Compound of tesseract and 16-cell
In 4-dimensional geometry, the tesseract 16-cell compound is a polytope compound composed of a regular tesseract and its dual, the regular 16-cell. Its convex hull is the regular 24-cell, which is self-dual.
| Tesseract 16-cell compound | |
|---|---|
| Type | Compound |
| Schläfli symbol | {4,3,3} ∪ {3,3,4} |
| Coxeter diagram | ∪ |
| Intersection | bitruncated tesseract |
| Convex hull | 24-cell |
| Polychora | 2: 1 tesseract 1 16-cell |
| Polyhedra | 24: 8 cubes 16 tetrahedra |
| Faces | 56: 24 squares 32 triangles |
| Edges | 56 |
| Vertices | 24 |
| Symmetry group | Hyperoctahedral symmetry [4,3,3], order 384 |
A compound polytope is a figure that is composed of several polytopes sharing a common center. The outer vertices of a compound can be connected to form a convex polytope called its convex hull. The compound is a facetting of the convex hull. In 4-polytope compounds constructed as dual pairs, cells and vertices swap positions and faces and edges swap positions. Because of this the number of cells and vertices are equal, as are faces and edges. Mid-edges of the tesseract cross mid-face in the 16-cell, and vice versa.
The tesseract 16-cell compound can be seen as the 4-dimensional analogue of a compound of cube and octahedron.
It is one of four compound polytopes which are obtained by combining a regular convex 4-polytope with its dual; the other three being the compound of two 5-cells, compound of two 24-cells and compound of 120-cell and 600-cell.