A tesseract or octachoron is a 4 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 our 3D world of limitations.
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
- 16 points (0D)
- 32 line segments (1D)
- 24 squares (2D)
- 8 cubes (3D)
- 1 tesseract (4D)
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 given by (±1,±1,±1,±1).
Equations
The surface of a tesseract can be given by the equation
Notations
- Toratopic notation:
- Tapertopic notation:
Related shapes
- Dual: Hexadecachoron
- Vertex figure: Tetrahedron, edge length
See Also
Regular polychora (+ tho) |
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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 |
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 |