# Tesseract

Tesseract | |
---|---|

Rank | 4 |

Type | Regular |

Space | Spherical |

Notation | |

Bowers style acronym | Tes |

Coxeter diagram | x4o3o3o () |

Schläfli symbol | {4,3,3} |

Tapertopic notation | 1111 |

Toratopic notation | IIII |

Bracket notation | [IIII] |

Elements | |

Cells | 8 cubes |

Faces | 24 squares |

Edges | 32 |

Vertices | 16 |

Vertex figure | Tetrahedron, edge length √2 |

Edge figure | cube 4 cube 4 cube 4 |

Measures (edge length 1) | |

Circumradius | 1 |

Edge radius | |

Face radius | |

Inradius | |

Hypervolume | 1 |

Dichoral angle | 90° |

Interior angles | At square: At edge: At vertex: |

Height | 1 |

Central density | 1 |

Number of pieces | 8 |

Level of complexity | 1 |

Related polytopes | |

Army | Tes |

Regiment | Tes |

Dual | Hexadecachoron |

Petrie dual | Petrial tesseract |

Conjugate | None |

Abstract properties | |

Flag count | 384 |

Net count | 261^{[1]} |

Euler characteristic | 0 |

Topological properties | |

Orientable | Yes |

Properties | |

Symmetry | B_{4}, order 384 |

Convex | Yes |

Nature | Tame |

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]

Segmentochoron display, cube atop cube

Cross-sections, cell-first

## 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 (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 (A1×A1×A1×A1 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 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:

- Cubic prism - 2 cubic bases and 6 square prism sides
- Square duoprism - 2 sets of 4 square prisms
- Square-rectangular duoprism - 4 identical cuboids, 2 pairs of different square prisms
- Rectangular duoprism - 2 sets of 4 cuboids
- Tesseractoid - 4 pairs of cuboids
- Rhombic duoprism - 8 identical rhombic prisms
- Tetrahedral antitegum - 8 identical triangular antitegums
- 8-3 gyrochoron - least symmetric isochoric version
- Rectangular trapezoprismatic prism - 2 rectangle trapezoprisms, 2 cuboids, and 4 rectangle frustums, isogonal
- Digonal duoantitegum - 8 identical stretched cubes
- Tetraswirlic octachoron - as first tetrahedral swirltegum (no metrical variations)

## 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 merging into octahedra.

Uniform polychoron compounds composed of tesseracts include:

- Great icositetrachoron (3)
- Great stellated tetracontoctachoron (6)
- Chirotriangular crystallic enneacontahexachoron ª12)
- Chirocubichoron (15)
- Compound of 18 tesseracts (18)
- Bitrigonal crystallic enneacontahexachoron (24)
- Dodecahedronary chirodiscubichoron (30)
- Chirodiscubichoron (30)
- Compound of 36 tesseracts (36)
- Dodecahedronary chirotriscubichoron (45)
- Chirotriscubichoron (45)
- Compound of 48 tesseracts (48)
- Dodecahedronary chirotetriscubichoron (60)
- Chirotetriscubichoron (60)
- Dodecahedronary cubichoron (75)
- Cubichoron (75)
- Dodecahedronary chirocubisnubachoron (300)
- Snub dodecahedronary pentiscubichoron (375)
- Dodecahedronary cubisnubachoron (600)
- Compound of 675 tesseracts (675)
- An infinite family of compounds of square duoprisms

Name | OBSA | CD diagram | Picture |
---|---|---|---|

Tesseract | tes | x4o3o3o | |

Truncated tesseract | tat | x4x3o3o | |

Rectified tesseract | rit | o4x3o3o | |

Tesseractihexadecachoron | tah | o4x3x3o | |

Rectified hexadecachoron = Icositetrachoron | ico | o4o3x3o | |

Truncated hexadecachoron | thex | o4o3x3x | |

Hexadecachoron | hex | o4o3o3x | |

Small rhombated tesseract | srit | x4o3x3o | |

Great rhombated tesseract | grit | x4x3x3o | |

Small rhombated hexadecachoron = Rectified icositetrachoron | rico | o4x3o3x | |

Great rhombated hexadecachoron = Truncated icositetrachoron | tico | o4x3x3x | |

Small disprismatotesseractihexadecachoron | sidpith | x4o3o3x | |

Prismatorhombated hexadecachoron | proh | x4x3o3x | |

Prismatorhombated tesseract | prit | x4o3x3x | |

Great disprismatotesseractihexadecachoron | gidpith | x4x3x3x |

## Isogonal derivatives[edit | edit source]

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

- Cube (8): Hexadecachoron
- Square (24): Icositetrachoron
- Edge (32): Rectified tesseract

## External links[edit | edit source]

- Bowers, Jonathan. "Category 1: Regular Polychora" (#2).

- Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".

- Bowers, Jonathan. "Tessic Isogonals".

- Klitzing, Richard. "tes".

- Quickfur. "Tesseract".

- Wikipedia Contributors. "Tesseract".
- Hi.gher.Space Wiki Contributors. "Geochoron".

## References[edit | edit source]

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

- Hypercubes
- B4 symmetry
- Tes regiment
- Schläfli type 4,3,3
- 4D duoprisms
- 4D duoantitegums
- 4D duotransitionalterprisms
- 4D duoprismatic swirltegums
- Tetrahedral swirlchora
- Square dihedral swirlchora
- Square hosohedral swirlchora
- Isogonal swirlchora
- Isochoric swirlchora
- Noble swirlchora
- Gyrochora
- Convex regular polychora
- 4D uniform prisms
- Convex segmentochora
- Quasiregular dual polychora
- Zonochora