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 its dual, the regular 16-cell. Its convex hull is the regular 24-cell, which is self-dual.

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.

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 24-cell

The convex hull of the tesseract 16-cell compound is the regular 24-cell. This makes the compound a faceting of the 24-cell.

The 24-cell is a rectified 16-cell and also the convex hull of a compound of three 16-cells. The tesseract is the convex hull of a compound of two 16-cells. Thus the tesseract 16-cell compound is a lower-symmetry form of the 24-cell, which is the whole package (the F₄ symmetry group).

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	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

This page was last edited on 19 January 2024, at 19:23