Jensen–Shannon divergence

Jensen–Shannon divergence
Schlegel diagram Centered on cuboctahedron tetrahedral cells shown
Type	Uniform polychoron
Schläfli symbol	r{4,3,3} 2r{3,31,1} h3{4,3,3}
Coxeter-Dynkin diagrams	Template:CDD Template:CDD Template:CDD = Template:CDD
Cells	24	8 (3.4.3.4) 16 (3.3.3)
Faces	88	64 {3} 24 {4}
Edges	96
Vertices	32
Vertex figure	(Elongated equilateral-triangular prism)
Symmetry group	BC4 [3,3,4], order 384 D4 [31,1,1], order 192
Properties	convex, edge-transitive
Uniform index	10 11 12

In geometry, the rectified tesseract, rectified 8-cell, or runcic tesseract is a uniform polychoron (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. It has half the vertices of a runcinated tesseract, with its Template:CDD construction.

It has two uniform constructions, as a rectified 8-cell t₁{4,3,3} and a cantellated demitesseract, t_0,2{3^1,1,1}, the second alternating with two types of tetrahedral cells.

Construction

The rectified tesseract may be constructed from the tesseract by truncating its vertices at the midpoints of its edges.

The Cartesian coordinates of the vertices of the rectified tesseract with edge length 2 is given by all permutations of:

$(0,\ \pm {\sqrt {2}},\ \pm {\sqrt {2}},\ \pm {\sqrt {2}})$

Images

Wireframe

16 tetrahedral cells

Projections

In the cuboctahedron-first parallel projection of the rectified tesseract into 3-dimensional space, the image has the following layout:

• The projection envelope is a cube.
• A cuboctahedron is inscribed in this cube, with its vertices lying at the midpoint of the cube's edges. The cuboctahedron is the image of two of the cuboctahedral cells.
• The remaining 6 cuboctahedral cells are projected to the square faces of the cube.
• The 8 tetrahedral volumes lying at the triangular faces of the central cuboctahedron are the images of the 16 tetrahedral cells, two cells to each image.

Alternative names

• Rit (Jonathan Bowers: for rectified tesseract)
• Ambotesseract (Neil Sloane & John Horton Conway)
• Rectified tesseract/Runcic tesseract (Norman W. Johnson)
  • Runcic 4-hypercube/8-cell/octachoron/4-measure polytope/4-regular orthotope
  • Rectified 4-hypercube/8-cell/octachoron/4-measure polytope/4-regular orthotope

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Template:PolyCell
• Template:KlitzingPolytopes