Petrial tesseract

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Petrial tesseract
Rank4
TypeRegular
SpaceSpherical
Notation
Schläfli symbol
Elements
Cells3 square duocombs
Faces24 squares
Edges32
Vertices16
Vertex figurePetrial tetrahedron, edge length 2
Deep holesSquares
Measures (edge length 1)
Circumradius1
Related polytopes
ArmyTes
RegimentTes
Petrie dualTesseract
κ ?Petrial alternative tesseract
Abstract & topological properties
Flag count384
Euler characteristic5
Schläfli type{4,4,3}
OrientableNo
Properties
SymmetryB4, order 384
ConvexNo
Net count1
Dimension vector(3,2,3,3)

The Petrial tesseract is a regular skew polychoron consisting of 3 square duocombs. It shares its faces, edges, and vertices with the tesseract, and it has a Petrial tetrahedron vertex figure. It is the Petrie dual of the tesseract[1], and it is also a flat polychoron.

A square duocomb in blue oriented in the xy direction
A square duocomb in red oriented in the yz direction
A square duocomb in yellow oriented in the xz direction
A projected view of each of the three cells in a different color. In reality these three cells occupy the same space, but they are shown side by side for clarity.

The Petrial tesseract is also the only known (concrete) regular polychoron that has duocombs as cells. It is also a quotient of the square tiling honeycomb. It is both locally toroidal and locally projective.

Vertex coordinates[edit | edit source]

The vertex coordinates of the Petrial tesseract are the same as those of the tesseract.

External links[edit | edit source]

References[edit | edit source]

  1. McMullen, Peter (2004). "Regular polytopes of Full Rank" (PDF). Discrete Computational Geometry: 20.