FANDOM


Тессеракт

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 would 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:

\{4,3,3\} - 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.

\{4,3\} \times \{\} - 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 = 8\left(3a + b\right)
  • surface area = 12a \left( a + b \right)
  • surcell volume = 2a^2 \left( a + 3b \right)
  • surteron bulk = a^3 b

When a=b, this becomes the symmetrical tesseract.

\{4\} \times \{\}^2 - 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 = 8 \left( 2a + b + c \right)
  • surface area = 4 \left( a^2 + 2 ab + 2 ac + bc \right)
  • surcell volume = 2a \left(  ab + ac + 2bc \right)
  • surteron bulk = a^2 b c

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.

\{\}^4 - 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 = 8\left( a + b + c + d \right)
  • surface area = 4 \left( ab + ac + ad + bc + bd + cd \right)
  • surcell volume = 2 \left( abc + abd + acd + bcd \right)
  • surteron bulk = abcd

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.

\{4\}^2 - 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 = 16 \left( a + b \right)
  • surface area = 4 \left( a^2 + 4 ab + b^2 \right)
  • surcell volume = 4ab \left( a + b \right)
  • surteron bulk = a^2 b^2

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.

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

See Also

Zeroth First Second Third Fourth Fifth Sixth Seventh Eighth Ninth Tenth Eleventh Twelfth Thirteenth Fourteenth Fifteenth Sixteenth
Simplex Point Line segment Triangle Tetrahedron Pentachoron Hexateron Heptapeton Octaexon Enneazetton Decayotton Hendecaxennon Dodecadakon Tridecahendon Tetradecadokon Pentadecatradakon Hexadecatedakon Heptdecapedakon
Hypercube Point Line segment Square Cube Tesseract Penteract Hexeract Hepteract Octeract Enneract Dekeract Hendekeract Dodekeract Tridekeract Tetradekeract Pentadekeract Hexadekeract
Cross Point Line segment Square Octahedron Hexadecachoron Pentacross Hexacross Heptacross Octacross Enneacross Dekacross Hendekacross Dodekacross Tridekacross Tetradekacross Pentadekacross Hexadekacross
Hypersphere Point Line segment Disk Ball Gongol Pentorb Hexorb Heptorb Octorb Enneorb Dekorb Hendekorb Dodekorb Tridekorb Tetradekorb Pentadekorb Hexadekorb
\{2,3,3\} \{3,3,3\} \{4,3,3\} \{5,3,3\} \{6,3,3\}
Tetrahedral hosochoron Pentachoron Tesseract Hecatonicosachoron Order-3 hexagonal tiling honeycomb
\{4,3,2\} \{4,3,3\} \{4,3,4\} \{4,3,5\} \{4,3,6\}
Cubic dichoron Tesseract Cubic honeycomb Order-5 cubic honeycomb Order-6 cubic honeycomb
Regular
t_0 \{4,3,3\}
Rectified
t_1 \{4,3,3\}
Birectified
t_2 \{4,3,3\}
Trirectified
t_3 \{4,3,3\}
Truncated
t_{0,1} \{4,3,3\}
Bitruncated
t_{1,2} \{4,3,3\}
Tritruncated
t_{2,3} \{4,3,3\}
Tesseract Rectified tesseract Icositetrachoron Hexadecachoron Truncated tesseract Bitruncated tesseract Truncated hexadecachoron
Cantellated
t_{0,2} \{4,3,3\}
Bicantellated
t_{1,3} \{4,3,3\}
Cantitruncated
t_{0,1,2} \{4,3,3\}
Bicantitruncated
t_{1,2,3} \{4,3,3\}
Runcinated
t_{0,3} \{4,3,3\}
Runcicantellated
t_{0,2,3} \{4,3,3\}
Runcitruncated
t_{0,1,3} \{4,3,3\}
Runcicantitruncated
t_{0,1,2,3} \{4,3,3\}
Cantellated tesseract Rectified icositetrachoron Cantitruncated tesseract Truncated icositetrachoron Runcinated tesseract Runcitruncated hexadecachoron Runcitruncated tesseract Omnitruncated tesseract
Regular
t_0 \{4,3,2\}
Rectified
t_1 \{4,3,2\}
Birectified
t_2 \{4,3,2\}
Trirectified
t_3 \{4,3,2\}
Truncated
t_{0,1} \{4,3,2\}
Bitruncated
t_{1,2} \{4,3,2\}
Tritruncated
t_{2,3} \{4,3,2\}
Cubic dichoron Rectified cubic dichoron Rectified octahedral hosochoron Octahedral hosochoron Truncated cubic dichoron Bitruncated cubic dichoron Octahedral prism
Cantellated
t_{0,2} \{4,3,2\}
Bicantellated
t_{1,3} \{4,3,2\}
Cantitruncated
t_{0,1,2} \{4,3,2\}
Bicantitruncated
t_{1,2,3} \{4,3,2\}
Runcinated
t_{0,3} \{4,3,2\}
Runcicantellated
t_{0,2,3} \{4,3,2\}
Runcitruncated
t_{0,1,3} \{4,3,2\}
Runcicantitruncated
t_{0,1,2,3} \{4,3,2\}
Cantellated cubic dichoron Cuboctahedral prism Cantitruncated cubic dichoron Truncated octahedral prism Tesseract Rhombicuboctahedral prism Truncated cubic prism Great rhombicuboctahedral prism
Regular
t_0 \{4,2,4\}
Rectified
t_1 \{4,2,4\}
Birectified
t_2 \{4,2,4\}
Trirectified
t_3 \{4,2,4\}
Truncated
t_{0,1} \{4,2,4\}
Bitruncated
t_{1,2} \{4,2,4\}
Tritruncated
t_{2,3} \{4,2,4\}
Square tetrachoron Rectified square tetrachoron Rectified square tetrachoron Square tetrachoron Truncated square tetrachoron Tesseract Truncated square tetrachoron
Cantellated
t_{0,2} \{4,2,4\}
Bicantellated
t_{1,3} \{4,2,4\}
Cantitruncated
t_{0,1,2} \{4,2,4\}
Bicantitruncated
t_{1,2,3} \{4,2,4\}
Runcinated
t_{0,3} \{4,2,4\}
Runcicantellated
t_{0,2,3} \{4,2,4\}
Runcitruncated
t_{0,1,3} \{4,2,4\}
Runcicantitruncated
t_{0,1,2,3} \{4,2,4\}
Tesseract Tesseract Square-octagonal duoprism Square-octagonal duoprism Tesseract Square-octagonal duoprism Square-octagonal duoprism Octagonal duoprism
Regular
t_0 \{4,2,2\}
Rectified
t_1 \{4,2,2\}
Birectified
t_2 \{4,2,2\}
Trirectified
t_3 \{4,2,2\}
Truncated
t_{0,1} \{4,2,2\}
Bitruncated
t_{1,2} \{4,2,2\}
Tritruncated
t_{2,3} \{4,2,2\}
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
t_{0,2} \{4,2,2\}
Bicantellated
t_{1,3} \{4,2,2\}
Cantitruncated
t_{0,1,2} \{4,2,2\}
Bicantitruncated
t_{1,2,3} \{4,2,2\}
Runcinated
t_{0,3} \{4,2,2\}
Runcicantellated
t_{0,2,3} \{4,2,2\}
Runcitruncated
t_{0,1,3} \{4,2,2\}
Runcicantitruncated
t_{0,1,2,3} \{4,2,2\}
Square dihedral prism Square dihedral prism Truncated square dihedral prism Tesseract Square dihedral prism Tesseract Truncated square dihedral prism Square-octagonal duoprism
Regular
t_0 \{2,2,2\}
Rectified
t_1 \{2,2,2\}
Birectified
t_2 \{2,2,2\}
Trirectified
t_3 \{2,2,2\}
Truncated
t_{0,1} \{2,2,2\}
Bitruncated
t_{1,2} \{2,2,2\}
Tritruncated
t_{2,3} \{2,2,2\}
Digonal dihedral dichoron Digonal dihedral dichoron Digonal dihedral dichoron Digonal dihedral dichoron Digonal dihedral prism Digonal dihedral prism Digonal dihedral prism
Cantellated
t_{0,2} \{2,2,2\}
Bicantellated
t_{1,3} \{2,2,2\}
Cantitruncated
t_{0,1,2} \{2,2,2\}
Bicantitruncated
t_{1,2,3} \{2,2,2\}
Runcinated
t_{0,3} \{2,2,2\}
Runcicantellated
t_{0,2,3} \{2,2,2\}
Runcitruncated
t_{0,1,3} \{2,2,2\}
Runcicantitruncated
t_{0,1,2,3} \{2,2,2\}
Digonal dihedral prism Digonal dihedral prism Digonal-square duoprism Digonal-square duoprism Digonal dihedral prism Digonal-square duoprism Digonal-square duoprism Tesseract

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.