Compound of tesseract and 16-cell

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

In 4-dimensional geometry, the tesseract 16-cell compound^[1] is a polytope compound composed of a regular tesseract and dual regular 16-cell. 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 the 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.

It can be seen as the 4-dimensional analogue of a compound of cube and octahedron.

This is one of four compound polytopes which is obtained through combining a regular convex 4-polytope with its dual; the other three being the compound of two 5-cells, and compound of 120-cell and 600-cell.

Construction[]

The 24 Cartesian coordinates of the vertices of the compound are:

8: (±2, 0, 0, 0), ( 0, ±2, 0, 0), ( 0, 0, ±2, 0), ( 0, 0, 0, ±2)

16: ( ±1, ±1, ±1, ±1)

These are the first two vertex sets of the stellations of a 16-cell.^[2]

Faceting[]

The convex hull is the self-dual regular 24-cell, which is also a rectified 16-cell. This makes it a faceting of the 24-cell.

The intersection of the tesseract and 16-cell compound is the uniform bitruncated tesseract: = ∩ .

Graphs in B4 Coxeter plane

Elements		Compound	Convex hull	Intersection
Tesseract	16-cell	Tesseract and 16-cell	Self dual 24-cell	Bitruncated tesseract
		∪	= =

See also[]

• Compound of 5-cube and 5-orthoplex

References[]

• Weisstein, Eric W. "Tesseract". MathWorld.
• Weisstein, Eric W. "16-Cell". MathWorld.

External links[]

• Compound of the Tesseract and Its Dual, the 16-Cell
• Rotating Compound of the Tesseract and Its Dual