In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a tesseract is the
four-dimensional
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called '' dimensions'' ...
analogue of the
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
; the tesseract is to the cube as the cube is to the
square. Just as the surface of the cube consists of six square
faces, the
hypersurface of the tesseract consists of eight cubical
cells
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
. The tesseract is one of the six
convex regular 4-polytopes.
The tesseract is also called an 8-cell, C
8, (regular) octachoron, octahedroid, cubic prism, and tetracube. It is the four-dimensional hypercube, or 4-cube as a member of the dimensional family of
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...
s or measure polytopes.
Coxeter labels it the
polytope. The term ''hypercube'' without a dimension reference is frequently treated as a synonym for this specific
polytope.
The ''
Oxford English Dictionary
The ''Oxford English Dictionary'' (''OED'') is the first and foundational historical dictionary of the English language, published by Oxford University Press (OUP). It traces the historical development of the English language, providing a c ...
'' traces the word ''tesseract'' to
Charles Howard Hinton
Charles Howard Hinton (1853 – 30 April 1907) was a British mathematician and writer of science fiction works titled ''Scientific Romances''. He was interested in higher dimensions, particularly the fourth dimension. He is known for coining t ...
's 1888 book ''
A New Era of Thought
''A New Era of Thought'' is a non-fiction work written by Charles Howard Hinton, published in 1888 and reprinted in 1900 by Swan Sonnenschein & Co. Ltd., London. ''A New Era of Thought'' is about the fourth dimension and its implications on human ...
''. The term derives from the
Greek ( 'four') and from ( 'ray'), referring to the four edges from each vertex to other vertices. Hinton originally spelled the word as ''tessaract''.
Geometry
As a
regular polytope with three
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
s folded together around every edge, it has
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...
with
hyperoctahedral symmetry of order 384. Constructed as a 4D
hyperprism
In geometry, a prism is a polyhedron comprising an polygon base, a second base which is a translated copy (rigidly moved without rotation) of the first, and other faces, necessarily all parallelograms, joining corresponding sides of the two ba ...
made of two parallel cubes, it can be named as a composite Schläfli symbol × , with symmetry order 96. As a 4-4
duoprism, a
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\t ...
of two
squares, it can be named by a composite Schläfli symbol ×, with symmetry order 64. As an
orthotope it can be represented by composite Schläfli symbol × × × or
4, with symmetry order 16.
Since each vertex of a tesseract is adjacent to four edges, the
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
of the tesseract is a regular
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...
. The
dual polytope of the tesseract is the
16-cell with Schläfli symbol , with which it can be combined to form the
compound of tesseract and 16-cell.
Each edge of a regular tesseract is of the same length. This is of interest when using tesseracts as the basis for a
network topology to link multiple processors in
parallel computing
Parallel computing is a type of computation in which many calculations or processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. There are several different f ...
: the distance between two nodes is at most 4 and there are many different paths to allow weight balancing.
Coordinates
The standard tesseract in
Euclidean 4-space
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
is given as the
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of the points (±1, ±1, ±1, ±1). That is, it consists of the points:
:
In this Cartesian frame of reference, the tesseract has radius 2 and is bounded by eight
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
s (''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.
Net
An unfolding of a
polytope 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
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
together with a
perfect matching
In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph , a perfect matching in is a subset of edge set , such that every vertex in the vertex set is adjacent to exactl ...
in its
complement).
Construction
The construction of
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...
s can be imagined the following way:
* 1-dimensional: Two points A and B can be connected to become a line, giving a new line segment AB.
* 2-dimensional: Two parallel line segments AB and CD separated by a distance of AB can be connected to become a square, with the corners marked as ABCD.
* 3-dimensional: Two parallel squares ABCD and EFGH separated by a distance of AB can be connected to become a cube, with the corners marked as ABCDEFGH.
* 4-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP separated by a distance of AB can be connected to become a tesseract, with the corners marked as ABCDEFGHIJKLMNOP. However, this parallel positioning of two cubes such that their 8 corresponding pairs of vertices are each separated by a distance of AB can only be achieved in a space of 4 or more dimensions.
The tesseract can be decomposed into smaller 4-polytopes. It is the convex hull of the compound of two
demitesseracts (
16-cells). It can also be
triangulated into 4-dimensional
simplices (
irregular 5-cells) that share their vertices with the tesseract. It is known that there are 92487256 such triangulations and that the fewest 4-dimensional simplices in any of them is 16.
The dissection of the tesseract into instances of its
characteristic simplex (a particular
orthoscheme with Coxeter diagram ) is the most basic direct construction of the tesseract possible. The
characteristic 5-cell of the 4-cube is a
fundamental region of the tesseract's defining
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
, the group which generates the
B4 polytopes. The tesseract's characteristic simplex directly ''generates'' the tesseract through the actions of the group, by reflecting itself in its own bounding facets (its ''mirror walls'').
Radial equilateral symmetry
The long radius (center to vertex) of the tesseract is equal to its edge length; thus its diagonal through the center (vertex to opposite vertex) is 2 edge lengths. Only a few uniform
polytopes
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an - ...
have this property, including the four-dimensional tesseract and
24-cell, the three-dimensional
cuboctahedron, and the two-dimensional
hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A '' regular hexagon'' has ...
. In particular, the tesseract is the only hypercube (other than a 0-dimensional point) that is ''radially equilateral''. The longest vertex-to-vertex diameter of an ''n''-dimensional hypercube of unit edge length is , so for the square it is , for the cube it is , and only for the tesseract it is , exactly 2 edge lengths.
Formulas
For a tesseract with side length :
* Hypervolume:
* Surface volume:
*
Face diagonal:
*
Cell diagonal:
*4-space diagonal:
As a configuration
This
configuration matrix represents the tesseract. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole tesseract. The nondiagonal numbers say how many of the column's element occur in or at the row's element. For example, the 2 in the first column of the second row indicates that there are 2 vertices in (i.e., at the extremes of) each edge; the 4 in the second column of the first row indicates that 4 edges meet at each vertex.
Projections
It is possible to project tesseracts into three- and two-dimensional spaces, similarly to projecting a cube into two-dimensional space.
The ''cell-first'' parallel
projection of the tesseract into three-dimensional space has a
cubical envelope. The nearest and farthest cells are projected onto the cube, and the remaining six cells are projected onto the six square faces of the cube.
The ''face-first'' parallel projection of the tesseract into three-dimensional space has a
cuboid
In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a c ...
al envelope. Two pairs of cells project to the upper and lower halves of this envelope, and the four remaining cells project to the side faces.
The ''edge-first'' parallel projection of the tesseract into three-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 six 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 three-dimensional space has a
rhombic dodecahedral envelope. Two vertices of the tesseract are projected to the origin. There are exactly two ways of
dissecting a rhombic dodecahedron into four congruent
rhombohedra
In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be use ...
, giving a total of eight possible rhombohedra, each a projected
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
of the tesseract. This projection is also the one with maximal volume. One set of projection vectors are ''u''=(1,1,-1,-1), ''v''=(-1,1,-1,1), ''w''=(1,-1,-1,1).
Tessellation
The tesseract, like all
hypercubes, tessellates
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
. The self-dual
tesseractic honeycomb consisting of 4 tesseracts around each face has
Schläfli symbol . Hence, the tesseract has a
dihedral angle of 90°.
The tesseract's
radial equilateral symmetry makes its tessellation the
unique regular body-centered cubic lattice of equal-sized spheres, in any number of dimensions.
Related polytopes and honeycombs
The tesseract is 4th in a series of
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...
:
The tesseract (8-cell) is the third in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).
As a uniform
duoprism, the tesseract exists in a
sequence of uniform duoprisms: ×.
The regular tesseract, along with the
16-cell, exists in a set of 15
uniform 4-polytopes with the same symmetry. The tesseract exists in a
sequence of regular 4-polytopes and honeycombs, with
tetrahedral vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
s, . The tesseract is also in a
sequence of regular 4-polytope and honeycombs, with
cubic cells
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
.
The
regular complex polytope In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.
A complex polytope may be understood as a collect ...
42, , in
has a real representation as a tesseract or 4-4
duoprism in 4-dimensional space.
42 has 16 vertices, and 8 4-edges. Its symmetry is
4 sub>2, order 32. It also has a lower symmetry construction, , or
4×
4, with symmetry
4 sub>4, order 16. This is the symmetry if the red and blue 4-edges are considered distinct.
In popular culture
Since their discovery, four-dimensional hypercubes have been a popular theme in art, architecture, and science fiction. Notable examples include:
* "
And He Built a Crooked House
or AND may refer to:
Logic, grammar, and computing
* Conjunction (grammar), connecting two words, phrases, or clauses
* Logical conjunction in mathematical logic, notated as "∧", "⋅", "&", or simple juxtaposition
* Bitwise AND, a boolea ...
",
Robert Heinlein's 1940 science fiction story featuring a building in the form of a four-dimensional hypercube. This and
Martin Gardner's "The No-Sided Professor", published in 1946, are among the first in science fiction to introduce readers to the
Moebius band
Moebius, Möbius or Mobius may refer to:
People
* August Ferdinand Möbius (1790–1868), German mathematician and astronomer
* Theodor Möbius (1821–1890), German philologist
* Karl Möbius (1825–1908), German zoologist and ecologist
* Pa ...
, the
Klein bottle
In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a ...
, and the hypercube (tesseract).
* ''
Crucifixion (Corpus Hypercubus)'', a 1954 oil painting by Salvador Dalí featuring a four-dimensional hypercube unfolded into a three-dimensional
Latin cross
A Latin cross or ''crux immissa'' is a type of cross in which the vertical beam sticks above the crossbeam, with the three upper arms either equally long or with the vertical topmost arm shorter than the two horizontal arms, and always with a mu ...
.
* The
Grande Arche
La Grande Arche de la Défense (; "The Great Arch of the Defense"), originally called La Grande Arche de la Fraternité (; "Fraternity"), is a monument and building in the business district of La Défense and in the commune of Puteaux, to the west ...
, a monument and building near Paris, France, completed in 1989. According to the monument's engineer,
Erik Reitzel
Erik Reitzel (10 May 1941 – 6 February 2012) was a Danish civil engineer who started work in 1964 and was for many years a professor at the Royal Danish Academy of Fine Arts and at the Technical University of Denmark, in the disciplines of beari ...
, the Grande Arche was designed to resemble the projection of a hypercube.
* ''
Fez'', a video game where one plays a character who can see beyond the two dimensions other characters can see, and must use this ability to solve platforming puzzles. Features "Dot", a tesseract who helps the player navigate the world and tells how to use abilities, fitting the theme of seeing beyond human perception of known dimensional space.
The word ''tesseract'' was later adopted for numerous other uses in popular culture, including as a plot device in works of science fiction, often with little or no connection to the four-dimensional hypercube of this article. See
Tesseract (disambiguation)
A tesseract is a four-dimensional analog of the cube.
Tesseract may also refer to:
Arts and entertainment Literature
* Tesseract, a concept for spacetime travel in the novel ''A Wrinkle in Time'', by Madeleine L'Engle (1962), and subsequent novel ...
.
See also
*
Mathematics and art
Notes
References
*
* F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss (1995) ''Kaleidoscopes: Selected Writings of H.S.M. Coxeter'', Wiley-Interscience Publication
** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'',
Mathematische Zeitschrift
''Mathematische Zeitschrift'' (German for ''Mathematical Journal'') is a mathematical journal for pure and applied mathematics published by Springer Verlag.
It was founded in 1918 and edited by Leon Lichtenstein together with Konrad Knopp, Erha ...
46 (1940) 380–407, MR 2,10]
** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'',
ath. Zeit. 188 (1985) 559-591** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'',
ath. Zeit. 200 (1988) 3-45*
John H. Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches o ...
, Heidi Burgiel, Chaim Goodman-Strass (2008) ''The Symmetries of Things'', (Chapter 26. pp. 409: Hemicubes: 1
n1)
*
T. Gosset (1900) ''On the Regular and Semi-Regular Figures in Space of n Dimensions'',
Messenger of Mathematics, Macmillan.
*
T. Proctor Hall
Thomas Proctor Hall (1858–1931) was a Canadian physician who wrote mathematics, chemistry, physics, theology, and science fiction.
T. Proctor Hall was born October 7, 1858 at Hornby, Ontario. He attended Woodstock College and University of Toro ...
(1893
"The projection of fourfold figures on a three-flat" American Journal of Mathematics 15:179–89.
*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
*
Victor Schlegel (1886) ''Ueber Projectionsmodelle der regelmässigen vier-dimensionalen Körper'', Waren.
External links
*
*
The TesseractRay traced images with hidden surface elimination. This site provides a good description of methods of visualizing 4D solids.
Marco Möller's Regular polytopes in ℝ
4 (German)
WikiChoron: Tesseractis an open source program for the
Apple Macintosh
The Mac (known as Macintosh until 1999) is a family of personal computers designed and marketed by Apple Inc. Macs are known for their ease of use and minimalist designs, and are popular among students, creative professionals, and software ...
(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 98A
Windows
Windows is a group of several proprietary graphical operating system families developed and marketed by Microsoft. Each family caters to a certain sector of the computing industry. For example, Windows NT for consumers, Windows Server for se ...
program that displays animated hypercubes, by
Rudy Rucker
Rudolf von Bitter Rucker (; born March 22, 1946) is an American mathematician, computer scientist, science fiction author, and one of the founders of the cyberpunk literary movement. The author of both fiction and non-fiction, he is best known ...
ken perlin's home pageA way to visualize hypercubes, by
Ken Perlin
Kenneth H. Perlin is a professor in the Department of Computer Science at New York University, founding director of the Media Research Lab at NYU, director of the Future Reality Lab at NYU, and the Director of the Games for Learning Institute. He ...
Some Notes on the Fourth Dimensionincludes animated tutorials on several different aspects of the tesseract, b
Davide P. Cervone
{{Polytopes
Algebraic topology
Four-dimensional geometry
008