# Rectified tesseract

The rectified tesseract, or rit, is a convex uniform polychoron that consists of 16 regular tetrahedra and 8 cuboctahedra. Two tetrahedra and three cuboctahedra join at each triangular prismatic vertex. As the name suggests, it can be obtained by rectifying the tesseract.

Rectified tesseract
Rank4
TypeUniform
Notation
Bowers style acronymRit
Coxeter diagramo4x3o3o ()
Elements
Cells16 tetrahedra, 8 cuboctahedra
Faces64 triangles, 24 squares
Edges96
Vertices32
Vertex figureSemi-uniform triangular prism, edge lengths 1 (base) and 2 (side)
Edge figuretet 3 co 4 co 3
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {6}}{2}}\approx 1.22474}$
Hypervolume${\displaystyle {\frac {23}{6}}\approx 3.83333}$
Dichoral anglesCo–3–tet: 120°
Co–4–co: 90°
Central density1
Number of external pieces24
Level of complexity3
Related polytopes
ArmyRit
RegimentRit
ConjugateNone
Abstract & topological properties
Flag count1152
Euler characteristic0
OrientableYes
Properties
SymmetryB4, order 384
Flag orbits3
ConvexYes
NatureTame

As the rectified tesseract, it is the square member of an infinite family of isogonal rectified duoprisms, and could be called the rectified square duoprism. In this representation it is also the convex hull of 2 oppositely oriented semi-uniform square duoprisms where the edges of one square are ${\displaystyle {\sqrt {2}}\approx 1.41421}$ times the length of those of the other square.

It is also the convex hull of two perpendicular digonal-square prismantiprismoids (transitional digonal double gyroprismantiprismoid) and is the first member of an infinite family of double prismantiprismoids. It also contains the vertices of two digonal-scalenohedral 8-3 double step prisms.

## Vertex coordinates

The vertices of a rectified tesseract of edge length 1 are given by all permutations of:

• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0\right)}$ .

Alternatively, they can be given under D4 symmetry as even sign changes and all permutations of:

• ${\displaystyle \left({\frac {3{\sqrt {2}}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right)}$ .

## Representations

A rectified tesseract has the following Coxeter diagrams:

• o4x3o3o (       ) (full symmetry)
• x3o3x *b3o (     ) (D4 symmetry, as small rhombated demitesseract)
• s4o3o3x (       ) (as runcic tesseract)
• s4x3o3o (       ) (similar to above)
• xxoo3oxxo3ooxx&#xt (A3 axial, tetrahedron-first)
• oqo4xox3ooo&#xt (B3 axial, cuboctahedron-first)
• qo oq4xo3oo&#zx (B3×A1 symmetry)
• ox4qo xo4oq&#zx (B2×B2 symmetry, rectified square duoprism)
• x(uo)x3o(oo)o3x(uo)x&#xt (A3 axial, cuboctahedron-first)
• oxuxo xoxox4oqoqo&#xt (B2×A1 axial, square-first)
• oqoqoqo oooxuxo3oxuxooo&#xt (A2×A1 symmetry, vertex-first)

## Variations

The rectified tesseract has the following general variations:

## Related polychora

The rectified tesseract is the colonel of a 5-member regiment. Other members of this regiment include the facetorectified tesseract, hexadecintercepted tesseract, small trisoctachoron, and great trisoctachoron. The first two of these have full B4 symmetry, while the latter two have D4 symmetry only.

When viewed in A3 axial symmetry, the rectified tesseract can be seen as a central truncated tetrahedral cupoliprism with 2 tetrahedron atop truncated tetrahedron segmentochora attached to its bases.

Uniform polychoron compounds composed of rectified tesseracts include:

### Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora: