Rectified tesseract

From Polytope Wiki
Jump to navigation Jump to search
Rectified tesseract
Schlegel half-solid rectified 8-cell.png
Bowers style acronymRit
Coxeter diagramo4x3o3o (CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png)
Cells16 tetrahedra, 8 cuboctahedra
Faces64 triangles, 24 squares
Vertex figureSemi-uniform triangular prism, edge lengths 1 (base) and 2 (side) Rectified 8-cell verf.png
Edge figuretet 3 co 4 co 3
Measures (edge length 1)
Dichoral anglesCo–3–tet: 120°
 Co–4–co: 90°
Central density1
Number of pieces24
Level of complexity3
Related polytopes
DualJoined hexadecachoron
Abstract properties
Flag count1152
Euler characteristic0
Topological properties
SymmetryB4, order 384

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 semi-uniform 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 digonal-square prismantiprismoids (transitional digonal double prismantiprismoid) 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.

Gallery[edit | edit source]

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 (D4 symmetry, as small rhombated demitesseract)
  • s4o3o3x (as runcic tesseract)
  • s4x3o3o (similar to above)
  • xxoo3oxxo3ooxx&#xt (A3 axial, tetrahedron-first)
  • oqo4xox3ooo&#xt (BC3 axial, cuboctahedron-first)
  • qo oq4xo3oo&#zx (BC3×A1 symmetry)
  • ox4qo xo4oq&#zx (BC2×BC2 symmetry, rectified square duoprism)
  • x(uo)x3o(oo)o3x(uo)x&#xt (A3 axial, cuboctahedron-first)
  • oxuxo xoxox4oqoqo&#xt (BC2×A1 axial, square-first)
  • oqoqoqo oooxuxo3oxuxooo&#xt (A2×A1 symmetry, vertex-first)

Variations[edit | edit source]

The rectified tesseract has the following general variations:

Related polychora[edit | edit source]

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 BC4 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:

o4o3o3o truncations
Name OBSA CD diagram Picture
Tesseract tes x4o3o3o
Schlegel wireframe 8-cell.png
Truncated tesseract tat x4x3o3o
Schlegel half-solid truncated tesseract.png
Rectified tesseract rit o4x3o3o
Schlegel half-solid rectified 8-cell.png
Tesseractihexadecachoron tah o4x3x3o
Schlegel half-solid bitruncated 8-cell.png
Rectified hexadecachoron = Icositetrachoron ico o4o3x3o
Schlegel half-solid rectified 16-cell.png
Truncated hexadecachoron thex o4o3x3x
Schlegel half-solid truncated 16-cell.png
Hexadecachoron hex o4o3o3x
Schlegel wireframe 16-cell.png
Small rhombated tesseract srit x4o3x3o
Schlegel half-solid cantellated 8-cell.png
Great rhombated tesseract grit x4x3x3o
Schlegel half-solid cantitruncated 8-cell.png
Small rhombated hexadecachoron = Rectified icositetrachoron rico o4x3o3x
Schlegel half-solid cantellated 16-cell.png
Great rhombated hexadecachoron = Truncated icositetrachoron tico o4x3x3x
Schlegel half-solid cantitruncated 16-cell.png
Small disprismatotesseractihexadecachoron sidpith x4o3o3x
Schlegel half-solid runcinated 8-cell.png
Prismatorhombated hexadecachoron proh x4x3o3x
Schlegel half-solid runcitruncated 8-cell.png
Prismatorhombated tesseract prit x4o3x3x
Schlegel half-solid runcitruncated 16-cell.png
Great disprismatotesseractihexadecachoron gidpith x4x3x3x
Schlegel half-solid omnitruncated 8-cell.png

Isogonal derivatives[edit | edit source]

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

External links[edit | edit source]

  • Klitzing, Richard. "rit".