices centered.

Irregular 5-cells

In the case of simplexes such as the 5-cell, certain irregular forms are in some sense more fundamental than the regular form. Although regular 5-cells cannot fill 4-space or the regular 4-polytopes, there are irregular 5-cells which do. These characteristic 5-cells are the fundamental domains of the different symmetry groups which give rise to the various 4-polytopes.

Orthoschemes

A 4-orthoscheme is a 5-cell where all 10 faces are right triangles. An orthoscheme is an irregular simplex that is the convex hull of a tree in which all edges are mutually perpendicular. In a 4-dimensional orthoscheme, the tree consists of four perpendicular edges connecting all five vertices in a linear path that makes three right-angled turns. The elements of an orthoscheme are also orthoschemes. Each tetrahedral cell of a 4-orthoscheme is a 3-orthoscheme, and each triangular face is a 2-orthoscheme.
Orthoschemes are the characteristic simplexes of the regular polytopes, because each regular polytope is generated by reflections in the bounding facets of its particular characteristic orthoscheme. For example, the special case of the 4-orthoscheme with equal-length perpendicular edges is the characteristic orthoscheme of the 4-cube, the 4-dimensional analogue of the 3-dimensional cube. If the three perpendicular edges of the 4-orthoscheme are of unit length, then all its edges are of length,,, or, precisely the chord lengths of the unit 4-cube. Therefore this 4-orthoscheme fits within the 4-cube, and the 4-cube can be dissected into instances of its characteristic orthoscheme.
A 3-orthoscheme is easily illustrated, but a 4-orthoscheme is more difficult to visualize. A 4-orthoscheme is a tetrahedral pyramid with a 3-orthoscheme as its base. It has four more edges than the 3-orthoscheme, joining the four vertices of the base to its apex. Pick out any one of the 3-orthoschemes of the six shown in the 3-cube illustration. Notice that it touches four of the cube's eight vertices, and those four vertices are linked by a 3-edge path that makes two right-angled turns. Imagine that this 3-orthoscheme is the base of a 4-orthoscheme, so that from each of those four vertices, an unseen 4-orthoscheme edge connects to a fifth apex vertex. Although the four additional edges all reach the same apex vertex, they will all be of different lengths. The first of them, at one end of the 3-edge orthogonal path, extends that path with a fourth orthogonal edge by making a third 90 degree turn and reaching perpendicularly into the fourth dimension to the apex. The second of the four additional edges is a diagonal of a cube face. The third additional edge is a diagonal of a 3-cube. The fourth additional edge is a long diameter of the tesseract itself, of length. It reaches through the exact center of the tesseract to the antipodal vertex, which is the apex. Thus the characteristic 5-cell of the 4-cube has four edges, three edges, two edges, and one edge.
The 4-cube can be dissected into 24 such 4-orthoschemes eight different ways, with six 4-orthoschemes surrounding each of four orthogonal tesseract long diameters. The 4-cube can also be dissected into 384 smaller instances of this same characteristic 4-orthoscheme, just one way, by all of its symmetry hyperplanes at once, which divide it into 384 4-orthoschemes that all meet at the center of the 4-cube.
More generally, any regular polytope can be dissected into g instances of its characteristic orthoscheme that all meet at the regular polytope's center. The number g is the order of the polytope, the number of reflected instances of its characteristic orthoscheme that comprise the polytope when a single mirror-surfaced orthoscheme instance is reflected in its own facets. More generally still, characteristic simplexes are able to fill uniform polytopes because they possess all the requisite elements of the polytope. They also possess all the requisite angles between elements. The characteristic simplexes are the genetic codes of polytopes: like a Swiss Army knife, they contain one of everything needed to construct the polytope by replication.
Every regular polytope, including the regular 5-cell, has its characteristic orthoscheme. There is a 4-orthoscheme which is the characteristic 5-cell of the regular 5-cell. It is a tetrahedral pyramid based on the characteristic tetrahedron of the regular tetrahedron. The regular 5-cell can be dissected into 120 instances of this characteristic 4-orthoscheme just one way, by all of its symmetry hyperplanes at once, which divide it into 120 4-orthoschemes that all meet at the center of the regular 5-cell.
The characteristic 5-cell of the regular 5-cell has four more edges than its base characteristic tetrahedron, which join the four vertices of the base to its apex. The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. If the regular 5-cell has unit radius and edge length, its characteristic 5-cell's ten edges have lengths,, around its exterior right-triangle face, plus,, , plus,,, . The 4-edge path along orthogonal edges of the orthoscheme is,,,, first from a regular 5-cell vertex to a regular 5-cell edge center, then turning 90° to a regular 5-cell face center, then turning 90° to a regular 5-cell tetrahedral cell center, then turning 90° to the regular 5-cell center.