Rectified tesseract
Rectified tesseract  

Rank  4 
Type  Uniform 
Notation  
Bowers style acronym  Rit 
Coxeter diagram  o4x3o3o () 
Elements  
Cells  16 tetrahedra, 8 cuboctahedra 
Faces  64 triangles, 24 squares 
Edges  96 
Vertices  32 
Vertex figure  Semiuniform triangular prism, edge lengths 1 (base) and √2 (side) 
Edge figure  tet 3 co 4 co 3 
Measures (edge length 1)  
Circumradius  
Hypervolume  
Dichoral angles  Co–3–tet: 120° 
Co–4–co: 90°  
Central density  1 
Number of external pieces  24 
Level of complexity  3 
Related polytopes  
Army  Rit 
Regiment  Rit 
Dual  Joined hexadecachoron 
Conjugate  None 
Abstract & topological properties  
Flag count  1152 
Euler characteristic  0 
Orientable  Yes 
Properties  
Symmetry  B_{4}, order 384 
Convex  Yes 
Nature  Tame 
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.
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 semiuniform square duoprisms where the edges of one square are times the length of those of the other square.
It is also the convex hull of two perpendicular digonalsquare prismantiprismoids (transitional digonal double prismantiprismoid) and is the first member of an infinite family of double prismantiprismoids. It also contains the vertices of two digonalscalenohedral 83 double step prisms.
Gallery[edit  edit source]

Wireframe

Net
Vertex coordinates[edit  edit source]
The vertices of a rectified tesseract of edge length 1 are given by all permutations of:
 .
Alternatively, they can be given under D4 symmetry as even sign changes and all permutations of:
 .
Representations[edit  edit source]
A rectified tesseract has the following Coxeter diagrams:
 o4x3o3o () (full symmetry)
 x3o3x *b3o () (D_{4} symmetry, as small rhombated demitesseract)
 s4o3o3x () (as runcic tesseract)
 s4x3o3o () (similar to above)
 xxoo3oxxo3ooxx&#xt (A_{3} axial, tetrahedronfirst)
 oqo4xox3ooo&#xt (B_{3} axial, cuboctahedronfirst)
 qo oq4xo3oo&#zx (B_{3}×A_{1} symmetry)
 ox4qo xo4oq&#zx (B_{2}×B_{2} symmetry, rectified square duoprism)
 x(uo)x3o(oo)o3x(uo)x&#xt (A_{3} axial, cuboctahedronfirst)
 oxuxo xoxox4oqoqo&#xt (B_{2}×A_{1} axial, squarefirst)
 oqoqoqo oooxuxo3oxuxooo&#xt (A_{2}×A_{1} symmetry, vertexfirst)
Variations[edit  edit source]
The rectified tesseract has the following general variations:
 Small rhombated demitesseract  half symmetry, isogonal, 2 types of tetrahedra, cuboctahedra as rhombitetratetrahedra
 Rectified square duoprism  no variations, cuboctahedra have symmetry of recctified square prisms, tetrahedra as tetragonal disphenoids
 Transitional digonal double prismantiprismoid  less symmetric isogonal variant
Related polychora[edit  edit source]
The rectified tesseract is the colonel of a 5member 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:
 Birectified demidistesseract (2)
 Rectified great icositetrachoron (3)
 Rectified great stellated tetracontoctachoron (6)
 Rectified dodecahedronary cubichoron (75)
 Rectified cubichoron (75)
Isogonal derivatives[edit  edit source]
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 Cuboctahedron (8): Hexadecachoron
 Tetrahedron (16): Tesseract
 Square (24): Icositetrachoron
 Triangle (64): Semiuniform small disprismatotesseractihexadecachoron
 Edge (96): Semiuniform small rhombated tesseract
External links[edit  edit source]
 Bowers, Jonathan. "Category 3: Triangular Rectates" (#42).
 Bowers, Jonathan. "Tessic Isogonals".
 Klitzing, Richard. "rit".
 Quickfur. "The Rectified Tesseract".
 Wikipedia contributors. "Rectified tesseract".