Rectified 8-cell, Schlegel diagram, Centered on cuboctahedron
tetrahedral cells shown

In geometry, the rectified tesseract, or rectified 8-cell is a uniform polychoron (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra.

It has two uniform constructions, as a rectified 8-cell t1{4,3,3} and a cantellated demitesseract, t0,2{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:

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 (Norman W. Johnson)
* Rectified 4-hypercube
* Rectified 8-cell
* Rectified octachoron
* Rectified 4-measure polytope
* Rectified 4-regular orthotope
* Runcic tesseract (Norman W. Johnson)
* Runcic 4-hypercube
* Runcic 8-cell
* Runcic octachoron
* Runcic 4-measure polytope
* Runcic 4-regular orthotope

See also

* Tesseract
* Uniform polychoron

Related uniform polytopes

Name	tesseract	rectified tesseract	truncated tesseract	cantellated tesseract	runcinated tesseract	bitruncated tesseract	cantitruncated tesseract	runcitruncated tesseract	omnitruncated tesseract
Coxeter-Dynkin diagram
Schläfli symbol
Schlegel diagram
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-Dynkin diagram
Schläfli symbol
Schlegel diagram

Images: Robert Webb's Great Stella software

Geometry