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Tesseract

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Tesseract
Hypercube
(8-cell)

Cell-first perspective projection into 3-dimensional space
Type Regular polychoron
Cells 8 (4.4.4)
Faces 24 {4}
Edges 32
Vertices 16
Vertex figure 4 (4.4.4)
(tetrahedron)
Schläfli symbol {4,3,3}
Symmetry group B4, [3,3,4]
Dual 16-cell
Properties convex
Stereographic projection

In geometry, the tesseract is the 4-dimensional analog of the (3-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. According to OED, the word tesseract was coined and first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from tesseres aktines = 'four rays' in Ionic Greek, referring to the four lines from each vertex to other vertices. Alternately, some people have called the same figure a 'tetracube'.

Generalizations of the cube to dimensions greater than three are called hypercubes or measure polytopes. The tesseract is the 4D hypercube.

Geometry

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:

A tesseract is bounded by eight hyperplanes (xi = ±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.

Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron. Thus the tesseract is given Schläfli symbol {4,3,3}. The dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol {3,3,4}.

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:

       

The first illustration shows how 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. The second picture accounts for the fact that each edge of a tesseract is of the same length. This picture also enables the human brain to find a multitude of cubes that are nicely interconnected. The third diagram finally orders 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.

The connection pattern of the tesseract's vertices is the same as that of a 4×4 square array drawn on a torus; each cell (representing a vertex of the tesseract) is adjacent to exactly four other cells. See geometry of the 4x4 square. 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. The 8 cells project onto volumes in the shape of parallelogrammic prisms, which are laid out in the hexagonal prism in a way analogous to how the faces of the 3D cube project onto 6 parallelograms in a hexagonal envelope under vertex-first projection.

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.

Tesseracts in art and literature

A net of a tesseract. View animation.

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

In one episode of The Adventures Of Jimmy Neutron: Boy Genius, Jimmy invents a 4-dimensional hypercube identical to the foldbox in Robert A. Heinlein's 1963 novel Glory Road.

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. A tribute to this house was built in the 3-dimensional online universe Second Life, where the house folds and changes as the user's avatar moves around within it (covered here). Stephen Baxter's short story collection 'Vacuum Diagrams' includes a similar structure, in which the internal layout of rooms in a house is deeply confusing to the characters exploring it.

Heinlein's 1963 novel Glory Road included the foldbox, a hyperdimensional packing case that was bigger inside than outside.

A hypercube is also used as the main deus ex machina of Robert J. Sawyer's book Factoring Humanity (ISBN 0-7653-0903-3 ), 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 treatment is extremely vague. In that book she uses the tesseract as a portal, a doorway which you can pass through and emerge far away from the starting point, as if the two distant points were brought together at one intersection (at the tesseract doorway) by the folding of space-time, enabling near-instantaneous transportation (though this description more closely matches a wormhole).

Henry Kuttner's 1943 short story Mimsy Were the Borogoves mentions a teaching toy from the far future with beads strung on wires in a tesseract-like frame. The children who play with this toy learn to work with higher dimensions. Similarly, characters in Mark Clifton's 1952 story Star Bright learn to travel by mental manipulations of tesseracts.

In Alex Garland's novel The Tesseract (1998), the term is used for the three-dimensional net of the four-dimensional hypercube rather than the hypercube itself. It is a metaphor for the characters' inability to understand the causes behind the events which shape their lives: they can only visualize the superficial world they inhabit.

In the novel Mythago Wood (and its sequels) by Robert Holdstock there is a landscape within the wood that is far larger on the inside than the outside. It distorts time and features its own distinct time zones. People and creatures are formed by some force in the wood interacting with the human subconscious mind and tapping into buried memories of ancient myths and events. From these the wood generates creatures and places within its own vast space.

The movie Cube 2: Hypercube focuses on eight strangers trapped inside a "hypercube", or a net of connected cubes.

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

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

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 and in the book 'The Wild West Witches' by Michael Molloy, it was used to store spare equipment.

The television series 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.

Crucifixion (Corpus Hypercubus) by Salvador Dalí (1954)

Another TV series, Strange Days at Blake Holsey High (also known as Black Hole High), features an episode ("The Tesseract") where Lucas gets trapped in a tesseract. Lucas calls someone he thinks will help (Corrine), but she gets sucked into the tesseract. Eventually, the school folds up in time as well as space. With the help of a "disappeared" teacher, Lucas unfolds the tesseract. The episode gave a good description of a tesseract (see [1]).

In the Infocom text adventure, Spellbreaker, the climax of the story features a tesseract constructed by various featureless white cubes, each of which has magical property corresponding to the base elements of the universe.

In Finnegans Wake by James Joyce, the main protagonist HCE transforms into a tesseract ("Wherefore let it hardly by any being thinking be said either or thought that the prisoner of that sacred edifice, were he an Ivor the Boneless or an Olaf the Hide, was at his best a onestone parable, a rude breathing on the void of to be, a venter hearing his own bauchspeech in backwords, or, more strictly, but tristurned initials, the cluekey to a worldroom beyond the roomwhorld, for scarce one, or pathetically few of his dode canal sammenliverscared seriously or for long to doubt with Kurt Iuld van Dijke (the gravitational pull perceived by certain fixed residents and the capture of uncertain comets chancedrifting through our system suggesting an authenticitatem of his aliquitudinis) the canonicity of his existence as a tesseract. Be still, O quick! Speak him dumb! Hush ye fronds of Ulma!").

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.

The Nextwave comic book by Marvel Comics features a vehicle called the Shockwave Rider that contains 5 tesseract zones inside of it.

In one of A. Bertram Chandler's later 'John Grimes' novels, a mention is made of a tesseract.

In Voivod's album Nothingface there is a track called "Into My Hypercube".

In the video game Starflight the tesseract was an artifact that would increase your ship's fuel efficiency.

In the game Chrono Cross, the Darkness Beyond Time was also reffered to as a tesseract.

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 2n 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 and the Connection Machine, the latter using the hypercube topology to connect groups of processors.

Hypercubes in information theory

Hypercubes are the logical representation of multidimensional data warehouses compiled for complex information cross-referencing. Instead of the usual two dimensional operational databases, Hypercubes allow you to cross reference a number of factors at once, structuring information into theoretical columns, rows and layers.

In the analysis of business data for example, the dimensions (or factors) for analysis might be the correlation between product, buyer segments and advertising budget, allowing us to compare multiple factors simultaneously. Hypercubes however are not limited to the number of dimensions they can use; as the analysis grows more precise, additional dimensions are added creating new layers within the Hypercubes. Although we show business data being analyzed in this example, it can be applied to many other fields of study as well.

The application of comparing multiple factors simultaneously during the analytical process makes the end users’ decisions much easier and more informed than with simple two-dimensional data analysis.

See also


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