Tesseract
| Tesseract Hypercube (8-cell) | |
|---|---|
 Schlegel diagram | |
| Type | Regular polychoron |
| 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 (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 “τεσσερες ακτινες” (“four rays”), referring to the four lines from each vertex to other vertices. Alternately, some people have called the same figure a “tetracube”.
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:
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The illustration on the left 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 center 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 diagram on the right 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.
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 (see the adjacent figure for an example of one of the 261 nets). 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).
Tesseracts in art and literature
In Edwin A. Abbott's novel Flatland, 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.
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.
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.
The movie Cube 2: Hypercube focuses on eight strangers trapped inside 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.
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.
Piers Anthony's novel Cube Route also features a tesseract.
Alex Garland's second book is called "Tesseract: a novel".
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; the Connection Machine, which uses the hypercube topology to connect groups of processors.
See also
References
- H. S. M. Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
External links
- Weisstein, Eric W. "Tesseract". MathWorld.
- Olshevsky, George. "Tesseract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- 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
- Geometry of the 4×4 square points out vertex-adjacency properties.
- Some Notes on the Fourth Dimension includes very good animated tutorials on several different aspects of the tesseract, by Davide P. Cervone