# Tesseract

(Redirected from 4-hypercube)
Tesseract
Rank4
TypeRegular
SpaceSpherical
Bowers style acronymTes
Info
Coxeter diagramx4o3o3o
Schläfli symbol{4,3,3}
Tapertopic notation1111
Toratopic notationIIII
Bracket notation[IIII]
SymmetryBC4, order 384
ArmyTes
RegimentTes
Elements
Vertex figureTetrahedron, edge length 2
Cells8 cubes
Faces24 squares
Edges32
Vertices16
Measures (edge length 1)
Edge radius${\displaystyle \frac{\sqrt3}{2} ≈ 0.86603}$
Face radius${\displaystyle \frac{\sqrt2}{2} ≈ 0.70711}$
Inradius${\displaystyle \frac12 = 0.5}$
Hypervolume1
Dichoral angle90°
Height1
Central density1
Euler characteristic0
Number of pieces8
Level of complexity1
Related polytopes
ConjugateTesseract
Properties
ConvexYes
OrientableYes
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, 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.

## 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:

• ${\displaystyle \left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac12\right).}$

## Surtope Angles

The surtope angles measure the proportion of space that the polytope measures at that point.

• A2 :30.00.00 1/4 = 90° margin angle
• A3 :15.00.00 1/8 = 90°E edge-angle
• A4 :07.60.00 1/16 vertex-angle

## Representations

A tesseract has the following Coxeter diagrams:

• x4o3o3o (full symmetry)
• x x4o3o (BC2×A1 symmetry, as cubic prism)
• x4o x4o (BC2×BC2 symmetry, square duoprism)
• x x x4o (BC2×A1×A1 symmetry, square prismatic prism)
• x x x x (A1×A1×A1×A1 symmetry, 4D hypercuboid)
• s4x2s4x
• xx4oo3oo&#x (BC3 axial, cube atop cube)
• xx xx4oo&#x (bases have BC2×A1 symmetry)
• xx xx xx&#x (bases have A1×A1×A1 symmetry)
• oqo xxx4ooo&#xt (BC2×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

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 merging into octahedra.

Two of the seven regular polychoron compounds are composed of tesseracts:

o4o3o3o truncations
Name OBSA Schläfli symbol CD diagram Picture
Tesseract tes {4,3,3}
Truncated tesseract tat t{4,3,3}
Rectified tesseract rit r{4,3,3}
Rectified hexadecachoron = Icositetrachoron ico r{3,3,4}