Rectified tesseract
File:Schlegel half-solid rectified 8-cell.png Schlegel diagram
Type
Uniform 4-polytope
Schläfli symbol
r{4,3,3} = { 4 3 , 3 } 1,1 }3 {4,3,3}
Coxeter-Dynkin diagrams
File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel nodes 11.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel node h.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png
Cells
24
8 (3.4.3.4 ) File:Cuboctahedron.png (3.3.3 ) File:Tetrahedron.png
Faces
88
64 {3} {4}
Edges
96
Vertices
32
Vertex figure
File:Rectified 8-cell verf.png File:Cantellated demitesseract verf.png
Symmetry group
B4 [3,3,4], order 3844 [31,1,1 ], order 192
Properties
convex , edge-transitive
Uniform index
10 12
File:Rectified tesseract net.png Net In geometry , the rectified tesseract , rectified 8-cell is a uniform 4-polytope (4-dimensional polytope ) bounded by 24 cells : 8 cuboctahedra , and 16 tetrahedra . It has half the vertices of a runcinated tesseract , with its File:CDel node h.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png runcic tesseract .
It has two uniform constructions, as a rectified 8-cell r{4,3,3} and a cantellated demitesseract , rr{3,31,1 }, the second alternating with two types of tetrahedral cells.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC8 .
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 , ± 2 , ± 2 , ± 2 ) Images 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 Related uniform polytopes Runcic cubic polytopes
Runcic n -cubes
n
4
5
6
7
8
[1+ ,4,3n-2 ]n-3,1 ]
[1+ ,4,32 ]1,1 ]
[1+ ,4,33 ]2,1 ]
[1+ ,4,34 ]3,1 ]
[1+ ,4,35 ]4,1 ]
[1+ ,4,36 ]5,1 ]
Runcic
File:Schlegel half-solid rectified 8-cell.png File:5-demicube t03 D5.svg File:6-demicube t03 D6.svg File:7-demicube t03 D7.svg File:8-demicube t03 D8.svg
Coxeter
File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node h1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel nodes 10ru.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
Schläfli
h3 {4,32 }
h3 {4,33 }
h3 {4,34 }
h3 {4,35 }
h3 {4,36 }
Tesseract polytopes
B4 symmetry polytopes
Name
tesseract
rectified
truncated
cantellated
runcinated
bitruncated
cantitruncated
runcitruncated
omnitruncated
Coxeter
File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel nodes 11.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel nodes 11.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png
Schläfli
{4,3,3}
t1
t0,1
t0,2
t0,3
t1,2
t0,1,2
t0,1,3
t0,1,2,3
Schlegel
File:Schlegel wireframe 8-cell.png File:Schlegel half-solid rectified 8-cell.png File:Schlegel half-solid truncated tesseract.png File:Schlegel half-solid cantellated 8-cell.png File:Schlegel half-solid runcinated 8-cell.png File:Schlegel half-solid bitruncated 8-cell.png File:Schlegel half-solid cantitruncated 8-cell.png File:Schlegel half-solid runcitruncated 8-cell.png File:Schlegel half-solid omnitruncated 8-cell.png
B4
File:4-cube t0.svg File:4-cube t1.svg File:4-cube t01.svg File:4-cube t02.svg File:4-cube t03.svg File:4-cube t12.svg File:4-cube t012.svg File:4-cube t013.svg File:4-cube t0123.svg
Name
16-cell
rectified
truncated
cantellated
runcinated
bitruncated
cantitruncated
runcitruncated
omnitruncated
Coxeter
File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel nodes.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel nodes.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel nodes.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel nodes 11.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel nodes 11.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel nodes 11.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png
Schläfli
{3,3,4}
t1
t0,1
t0,2
t0,3
t1,2
t0,1,2
t0,1,3
t0,1,2,3
Schlegel
File:Schlegel wireframe 16-cell.png File:Schlegel half-solid rectified 16-cell.png File:Schlegel half-solid truncated 16-cell.png File:Schlegel half-solid cantellated 16-cell.png File:Schlegel half-solid runcinated 16-cell.png File:Schlegel half-solid bitruncated 16-cell.png File:Schlegel half-solid cantitruncated 16-cell.png File:Schlegel half-solid runcitruncated 16-cell.png File:Schlegel half-solid omnitruncated 16-cell.png
B4
File:4-cube t3.svg File:24-cell t0 B4.svg File:4-cube t23.svg File:24-cell t1 B4.svg File:4-cube t03.svg File:4-cube t12.svg File:4-cube t123.svg File:4-cube t023.svg File:4-cube t0123.svg
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 , edited 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) 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 11 , George Olshevsky.Klitzing, Richard. "4D uniform polytopes (polychora) o4x3o3o - rit" . 
