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. (The 5 vertices form 5 tetrahedral
cells face-bonded to each other, with a total of 10 edges and 10 triangular faces.) 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 (just as the elements of a regular simplex are also regular simplexes). Each tetrahedral cell of a 4-orthoscheme is a
3-orthoscheme, and each triangular face is a 2-orthoscheme (a right triangle). 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 (also called the
tesseract or
8-cell), 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 (the lengths of the 4-cube's edges and its various diagonals). Therefore this 4-orthoscheme fits within the 4-cube, and the 4-cube (like every regular convex polytope) can be
dissected into instances of its characteristic orthoscheme. . Three are left-handed and three are right handed. A left and a right meet at each square face.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 (the fifth vertex of the 5-cell). 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 (which is outside the 3-cube and does not appear in the illustration at all). 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 (not of the illustrated 3-cube, but of another of the tesseract's eight 3-cubes). The third additional edge is a diagonal of a 3-cube (again, not the original illustrated 3-cube). The fourth additional edge (at the other end of the orthogonal path) is a
long diameter of the tesseract itself, of length . It reaches through the exact center of the tesseract to the
antipodal vertex (a vertex of the opposing 3-cube), 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 (from 90 degrees on down). 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 (4-orthoscheme) of the regular 5-cell has four more edges than its base characteristic tetrahedron (3-orthoscheme), which join the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 5-cell). 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 \sqrt{\tfrac{5}{2}}, its characteristic 5-cell's ten edges have lengths \sqrt{\tfrac{1}{10}}, \sqrt{\tfrac{1}{30}}, \sqrt{\tfrac{2}{15}} around its exterior right-triangle face (the edges opposite the
characteristic angles 𝟀, 𝝉, 𝟁), plus \sqrt{\tfrac{3}{20}}, \sqrt{\tfrac{1}{20}}, \sqrt{\tfrac{1}{60}} (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the
characteristic radii of the regular tetrahedron), plus \sqrt{1}, \sqrt{\tfrac{3}{8}}, \sqrt{\tfrac{1}{6}}, \sqrt{\tfrac{1}{16}} (edges which are the characteristic radii of the regular 5-cell). The 4-edge path along orthogonal edges of the orthoscheme is \sqrt{\tfrac{1}{30}}, \sqrt{\tfrac{2}{15}}, \sqrt{\tfrac{1}{60}}, \sqrt{\tfrac{1}{16}}, 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.
Isometries There are many lower symmetry forms of the 5-cell, including these found as uniform polytope
vertex figures: The
tetrahedral pyramid is a special case of a
5-cell, a
polyhedral pyramid, constructed as a regular
tetrahedron base in a 3-space
hyperplane, and an
apex point
above the hyperplane. The four
sides of the pyramid are made of
triangular pyramid cells. Many
uniform 5-polytopes have
tetrahedral pyramid vertex figures with
Schläfli symbols ( )∨{3,3}. Other uniform 5-polytopes have irregular 5-cell vertex figures. The symmetry of a vertex figure of a
uniform polytope is represented by removing the ringed nodes of the Coxeter diagram. == Construction ==