Compound of tesseract and 16-cell

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.

Tesseract 16-cell compound
TypeCompound
Schläfli symbol{4,3,3} ∪ {3,3,4}
Coxeter diagram ∪
Intersectionbitruncated tesseract
Convex hull24-cell
Polychora2:
1 tesseract
1 16-cell
Polyhedra24:
8 cubes
16 tetrahedra
Faces56:
24 squares
32 triangles
Edges56
Vertices24
Symmetry groupHyperoctahedral symmetry
[4,3,3], order 384

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, 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 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
ElementsCompoundConvex hullIntersection

Tesseract
16-cell
Tesseract and 16-cell
Self dual 24-cell
Bitruncated tesseract
∪ = =

See also

References
  1. Klitzing, Richard. "Compound polytopes".
  2. The Stellated Forms of the Sixteen-Cell B. L. Chilton The American Mathematical Monthly Vol. 74, No. 4 (Apr., 1967), pp. 372–378
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