Tesseract

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Tesseract
Schlegel wireframe 8-cell.png
Rank4
TypeRegular
SpaceSpherical
Notation
Bowers style acronymTes
Coxeter diagramx4o3o3o (CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png)
Schläfli symbol{4,3,3}
Tapertopic notation1111
Toratopic notationIIII
Bracket notation[IIII]
Elements
Cells8 cubes
Faces24 squares
Edges32
Vertices16
Vertex figureTetrahedron, edge length 2 8-cell verf.png
Edge figurecube 4 cube 4 cube 4
Petrie polygons8 octagonal-square coils
Measures (edge length 1)
Circumradius1
Edge radius
Face radius
Inradius
Hypervolume1
Dichoral angle90°
Interior anglesAt square:
At edge:
At vertex:
Height1
Central density1
Number of external pieces8
Level of complexity1
Related polytopes
ArmyTes
RegimentTes
DualHexadecachoron
Petrie dualPetrial tesseract
ConjugateNone
Abstract & topological properties
Flag count384
Euler characteristic0
OrientableYes
Properties
SymmetryB4, order 384
ConvexYes
Net count261[1]
NatureTame

The tesseract, or tes, also sometimes called the 8-cell or octachoron, is one of the 6 convex regular polychora. It has 8 cubes as cells, joining 3 to an edge and 4 to a vertex. It is the 4-dimensional hypercube.

It is also the uniform cubic prism (and thus also a segmentochoron designated K-4.20 on Richard Klitzing's list), uniform square duoprism, digonal duoantitegum, digonal diswirltegum, digonal duotransitionalterprism, and the 8-3 gyrochoron. It is the first in an infinite family of isochoric tetrahedral swirlchora, the first in an infinite family of isogonal square dihedral swirlchora and also the first in an infinite family of isochoric square hosohedral swirlchora.

It is one of the three regular polychora that can tile 4D space, similar to hypercubes of any other dimension. The tiling is the tesseractic tetracomb.

The tesseract has the same circumradius as its edge length. This relates to the fact that it is the vertex figure of the Euclidean icositetrachoric tetracomb.

Gallery[edit | edit source]

Naming[edit | edit source]

The name tesseract comes from the Greek τέσσαρα (4) and ἀκτίς (ray), referring to the four line segments meeting at each vertex. It was coined by Charles Howard Hinton. Other names include

  • Tessaract, an alternate spelling. Hinton spelled the word inconsistently, but "tesseract" is the spelling that is generally considered correct in the present day.
  • 8-cell or octachoron, referring to the number of cells. Octahedroid is sometimes also used.
  • 4-cube or sometimes tetracube, because it is the 4-dimensional hypercube.

Vertex coordinates[edit | edit source]

The vertices of a tesseract of edge length 1, centered at the origin, are given by:

  • .

Representations[edit | edit source]

A tesseract has the following Coxeter diagrams:

  • x4o3o3o (CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png) (full symmetry)
  • x x4o3o (CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png) (B2×A1 symmetry, as cubic prism)
  • x4o x4o (CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png) (B2×B2 symmetry, square duoprism)
  • x x x4o (CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png) (B2×A1×A1 symmetry, square prismatic prism)
  • x x x x (CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png) (A1×A1×A1×A1 symmetry, 4D hypercuboid)
  • s4x2s4x (CDel node h.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node 1.png)
  • xx4oo3oo&#x (B3 axial, cube atop cube)
  • xx xx4oo&#x (bases have B2×A1 symmetry)
  • xx xx xx&#x (bases have A1×A1×A1 symmetry)
  • oqo xxx4ooo&#xt (B2×A1 symmetry, square-first).
  • oqo xxx xxx&#xt (A1×A1×A1 axial, square-first)
  • xxxx oqoo3ooqo&#xt (A2×A1 axial, edge-first)
  • oqooo3ooqoo3oooqo&#xt (A3 axial, vertex-first, tetrahedral antitegum)
  • qo3oo3oq *b3oo&#zx (D4 subsymmetry, hull of 2 opposite demitesseracts/hexadecachora)
  • xx qo3oo3oq&#zx (A3×A1 symmetry, prism of hull of 2 tetrahedra)
  • xx4oo qo oq&#zx (as square/rhombic duoprism)
  • xx xx qo oq&#zx (as rectangular/rhombic duoprism)
  • qqoo ooqq qoqo oqoq&#zx (as rhombic/rhombic duoprism)

Variations[edit | edit source]

Besides the regular tesseract, there are various other polychora with 24 quadriateral faces and 8 hexahedral cells with lower symmetry. These include:

Related polychora[edit | edit source]

A tesseract can be decomposed into 8 CRF cubic pyramids. If these cubic pyramids are attached to the cells of a tesseract the result is a regular icositetrachoron, as adjacent square pyramids merge into octahedra.

Uniform polychoron compounds composed of tesseracts include:

o4o3o3o truncations
Name OBSA CD diagram Picture
Tesseract tes x4o3o3o (CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png)
Schlegel wireframe 8-cell.png
Truncated tesseract tat x4x3o3o (CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png)
Schlegel half-solid truncated tesseract.png
Rectified tesseract rit o4x3o3o (CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png)
Schlegel half-solid rectified 8-cell.png
Tesseractihexadecachoron tah o4x3x3o (CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png)
Schlegel half-solid bitruncated 8-cell.png
Rectified hexadecachoron = Icositetrachoron ico o4o3x3o (CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png)
Schlegel half-solid rectified 16-cell.png
Truncated hexadecachoron thex o4o3x3x (CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png)
Schlegel half-solid truncated 16-cell.png
Hexadecachoron hex o4o3o3x (CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png)
Schlegel wireframe 16-cell.png
Small rhombated tesseract srit x4o3x3o (CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png)
Schlegel half-solid cantellated 8-cell.png
Great rhombated tesseract grit x4x3x3o (CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png)
Schlegel half-solid cantitruncated 8-cell.png
Small rhombated hexadecachoron = Rectified icositetrachoron rico o4x3o3x (CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png)
Schlegel half-solid cantellated 16-cell.png
Great rhombated hexadecachoron = Truncated icositetrachoron tico o4x3x3x (CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png)
Schlegel half-solid cantitruncated 16-cell.png
Small disprismatotesseractihexadecachoron sidpith x4o3o3x (CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png)
Schlegel half-solid runcinated 8-cell.png
Prismatorhombated hexadecachoron proh x4x3o3x (CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png)
Schlegel half-solid runcitruncated 8-cell.png
Prismatorhombated tesseract prit x4o3x3x (CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png)
Schlegel half-solid runcitruncated 16-cell.png
Great disprismatotesseractihexadecachoron gidpith x4x3x3x (CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png)
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. "tes".

References[edit | edit source]

  1. Turney, Peter. "Unfolding the Tesseract". Retrieved 2022-12-03.