TESSERACT

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Tesseract
(8-cell)
''4-cube''
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|bgcolor=#ffffff align=center colspan=2|
Schlegel diagram
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|bgcolor=#e7dcc3|Type||Regular polychoron
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|bgcolor=#e7dcc3|Family||hypercube
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|bgcolor=#e7dcc3|Cells||8 (4.4.4)
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|bgcolor=#e7dcc3|Faces||24 {4}
|-
|bgcolor=#e7dcc3|Edges||32
|-
|bgcolor=#e7dcc3|Vertices||16
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|bgcolor=#e7dcc3|Vertex figure||(3.3.3)
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|bgcolor=#e7dcc3|Schläfli symbols|| {4,3,3}
{4,3}x{}
{4}x{4}
{4}x{}x{}
{}x{}x{}x{}
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|bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||




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|bgcolor=#e7dcc3|Symmetry group||B₄, [3,3,4]
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|bgcolor=#e7dcc3|Dual||16-cell
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|bgcolor=#e7dcc3|Properties||convex
|}
In geometry, the 'tesseract', also called '8-cell' or 'octachoron', is the four-dimensional analog of the (three-dimensional) cube, where motion along the fourth dimension is often a representation for bounded transformations of the cube through time. The tesseract is to the cube as the cube is to the square; or, more formally, the tesseract can be described as a regular convex 4-polytope whose boundary consists of eight cubical cells.
A generalization of the cube to dimensions greater than three is called a “hypercube”, “n-cube” or “measure polytope”. The tesseract is the four-dimensional hypercube or 4-cube.
According to the Oxford English Dictionary, the word ''tesseract'' was coined and first used in 1888 by Charles Howard Hinton in his book ''A New Era of Thought'', from the Ionic Greek “” (“four rays”), referring to the four lines from each vertex to other vertices. Some people have called the same figure a “tetracube”, and also simply a "hypercube" (although a hypercube can be a cube of any dimension).

Contents

Geometry

Projections to 2 dimensions

Projections to 3 dimensions

Unfolding the tesseract

Image gallery

Tesseracts in art and literature

Hypercubes in computer architecture

See also

References

External links

Geometry

The tesseract can be constructed in a number of different ways. As a regular polytope constructed by three cubes folded together around every edge, it has Schläfli symbol ''{4,3,3}''. Constructed as a 4D hyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol ''{4,3}x{ }''. As a duoprism, a Cartesian product of two squares, it can be named by a composite Schläfli symbol ''{4}x{4}''.
Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron. The dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol {3,3,4}.
The standard tesseract in Euclidean 4-space is given as the convex hull of the points (±1, ±1, ±1, ±1). That is, it consists of the points:
:$\{(x\_1,x\_2,x\_3,x\_4)\; in\; mathbb\; R^4\; ,:,\; -1\; leq\; x\_i\; leq\; 1\; \}.$
A tesseract is bounded by eight hyperplanes (''x''_i = ±1). Each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. Three cubes and three squares intersect at each edge. There are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes, 24 squares, 32 edges, and 16 vertices.

Projections to 2 dimensions

The construction of a hypercube can be imagined the following way:

★ 1-dimensional: Two points A and B can be connected to a line, giving a new line AB.

★ 2-dimensional: Two parallel lines AB and CD can be connected to become a square, with the corners marked as ABCD.

★ 3-dimensional: Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH.

★ 4-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP.
This structure is not easily imagined but it is possible to project tesseracts into three- or two-dimensional spaces. Furthermore, projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices. In this fashion, one can obtain pictures that no longer reflect the spatial relationships within the tesseract, but which illustrate the connection structure of the vertices, such as in the following examples:
A tesseract is in principle obtained by combining two cubes. The scheme is similar to the construction of a cube from two squares: juxtapose two copies of the lower dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length. A multitude of cubes that are nicely interconnected. The vertices of the tesseract with respect to the distance along the edges, with respect to the bottom point. This view is of interest when using tesseracts as the basis for a network topology to link multiple processors in parallel computing: the distance between two nodes is at most 4 and there are many different paths to allow weight balancing.
Tesseracts are also bipartite graphs, just as a path, square, cube and tree are.

Projections to 3 dimensions

The ''cell-first'' parallel projection of the tesseract into 3-dimensional space has a cubical envelope. The nearest and farthest cells are projected onto the cube, and the remaining 6 cells are projected onto the 6 square faces of the cube.
The ''face-first'' parallel projection of the tesseract into 3-dimensional space has a cuboidal envelope. Two pairs of cells project to the upper and lower halves of this envelope, and the 4 remaining cells project to the side faces.
The ''edge-first'' parallel projection of the tesseract into 3-dimensional space has an envelope in the shape of a hexagonal prism. Six cells project onto rhombic prisms, which are laid out in the hexagonal prism in a way analogous to how the faces of the 3D cube project onto 6 rhombs in a hexagonal envelope under vertex-first projection. The two remaining cells project onto the prism bases.
The ''vertex-first'' parallel projection of the tesseract into 3-dimensional space has a rhombic dodecahedral envelope. There are exactly two ways of decomposing a rhombic dodecahedron into 4 congruent parallelepipeds, giving a total of 8 possible parallelepipeds. The images of the tesseract's cells under this projection are precisely these 8 parallelepipeds. This projection is also the one with maximal volume.

Unfolding the tesseract

The tesseract can be unfolded into eight cubes, just as the cube can be unfolded into six squares. An unfolding of a polyhedron is called a net. There are 261 distinct nets of the tesseract. The unfoldings of the tesseract can be counted by mapping the nets to ''paired trees'' (a tree together with a perfect matching in its complement).

Image gallery

{| class="prettytable" width="610"
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Stereographic projection
(Edges are projected onto the 3-sphere)
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A 3D projection of an 8-cell performing a simple rotation about a plane which bisects the figure from front-left to back-right and top to bottom.
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A 3D projection of an 8-cell performing a double rotation about two orthogonal planes.
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|colspan=3 valign=top align=center|Orthogonal projection
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|align=center|
|align=center|
|align=center|
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A net of a tesseract.
()
|colspan=2 valign=top align=center|
A stereoscopic 3D projection of a tesseract.
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Tesseracts in art and literature

'Books/print'

★ Carl Sagan describes the tesseract in great detail using layman's terms in .

★ In Edwin A. Abbott's novel ''Flatland'', 1884, a hypercube is imagined by the narrator.

★ Robert A. Heinlein mentioned hypercubes in at least three of his science fiction stories. In ''“—And He Built a Crooked House—”'' (1940), he described a house built as a net (i.e., an unfolding of the cells into three-dimensional space) of a tesseract. It collapsed, becoming a real 4-dimensional tesseract. Heinlein's 1963 novel ''Glory Road'' included the ''foldbox'', a hyperdimensional packing case that was bigger inside than outside.

★ Hypercubes and all kinds of multi-dimensional space and structures star prominently in many books by Rudy Rucker.

★ A hypercube is used as the main deus ex machina of Robert J. Sawyer's book ''Factoring Humanity'', even appearing on its North American cover.

★ The tesseract is mentioned in the children's fantasy novel ''A Wrinkle in Time'', by Madeleine L'Engle, as a way of introducing the concept of higher dimensions, but the description more closely matches a wormhole.

★ Piers Anthony's novel ''Cube Route'' also features a tesseract.

★ Alex Garland's second book is called "Tesseract: a novel".

★ The DC Comics crossover ''DC One Million'' showed a future Earth in which cities occupied extradimensional areas called tesseracts, leaving the planet's surface unspoiled. Similar technology was used for Superman's current Fortress of Solitude, and was used as storage space in the headquarters of the original incarnation (pre-Zero Hour) of the Legion of Super-Heroes.

★ Ian Irvine's Sci-Fi Fantasy 'Tetrarch' Book two of the 'The Well of Echoes Quartet'.

★ David Lubar's 'Sleeping Freshmen Never Lie'. Tesseract meaning "spiraling into another dimension."

★ Lewis Padgett's classic short story, 'Mimsy Were the Borogoves' features two children who construct a tesseract using information from the future. They ultimately disappear into another dimension.
'Visual arts'

★ The painting ''Crucifixion (Corpus Hypercubus)'', by Salvador Dalí, 1954, depicts the crucified Jesus upon the net of a hypercube. It is featured at the Metropolitan Museum of Art in New York, USA.
'Television and movies'

★ The television program ''Andromeda'' makes use of ''tesseract generators'' as a plot device. These are primarily intended to manipulate space (also referred to as phase shifting) but often cause problems with time as well.

★ A character in the television program ''Numb3rs'' shows a model of a tesseract in the second-season episode ''Rampage'', during a discussion of using a 4-dimensional perspective to analyze an event.

★ The TV programme Strange Days at Blake Holsey High has an episode where the school campups transforms into a self-folding hypercube.

★ The movie '' focuses on eight strangers trapped inside a net of connected cubes.

★ The movie ''The Last Mimzy'' mentions tesseracts in a list of other geometrical shapes when the children are dreaming about the bridge across the universe, most likely in homage to ''A Wrinkle in Time''.
'Business'

★ ''Tesseract Books'' was a prominent publisher of Canadian science fiction books. The company is now an imprint of Hades Publishing Inc.
'Video Games'

★ ''Starflight'' included a tesseract as an artifact which could be found by exploring planet surfaces.

Hypercubes in computer architecture

In computer science, the term hypercube refers to a specific type of parallel computer, whose processors, or processing elements (PEs), are interconnected in the same way as the vertices of a hypercube.
Thus, an ''n''-dimensional hypercube computer has 2^''n'' PEs, each directly connected to ''n'' other PEs.
Examples include the nCUBE machines used to win the first Gordon Bell Prize, the Caltech Cosmic Cube; the Connection Machine, which uses the hypercube topology to connect ''groups'' of processors.
The purpose of this it to function as an exponetial expansion of parallel processing; in computer farms of this type the computing power of the individual processors is not merely added but multiplied, leading to an exponential growth in processing power.

See also

★ Fourth dimension

★ Hypercube - The dimensional family of polytopes:

★
★ Square - 2 dimensions

★
★ Cube - 3 dimensions

★
★ Penteract - 5 dimensions

★
★ Hexeract - 6 dimensions

★
★ Hepteract - 7 dimensions

★
★ Octeract - 8 dimensions

★
★ ...

★ Uniform polychoron#Tesseract family - uniform polychora based on tesseract

★ Other regular polytopes

★
★ Simplex - simplest polytope family

★
★ Cross-polytope - The dual family to hypercubes

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★ Demihypercube - a family of alternated hypercubes

★
★ List of regular polytopes

★ 3-sphere

★ Metatron's Cube

★ Logical connectives

References

★ H. S. M. Coxeter, ''Regular Polytopes'', 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.

External links

★

★

★ Der 8-Zeller (8-cell) Marco Möller's Regular polytopes in R⁴ (German)

★ WikiChoron: Tesseract

★ HyperSolids is an open source program for the Apple Macintosh ''(Mac OS X and higher)'' which generates the five regular solids of three-dimensional space and the six regular hypersolids of four-dimensional space.

★ Hypercube 98 A Windows program that displays animated hypercubes, by Rudy Rucker

★ ken perlin's home page A way to visualize hypercubes, by Ken Perlin

★ Some Notes on the Fourth Dimension includes very good animated tutorials on several different aspects of the tesseract, by Davide P. Cervone

★ Tesseract animation with hidden volume elimination