Төрт өлшемді тетраэдр және Паскаль үшбұрышының бесінші элементтерінің тізбегі

Four dimensional analogue of the tetrahedron
the sequence of fifth element numbers of Pascal's triangle

In geometry, the 5 cell is the convex 4 polytope with Schläfli symbol {3,3,3}. It is a 5 vertex four dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is the 4 simplex (Coxeter's polytope), 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 (the four dimensional analogues of the Platonic solids). 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.

The 5 cell is the 4 dimensional simplex, the simplest possible 4 polytope. As such it is the first in the sequence of 6 convex regular 4 polytopes (in order of size and complexity). A 5 cell is formed by any five points which are not all in the same hyperplane (as a tetrahedron is formed by any four points which are not all in the same plane, and a triangle is formed by any three points which are not all in the same line). 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.

When a net of five tetrahedra is folded up in 4 dimensional space such that each tetrahedron is face bonded to the other four, the resulting 5 cell has a total of 5 vertices, 10 edges and 10 faces. Four edges meet at each vertex, and three tetrahedral cells meet at each edge. The 5 cell is self dual (as are all simplexes), and its vertex figure is the tetrahedron. Its maximal intersection with 3 dimensional space is the triangular prism. Its dihedral angle is cos−1 , or approximately 75.52°. The convex hull of two 5 cells in dual configuration is the disphenoidal 30 cell, dual of the bitruncated 5 cell.

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 5 cell vertices. There are only two ways to make a circuit of the 5 cell through all 5 vertices along 5 edges, so there are two discrete Hopf fibrations of the great digons of the 5 cell. Each of the two fibrations corresponds to a left right pair of isoclinic rotations which each rotate all 5 vertices in a circuit of period 5. The 5 cell has only two distinct period 5 isoclines (those circles through all 5 vertices), each of which acts as the single isocline of a right rotation and the single isocline of a left rotation in two different fibrations. Below, a spinning 5 cell is visualized with the fourth dimension squashed and displayed as colour. The Clifford torus is depicted in its rectangular (wrapping) form.

A 5 cell can be constructed as a Boerdijk–Coxeter helix of five chained tetrahedra, folded into a 4 dimensional ring. The 10 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex, although folding into 4 dimensions causes edges to coincide. The purple edges form a regular pentagon which is the Petrie polygon of the 5 cell. The blue edges connect every second vertex, forming a pentagram which is the Clifford polygon of the 5 cell. The pentagram's blue edges are the chords of the 5 cell's isocline, the circular rotational path its vertices take during an isoclinic rotation, also known as a Clifford displacement.

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 (two tetrahedra joined face to face) with the two opposite vertices centered. Projections to 3 dimensionsThe vertex first projection of the 5 cell into 3 dimensions has a tetrahedral projection envelope. The closest vertex of the 5 cell projects to the center of the tetrahedron, as shown here in red. The farthest cell projects onto the tetrahedral envelope itself, while the other 4 cells project onto the 4 flattened tetrahedral regions surrounding the central vertex. The edge first projection of the 5 cell into 3 dimensions has a triangular dipyramidal envelope. The closest edge (shown here in red) projects to the axis of the dipyramid, with the three cells surrounding it projecting to 3 tetrahedral volumes arranged around this axis at 120 degrees to each other. The remaining 2 cells project to the two halves of the dipyramid and are on the far side of the pentatope. The face first projection of the 5 cell into 3 dimensions also has a triangular dipyramidal envelope. The nearest face is shown here in red. The two cells that meet at this face project to the two halves of the dipyramid. The remaining three cells are on the far side of the pentatope from the 4D viewpoint, and are culled from the image for clarity. They are arranged around the central axis of the dipyramid, just as in the edge first projection. The cell first projection of the 5 cell into 3 dimensions has a tetrahedral envelope. The nearest cell projects onto the entire envelope, and, from the 4D viewpoint, obscures the other 4 cells; hence, they are not rendered here.

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.

The compound of two 5 cells in dual configurations can be seen in this A5 Coxeter plane projection, with a red and blue 5 cell vertices and edges. This compound has [[3,3,3]] symmetry, order 240. The intersection of these two 5 cells is a uniform bitruncated 5 cell. = ∩
This compound can be seen as the 4D analogue of the 2D hexagram {} and the 3D compound of two tetrahedra.

The pentachoron (5 cell) is the simplest of 9 uniform polychora constructed from the [3,3,3] Coxeter group. It is in the {p,3,3} sequence of regular polychora with a tetrahedral vertex figure: the tesseract {4,3,3} and 120 cell {5,3,3} of Euclidean 4 space, and the hexagonal tiling honeycomb {6,3,3} of hyperbolic space. It is one of three {3,3,p} regular 4 polytopes with tetrahedral cells, along with the 16 cell {3,3,4} and 600 cell {3,3,5}. The order 6 tetrahedral honeycomb {3,3,6} of hyperbolic space also has tetrahedral cells. It is self dual like the 24 cell {3,4,3}, having a palindromic {3,p,3} Schläfli symbol.