# Tesseract

Tesseract
Rank4
TypeRegular
Notation
Bowers style acronymTes
Coxeter diagramx4o3o3o ()
Schläfli symbol{4,3,3}
Tapertopic notation1111
Toratopic notationIIII
Bracket notation[IIII]
Elements
Cells8 cubes
Faces24 squares
Edges32
Vertices16
Vertex figureTetrahedron, edge length 2
Edge figurecube 4 cube 4 cube 4
Petrie polygons24 octagonal-square coils
Measures (edge length 1)
Edge radius${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Face radius${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Inradius${\displaystyle {\frac {1}{2}}=0.5}$
Hypervolume1
Dichoral angle90°
Interior anglesAt square: ${\displaystyle {\frac {1}{4}}}$
At edge: ${\displaystyle {\frac {1}{8}}}$
At vertex: ${\displaystyle {\frac {1}{16}}}$
Height1
Central density1
Number of external pieces8
Level of complexity1
Related polytopes
ArmyTes
RegimentTes
Petrie dualPetrial tesseract
κ ?Alternative tesseract
ConjugateNone
Abstract & topological properties
Flag count384
Euler characteristic0
OrientableYes
SkeletonQ 4
Properties
SymmetryB4, order 384
Flag orbits1
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.

## Naming

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

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)}$.

## Representations

A tesseract has the following Coxeter diagrams:

• x4o3o3o () (full symmetry)
• x x4o3o () (B2×A1 symmetry, as cubic prism)
• x4o x4o () (B2×B2 symmetry, square duoprism)
• x x x4o () (B2×A1×A1 symmetry, square prismatic prism)
• x x x x () (K4 symmetry, 4D hypercuboid)
• s4x2s4x ()
• xx4oo3oo&#x (B3 axial, cube atop cube)
• xx xx4oo&#x (bases have B2×A1 symmetry)
• xx xx xx&#x (bases have K3 symmetry)
• oqo xxx4ooo&#xt (B2×A1 symmetry, square-first).
• oqo xxx xxx&#xt (K3 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

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

## Related polychora

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:

## Isogonal derivatives

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