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Verse and Dimensions Wikia

Тессеракт

A tesseract or octachoron is a fourth-dimensional hypercube. Since the number of dimensions is a square number, the diagonal length of a tesseract is an integer - in this case, 2. Its Bowers acronym is "tes". It is one of the three regular polychora that can tile 4-dimensional space, forming the tesseractic tetracomb. Under the elemental naming scheme it is called a geochoron.

Tesseract Rubik's cubes can found online, but cannot be built in the 3D world.

Hypercube Products[]

The tesseract can be expressed as a hypercube product, potentially with less symmetry than the uniform and regular ideal tesseract, in five different ways:

- tesseract[]

As a tesseract, the hypervolumes can be expressed in terms of a single variable, the edge length l. This is the most symmetrical variant of the tesseract.

- cube prism[]

As a cube prism, the hypervolumes require two lengths to express: the edge length a of the cube, and the height b of the prism.

  • edge length =
  • surface area =
  • surcell volume =
  • surteron bulk =

When a=b, this becomes the symmetrical tesseract.

- square prism prism[]

As a square prism prism, the hypervolumes require three lengths to express: the edge length a of the square, and the seperate heights b and c of the two prisms.

  • edge length =
  • surface area =
  • surcell volume =
  • surteron bulk =

When a=b xor a=c, this becomes the cubic prism. When b=c, this becomes the square duoprism. When a=b=c, this becomes the symmetrical tesseract.

- line prism prism prism[]

As a line prism prism prism, the hypervolumes require four lengths to express. This is the least symmetrical variant of the tesseract.

  • edge length =
  • surface area =
  • surcell volume =
  • surteron bulk =

When a=b and c=d, a=c and b=d, xor a=d and b=c, this becomes the square duoprism. When a=b=c, b=c=d, a=c=d xor a=b=d, this becomes the cubic prism. When a=b, a=c, a=d, b=c, b=d xor c=d, this becomes the square prism prism. When a=b=c=d, this becomes the symmetrical tesseract.

- square duoprism[]

As a square duoprism, the hypervolumes require two lengths to express: the seperate edge lengths a and b of the two squares.

  • edge length =
  • surface area =
  • surcell volume =
  • surteron bulk =

When a=b, this becomes the symmetrical tesseract.

Properties[]

The tesseract can be exactly decomposed into eight cubic pyramids with unit side length. This is because the distance between a vertex and the center is the same as the edge length. If these pyramids are joined to the cubes of the tesseract the result is the icositetrachoron - the square pyramidal cells merge into octahedra.

Symbols[]

Dynkin symbols of the tesseract include:

  • x4o3o3o (regular)
  • x x4o3o (cubic prism)
  • x4o x4o (square duoprism)
  • x x x4o (square diprism)
  • x x x x (tesseractic block)
  • xx4oo3oo&#xt, xx xx4oo&#xt, xx xx xx&#xt (as cube atop cube)
  • oqooo3ooqoo3oooqo&#xt (vertex first)
  • xxx4ooo oqo&#xt, xxx xxx oqo&#xt (square first)
  • xxxx oqoo3ooqo&#xt (edge first)
  • qo3oo3oq *b3oo&#zx (sum of two demitesseracts)
  • xx xx qo oq&#zx (rhombic diprism)
  • xx qo3oo3oq&#zx *prism of sum of two tetrahedra)

Structure and Sections[]

Structure[]

The tesseract is composed of 8 cubic cells. Two of these cubes line in parallel 3-D spaces, while the remaining six connect the faces of the cubes. Four cubes meet at each vertex.

In cube-first position, it is a sequence of identical cubes. In square-centered orientation, it is a square which expands to a square prism and back. When seen line-first it is a line that expands to a triangular prism, then turns to a hexagonal prism, and then back. Finally in corner first orientation, it goes through the entire tetrahedral truncation series, from point to tetrahedron to octahedron in the middle and then back.

Hypervolumes[]

Subfacets[]

Radii[]

  • Vertex radius:
  • Edge radius:
  • Face radius:
  • Cell radius:

Angles[]

  • Dichoral angle: 90º

Vertex coordinates[]

The vertices of a tesseract with side 2 can be denoted on a 4D Cartesian plane by (±1,±1,±1,±1).

Equations[]

The surface of a tesseract can be graphed by the equation

Notations[]

  • Toratopic notation:
  • Tapertopic notation:

Related shapes[]

See Also[]

Regular polychora (+ tho)
Convex regular polychora: pen · tes · hex · ico · hi · ex

Self-intersecting regular polychora: fix · gohi · gahi · sishi · gaghi · gishi · gashi · gofix · gax · gogishi

Tesseractihemioctachoron: tho

Dimensionality Negative One Zero One Two Three Four Five Six Seven Eight Nine Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen ... Aleph null
Simplex

Null polytope


Point


Line segment


Triangle

Tetrahedron

Pentachoron

Hexateron

Heptapeton

Octaexon

Enneazetton

Decayotton

Hendecaxennon

Dodecadakon

Tridecahendon

Tetradecadokon

Pentadecatradakon

Hexadecatedakon

Heptadecapedakon

... Omegasimplex
Cross

Square

Octahedron

Hexadecachoron

Pentacross

Hexacross

Heptacross

Octacross

Enneacross

Dekacross

Hendekacross

Dodekacross

Tridekacross

Tetradekacross

Pentadekacross

Hexadekacross

... Omegacross
Hydrotopes

Pentagon

Icosahedron

Hexacosichoron

Order-5 pentachoric tetracomb

Order-5 hexateric pentacomb

...
Hypercube

Square

Cube

Tesseract

Penteract

Hexeract

Hepteract

Octeract

Enneract

Dekeract

Hendekeract

Dodekeract

Tridekeract

Tetradekeract

Pentadekeract

Hexadekeract

... Omegeract
Cosmotopes

Pentagon

Dodecahedron

Hecatonicosachoron

Order-3 hecatonicosachoric tetracomb

Order-3-3 hecatonicosachoric pentacomb

...
Hyperball

Disk

Ball

Gongol

Pentorb

Hexorb

Heptorb

Octorb

Enneorb

Dekorb

Hendekorb

Dodekorb

Tridekorb

Tetradekorb

Pentadekorb

Hexadekorb

... Omegaball

Tetrahedral hosochoron Pentachoron Tesseract Hecatonicosachoron Order-3 hexagonal tiling honeycomb
Cubic dichoron Tesseract Cubic honeycomb Order-5 cubic honeycomb Order-6 cubic honeycomb
Regular
Rectified
Birectified
Trirectified
Truncated
Bitruncated
Tritruncated
Tesseract Rectified tesseract Icositetrachoron Hexadecachoron Truncated tesseract Bitruncated tesseract Truncated hexadecachoron
Cantellated
Bicantellated
Cantitruncated
Bicantitruncated
Runcinated
Runcicantellated
Runcitruncated
Runcicantitruncated
Cantellated tesseract Rectified icositetrachoron Cantitruncated tesseract Truncated icositetrachoron Runcinated tesseract Runcitruncated hexadecachoron Runcitruncated tesseract Omnitruncated tesseract
Regular
Rectified
Birectified
Trirectified
Truncated
Bitruncated
Tritruncated
Cubic dichoron Rectified cubic dichoron Rectified octahedral hosochoron Octahedral hosochoron Truncated cubic dichoron Bitruncated cubic dichoron Octahedral prism
Cantellated
Bicantellated
Cantitruncated
Bicantitruncated
Runcinated
Runcicantellated
Runcitruncated
Runcicantitruncated
Cantellated cubic dichoron Cuboctahedral prism Cantitruncated cubic dichoron Truncated octahedral prism Tesseract Rhombicuboctahedral prism Truncated cubic prism Great rhombicuboctahedral prism
Regular
Rectified
Birectified
Trirectified
Truncated
Bitruncated
Tritruncated
Square tetrachoron Rectified square tetrachoron Rectified square tetrachoron Square tetrachoron Truncated square tetrachoron Tesseract Truncated square tetrachoron
Cantellated
Bicantellated
Cantitruncated
Bicantitruncated
Runcinated
Runcicantellated
Runcitruncated
Runcicantitruncated
Tesseract Tesseract Square-octagonal duoprism Square-octagonal duoprism Tesseract Square-octagonal duoprism Square-octagonal duoprism Octagonal duoprism
Regular
Rectified
Birectified
Trirectified
Truncated
Bitruncated
Tritruncated
Square dihedral dichoron Rectified square dihedral dichoron Rectified square hosohedral hosochoron Square hosohedral hosochoron Truncated square dihedral dichoron Square dihedral prism Square hosohedral prism
Cantellated
Bicantellated
Cantitruncated
Bicantitruncated
Runcinated
Runcicantellated
Runcitruncated
Runcicantitruncated
Square dihedral prism Square dihedral prism Truncated square dihedral prism Tesseract Square dihedral prism Tesseract Truncated square dihedral prism Square-octagonal duoprism
Regular
Rectified
Birectified
Trirectified
Truncated
Bitruncated
Tritruncated
Digonal dihedral dichoron Digonal dihedral dichoron Digonal dihedral dichoron Digonal dihedral dichoron Digonal dihedral prism Digonal dihedral prism Digonal dihedral prism
Cantellated
Bicantellated
Cantitruncated
Bicantitruncated
Runcinated
Runcicantellated
Runcitruncated
Runcicantitruncated
Digonal dihedral prism Digonal dihedral prism Digonal-square duoprism Digonal-square duoprism Digonal dihedral prism Digonal-square duoprism Digonal-square duoprism Tesseract
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