# Rectified tesseract

Rectified tesseract

Schlegel diagram
Centered on cuboctahedron
tetrahedral cells shown
Type Uniform 4-polytope
Schläfli symbol r{4,3,3}
2r{3,31,1}
h3{4,3,3}
Coxeter-Dynkin diagrams

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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 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 construction.

It has two uniform constructions, as a rectified 8-cell r{4,3,3} and a cantellated demitesseract, rr{31,1,1}, 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

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph
Dihedral symmetry [12/3] [4]
 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

## Related uniform polytopes

Dimensional family of runcic n-cubes
n 4 5 6 7
[1+,4,3n-2]
= [3,3n-3,1]
[1+,4,32]
= [3,31,1]
[1+,4,33]
= [3,32,1]
[1+,4,34]
= [3,33,1]
[1+,4,35]
= [3,34,1]
Cantic
figure
Coxeter
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Schläfli h3{4,32} h3{4,33} h3{4,34} h3{4,35}
Name tesseract rectified
tesseract
truncated
tesseract
cantellated
tesseract
runcinated
tesseract
bitruncated
tesseract
cantitruncated
tesseract
runcitruncated
tesseract
omnitruncated
tesseract
Coxeter
diagram

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Schläfli
symbol
{4,3,3} t1{4,3,3}
r{4,3,3}
t0,1{4,3,3}
t{4,3,3}
t0,2{4,3,3}
rr{4,3,3}
t0,3{4,3,3} t1,2{4,3,3}
2t{4,3,3}
t0,1,2{4,3,3}
tr{4,3,3}
t0,1,3{4,3,3} t0,1,2,3{4,3,3}
Schlegel
diagram
B4

Name 16-cell rectified
16-cell
truncated
16-cell
cantellated
16-cell
runcinated
16-cell
bitruncated
16-cell
cantitruncated
16-cell
runcitruncated
16-cell
omnitruncated
16-cell
Coxeter
diagram

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Schläfli
symbol
{3,3,4} t1{3,3,4}
r{3,3,4}
t0,1{3,3,4}
t{3,3,4}
t0,2{3,3,4}
rr{3,3,4}
t0,3{3,3,4} t1,2{3,3,4}
2t{3,3,4}
t0,1,2{3,3,4}
tr{3,3,4}
t0,1,3{3,3,4} t0,1,2,3{3,3,4}
Schlegel
diagram
B4