Tesseract

Tesseract
(OFF file)
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	$\frac{\sqrt3}{2} ≈ 0.86603$
Face radius	$\frac{\sqrt2}{2} ≈ 0.70711$
Inradius	$\frac12 = 0.5$
Hypervolume	1
Dichoral angle	90°
Interior angles	At square: $\frac14$ At edge: $\frac18$ At vertex: $\frac1{16}$
Height	1
Central density	1
Number of external pieces	8
Level of complexity	1
Related polytopes
Army	Tes
Regiment	Tes
Dual	Hexadecachoron
Petrie dual	Petrial tesseract
Conjugate	None
Abstract & topological properties
Flag count	384
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B4, order 384
Convex	Yes
Net count	261
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

• 

  Tesseract in simple rotation

• 

  Tesseract in double rotation

• 

  Segmentochoron display, cube atop cube

• 

  Net

• 

  Cross-sections, cell-first

• 

  Cross-section animation, vertex-first

• 

  Wireframe, cell, net

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:

• $\left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac12\right).$

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 (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

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

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:

• Great icositetrachoron (3)
• Great stellated tetracontoctachoron (6)
• Chirotriangular crystallic enneacontahexachoron (12)
• Decachoric compound of 15 tesseracts (15)
• 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

Isogonal derivatives

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

• 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".

• Hartley, Michael. "{4,3,3}*384".