Tesseract (8-cell) 4-cube
Schlegel diagram
Type	Regular polychoron
Family	hypercube
Cells	8 (4.4.4)
Faces	24 {4}
Edges	32
Vertices	16
Vertex figure	(3.3.3)
Schläfli symbols	{4,3,3} {4,3}x{} {4}x{4} {4}x{}x{} {}x{}x{}x{}
Coxeter-Dynkin diagrams
Symmetry group	B4, [3,3,4]
Dual	16-cell
Properties	convex

In geometry, the tesseract, also called 8-cell or octachoron, is the four-dimensional analog of the cube, which is in turn the three dimensional analog of the square. 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. Tesseract may mean: Tesseract — the 4-dimensional analogue of the cube. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 294 KB, MIME type: image/png)tesseract [Schlegel diagram]], wireframe. ... Examples colored by the number of sides on each face. ... In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which is both a regular and convex. ... A square A projection of a cube (into a two-dimensional image) A projection of a hypercube (into a two-dimensional image) In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... For other uses, see Square. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In mathematics, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations. ... Coxeter groups in the plane with equivalent diagrams. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. ... In geometry, a 16-cell, or orthoplex, is a regular convex polychoron, or polytope existing in four dimensions. ... Look up Convex set in Wiktionary, the free dictionary. ... For other uses, see Geometry (disambiguation). ... For other uses, see Fourth dimension (disambiguation). ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Look up square in Wiktionary, the free dictionary. ... For other uses, see Square. ... In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which is both regular and convex. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Cubic honeycomb - four cubic cells per edge hypercube - three cubic cells per edge In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object. ...

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. A square A projection of a cube (into a two-dimensional image) A projection of a hypercube (into a two-dimensional image) In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ...

Rotating shadow of a tesseract rotating on a single axis

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 Greek “τέσσερεις ακτίνες” (“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 of any dimension). Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... OED stands for Oxford English Dictionary Office of Enrollment & Discipline This page concerning a three-letter acronym or abbreviation is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Charles Howard Hinton (1853-1907) was a British mathematician and writer of science fiction works that he called scientific romances. ...

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}. A dodecahedron, one of the five Platonic solids. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... In mathematics, the Schläfli symbol is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations. ... A duoprism is a 4-dimensional figure resulting from the Cartesian product of two polygons in the 2-dimensional Euclidean space. ... In mathematics, the Cartesian product is a direct product of sets. ... For other uses, see Square. ...

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}. In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the others. ... In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ...

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: Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Convex hull: elastic band analogy In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X. // For planar objects, i. ...

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. A hyperplane is a concept in geometry. ...

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. For other uses of topology, see topology (disambiguation). ... Parallel computing is the simultaneous execution of the same task (split up and specially adapted) on multiple processors in order to obtain results faster. ...

Tesseracts are also bipartite graphs, just as a path, square, cube and tree are. In the mathematical field of graph theory, a bipartite graph is a special graph where the set of vertices can be divided into two disjoint sets and such that no edge has both end-points in the same set. ...

Projections to 3 dimensions

The rhombic dodecahedron forms the hull of the vertex-first projection of a tesseract to 3 dimensions

Projection envelopes of the tesseract. (Each cell is drawn with different color faces, inverted cells are undrawn)

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. Image File history File links Hypercubeorder. ... Image File history File links Hypercubeorder. ... Image File history File links Size of this preview: 510 × 599 pixelsFull resolution (1540 × 1810 pixel, file size: 465 KB, MIME type: image/png)tesseract {4,3,3} - solid, orthogonal projections File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links Size of this preview: 510 × 599 pixelsFull resolution (1540 × 1810 pixel, file size: 465 KB, MIME type: image/png)tesseract {4,3,3} - solid, orthogonal projections File historyClick on a date/time to view the file as it appeared at that time. ... Graphical projection in the visual sciences is an imaging procedure the protocols of which preclude the necessity of mathematical calculation. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ...

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. The rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. ... In geometry, a parallelepiped (now usually pronounced , traditionally[1] in accordance with its etymology in Greek παραλληλ-επίπεδον, a body having parallel planes) is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ...

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 polytope is called a net. There are 261 distinct nets of the tesseract.^[1] 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). Categories: Polyhedra | Stub ... A labeled tree with 6 vertices and 5 edges In graph theory, a tree is a graph in which any two vertices are connected by exactly one path. ... Dheeraj Gedam This article is about mathematical matchings. ... In graph theory the complement or inverse of a graph is a graph on the same vertices such that two vertices of are adjacent if and only if they are not adjacent in . ...

Image gallery

Stereographic projection (Edges are projected onto the 3-sphere)	Simple edge graph	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.	A 3D projection of an 8-cell performing a double rotation about two orthogonal planes.
Orthogonal projection
A net of a tesseract. ( View animation.)	A stereoscopic 3D projection of a tesseract.

Image File history File links Download high-resolution version (1000x1000, 230 KB) Summary Stereographic projection of the 8-cell, a 4-dimensional polytope. ... Stereographic projection of a circle of radius R onto the x axis. ... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ... Image File history File links No higher resolution available. ... SO(4) is the symbol used in mathematics for the group of rotations about a fixed point in four-dimensional Euclidean space (for short, the 4D rotation group). ... Image File history File links Tesseract. ... SO(4) is the symbol used in mathematics for the group of rotations about a fixed point in four-dimensional Euclidean space (for short, the 4D rotation group). ... In geometry, an orthogonal projection of a k-dimensional object onto a d-dimensional hyperplane (d < k) is obtained by intersections of (k − d)- dimensional hyperplanes drawn through the points of an object orthogonally to the d-hyperplane. ... Image File history File links Hypercubecubes. ... Image File history File links Hypercubeorder. ... Image File history File links Hypercubestar. ... Image File history File links Tesseract2. ... Categories: Polyhedra | Stub ... Image File history File links Hcube_fold. ... Image File history File links 3D_stereographic_projection_tesseract. ... Stereogram may also refer to an integrated high fidelity system or music centre. ...

Tesseracts in popular culture

Trivia sections are discouraged under Wikipedia guidelines. The article could be improved by integrating relevant items and removing inappropriate ones.

Crucifixion (Corpus Hypercubus) (1954)

Books/print Image File history File links Broom_icon. ... Image File history File links Download high-resolution version (726x1139, 130 KB)Image taken from http://www. ... Image File history File links Download high-resolution version (726x1139, 130 KB)Image taken from http://www. ...

• Madeline L'Engle's novel A Wrinkle in Time uses tesseracts as a way for Meg Murry and her companions to travel to other planets and dimensions, however the description more closely matches a wormhole.
• Carl Sagan describes the tesseract in great detail using layman's terms in Cosmos, episode 10.
• 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.
• 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 second most recent 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.
• Umberto Eco references tesseracts in Foucault's Pendulum

Visual arts Madeleine LEngle (b. ... For the movie adaptation, see A Wrinkle in Time (film) . A Wrinkle in Time is a science fantasy[1] novel by Madeleine LEngle, written between 1959 and 1960[2] and published in 1962 after at least 26 rejections by publishers[3] because it was, in LEngles words... For other uses, see Wormhole (disambiguation). ... Insert non-formatted text here Carl Edward Sagan (November 9, 1934 – December 20, 1996) was an American astronomer and astrobiologist and a highly successful popularizer of astronomy, astrophysics, and other natural sciences. ... Cosmos: A Personal Voyage was the name of a thirteen part television series produced by Carl Sagan and Ann Druyan which was first broadcast by the Public Broadcasting Service in 1980. ... Edwin Abbott Abbott Edwin Abbott Abbott (December 20, 1838 – October 12, 1926), English schoolmaster and theologian, is best known as the author of the mathematical satire and religious allegory Flatland (1884). ... For various uses of the term Flatlander, see Flatlander (disambiguation) Flatland: A Romance of Many Dimensions is a 1884 novella by Edwin Abbott Abbott, still popular among mathematics and computer science students, and considered useful reading for people studying topics such as the concept of other dimensions. ... Robert Anson Heinlein (July 7, 1907 – May 8, 1988) was one of the most popular, influential, and controversial authors of hard science fiction. ... Science fiction is a form of speculative fiction principally dealing with the impact of imagined science and technology, or both, upon society and persons as individuals. ... “—And He Built a Crooked House—” is a science fiction short story by Robert A. Heinlein first published in Astounding Science Fiction in March 1941, and reprinted in the collection The Unpleasant Profession of Jonathan Hoag in 1959. ... Glory Road is a fantasy novel by Robert A. Heinlein published in 1963. ... Rudy Rucker, Fall 2004, photo by Georgia Rucker. ... For other uses, see Deus ex machina (disambiguation). ... Robert J. Sawyer is a Canadian hard science fiction writer, born in Ottawa in 1960 and now resident in Mississauga. ... Piers Anthony Dillingham Jacob (born August 6, 1934 in Oxford, England) is an American writer in the science fiction and fantasy genres, publishing under the name Piers Anthony. ... Cube Route is the twenty-seventh book of the Xanth series by Piers Anthony. ... Alex Garland (born 1970) is a British novelist and screenwriter. ... DC Comics is an American comic book and related media company. ... It has been suggested that Gaming crossovers be merged into this article or section. ... DC One Million was a crossover event published by DC Comics in 1998. ... Superman is a fictional character and comic book superhero , originally created by American writer Jerry Siegel and Canadian artist Joe Shuster and published by DC Comics. ... The Fortress of Solitude is the occasional headquarters of Superman in DC Comics. ... Zero Hour: Crisis in Time was a 1994 comic book miniseries and crossover storyline that ran in DC Comics. ... LSH redirects here. ... Ian Irvine (born 1950) is an Australian fantasy/eco-thriller author and marine scientist. ... David Lubar is an author, Electronic game programmer and designer who has written numerous books for teens. ... Lewis Padgett was the joint pseudonym of the science-fiction authors and spouses Henry Kuttner and C. L. Moore. ... Mimsy Were The Borogoves is a short story (now being made into a feature-length film titled The Last Mimzy) by Lewis Padgett originally published in 1943. ... Umberto Eco (born January 5, 1932) is an Italian medievalist, semiotician, philosopher and novelist, best known for his novel The Name of the Rose (Il nome della rosa) and his many essays. ... Foucaults Pendulum (original title: Il pendolo di Foucault) is a novel by Italian novelist and philosopher Umberto Eco. ...

• 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 Salvador Domingo Felipe Jacinto Dalí i Domènech, 1st Marquis of Púbol (May 11, 1904 – January 23, 1989), was a Spanish surrealist painter of Catalan descent born in Figueres, Catalonia (Spain). ... This article is about Jesus of Nazareth. ... Metropolitan Museum of Art New York Elevation The Metropolitan Museum of Art, often referred to simply as the Met, is one of the worlds largest and most important art museums. ... This article is about the state. ...

• 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 campus transforms into a self-folding hypercube.
• The movie Cube 2: Hypercube focuses on eight strangers trapped inside a net of connected cubes or perhaps some sort of tesseract which shifts in the direction of the 8 strangers movements in any directions, making a seemingly endless continuum of singular 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, as does the short story on which it is based, 'Mimsy Were the Borogoves' (listed above). This may also be in homage to A Wrinkle in Time.

Business A television program (US), television programme (UK) or simply television show is a segment of programming in television broadcasting. ... Gene Roddenberrys Andromeda is an American science fiction television series, based on unused material by Gene Roddenberry developed by Robert Hewitt Wolfe, and produced posthumously by his widow, Majel Roddenberry. ... Numb3rs (also capitalized as NUMB3RS and pronounced as Numbers) is an American television show produced by brothers Ridley Scott and Tony Scott. ... Strange Days at Blake Holsey High, also known as Black Hole High, is a Canadian science fiction television program which first aired in North America in October 2002 on Discovery Kids on NBC and Discovery Kids Channel. ... Categories: Movie stubs | 2002 films | Science fiction films ... The Last Mimzy is a 2007 science fiction family film directed by Bob Shaye. ... Mimsy Were The Borogoves is a short story (now being made into a feature-length film titled The Last Mimzy) by Lewis Padgett originally published in 1943. ... For the movie adaptation, see A Wrinkle in Time (film) . A Wrinkle in Time is a science fantasy[1] novel by Madeleine LEngle, written between 1959 and 1960[2] and published in 1962 after at least 26 rejections by publishers[3] because it was, in LEngles words...

• 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.

Games See interstellar travel for travel between the stars. ...

• A tesseract forms the basis of the fantasy Advanced Dungeons & Dragons module Baba Yaga's Hut, which appeared in an early issue of Dragon Magazine, with the tesseract existing as the interior of the titular Hut.

For other uses, see Dungeons & Dragons (disambiguation). ... Yaga can refer to: Yajna (Hindu mythology) Baba Yaga (Russian mythology) Yaga (clothing company) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

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
  • Demihypercube - a family of alternated hypercubes
  • List of regular polytopes
• 3-sphere
• Metatron's Cube
• Logical connectives
• Ludwig Schläfli - Polytopes

For other uses, see Fourth dimension (disambiguation). ... A square A projection of a cube (into a two-dimensional image) A projection of a hypercube (into a two-dimensional image) In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ... For other uses, see Square. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... A penteract is a name for a five dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract hypercells. ... A hexeract is a name for a six dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 penteract 5-faces. ... A projection of a cube (into a two-dimensional image) A projection of a tesseract (into a two-dimensional image) A tesseract projection In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ... An octeract is an eight-dimensional hypercube with 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 penteract 5-faces, 112 hexeract 6-faces, and 16 hepteract 7-faces. ... Schlegel diagram for the truncated 120-cell with tetrahedral cells visible. ... A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ... In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ... In geometry, demihypercubes (also called half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube. ... This page lists the regular polytopes in Euclidean space. ... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ... Metatron (Hebrew מטטרון or מיטטרון), is the name of an angel in Judaism and some branches of Christianity. ... In logic, a logical connective is a syntactic operation on sentences, or the symbol for such an operation, that corresponds to a logical operation on the logical values of those sentences. ... This article needs to be cleaned up to conform to a higher standard of quality. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

References

1. ^ Unfolding an 8-cell.

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

Stereographic projection of the 120-cell, a 4-dimensional regular polytope. ...

External links

• Eric W. Weisstein, Tesseract at MathWorld.
• Olshevsky, George, Tesseract at Glossary for Hyperspace.
• 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 via the Internet archive 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
• A Study of Dimensions by Bill Price