pentachora

A pentachoron is the four-dimensional analog of a tetrahedron, just as a tetrahedron is the three-dimensional analog of a triangle. It is a regular convex polytope in four dimensions. The pentachoron is bounded by five tetrahedra, which are its threedimensional "faces." It has 5 vertices, 10 edges, and 10 faces. The number of cells (3D faces) is 5, the number of faces (2D faces) is 10, the number of edges is 10, and the number of vertices is 5. Its symmetry group is the same as that of the tetrahedron.

The pentachoron can be visualized by considering its projection into three dimensions. This projection can appear as a smaller tetrahedron enclosed within a larger tetrahedron, with vertices connected. Alternatively, it can be seen as five tetrahedra meeting at a central point. The construction of a pentachoron can be understood by taking a tetrahedron and expanding it, then connecting the vertices of the original tetrahedron to the corresponding vertices of the expanded tetrahedron.

In a coordinate system, the vertices of a regular pentachoron centered at the origin can be represented by vectors. For example, one common representation involves the vertices related to the roots of unity. The pentachoron is a fundamental shape in the study of higher-dimensional geometry and is also known as a 5-cell.