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Tesseract

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For other uses, see Tesseract (disambiguation).

Tesseract

8-cell

4-cube

Schlegel diagram

Type Convex regular 4-polytope

Schlfli symbol {4,3,3}

{4,3}{ }

{4}{4}

{4}{ }{ }

{ }{ }{ }{ }

Coxeter-Dynkin

diagram


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Cells 8 (4.4.4)

Faces 24 {4}

Edges 32

Vertices 16

Vertex figure

Tetrahedron

Petrie polygon octagon

Coxeter group C4, [3,3,4]

Dual 16-cell

Properties convex, isogonal, isotoxal,

isohedral

Uniform index 10

In geometry, the tesseract, also called an 8-cell orregular octachoron orcubic prism, isthe four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the

square. Just as the surface of the cube consists of 6 square faces, the hypersurface of thetesseract consists of 8 cubical cells. The tesseract is one of the six convex regular 4-

polytopes.

A generalization of the cube to dimensions greater than three is called a "hypercube", "n-

cube" or "measurepolytope". The tesseract is the four-dimensional hypercube, or4-cube.

According to the Oxford English Dictionary, the word tesseractwas coined and first usedin 1888 by Charles Howard Hinton in his bookA New Era of Thought, from the Greek

("four rays"), referring to the four lines from each vertex to other

vertices.[1]

In this publication, as well as some of Hinton's later work, the word was


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occasionally spelled "tessaract." Some people have called the same figure a tetracube, andalso simply a hypercube (although the term hypercube is also used with dimensions greater

than 4).

Contents

1 Geometryo 1.1 Projections to 2 dimensionso 1.2 Parallel projections to 3 dimensions

2 Image galleryo 2.1 Perspective projectionso 2.2 2D orthographic projections

3 Related uniform polytopes 4 See also 5 Notes 6 References 7 External links

Geometry[edit]

The tesseract can be constructed in a number of ways. As a regular polytope with three

cubes folded together around every edge, it has Schlfli symbol {4,3,3} with

hyperoctahedral symmetry of order 384. Constructed as a 4D hyperprism made of twoparallel cubes, it can be named as a composite Schlfli symbol {4,3} { }, with symmetry

order 96. As a duoprism, a Cartesian product of two squares, it can be named by a

composite Schlfli symbol {4}{4}, with symmetry order 64. As an orthotope it can berepresented by composite Schlfli symbol { } { } { } { } or { }

4, with symmetry order

16.

Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract isa regulartetrahedron. The dual polytope of the tesseract is called the hexadecachoron, or

16-cell, with Schlfli 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:

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 ateach 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[edit]


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A diagram showing how to create a tesseract from a point

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 linesegment AB.

2-dimensional: Two parallel line segments AB and CD can be connected tobecome a square, with the corners marked as ABCD.

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

4-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP can beconnected 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 moreinstructive 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 theconstruction of a cube from two squares: juxtapose two copies of the lower dimensionalcube and connect the corresponding vertices. Each edge of a tesseract is of the same length.

This view is of interest when using tesseracts as the basis for a network topology to link

multiple processors inparallel computing: the distance between two nodes is at most 4 and

there are many different paths to allow weight balancing.

Tesseracts are alsobipartite graphs, just as a path, square, cube and tree are.


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Parallel projections to 3 dimensions[edit]

The rhombic dodecahedron forms the convex hull of the tesseract's vertex-first parallel-

projection. The number of vertices in the layers of this projection is 1 4 6 4 1 - the fourthrow in Pascal's triangle.

Parallel projection envelopes of the tesseract (each cell is drawn with different color faces,inverted cells are undrawn)

The cell-firstparallelprojection 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.

Theface-firstparallel 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-firstparallel projection of the tesseract into 3-dimensional space has an envelopein the shape of a hexagonal prism. Six cells project onto rhombic prisms, which are laid out


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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-firstparallel projection of the tesseract into 3-dimensional space has a rhombic

dodecahedral envelope. There are exactly two ways of decomposing a rhombicdodecahedron into 4 congruentparallelepipeds, giving a total of 8 possible parallelepipeds.

The images of the tesseract's cells under this projection are precisely these 8parallelepipeds. This projection is also the one with maximal volume.

Image gallery[edit]

The tesseract can be unfolded into eight cubes into3D space, just as the cube can be unfolded into six

squares into 2D space (view animation). An

unfolding of a polytope is called a net. There are 261distinct nets of the tesseract.

[2]The unfoldings of the

tesseract can be counted by mapping the nets to

paired trees (a tree together with aperfect matching

in its complement).

Stereoscopic 3D projection of a tesseract (paral)

Perspective projections[edit]


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A 3D projection of an 8-cell

performing a simple rotation

about a plane which bisects thefigure 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

Perspective with hidden

volume elimination. The redcorner is the nearest in 4D and

has 4 cubical cells meeting

around it.

The tetrahedron forms the convex hull of the tesseract's vertex-centered central projection. Four of 8 cubic cells are shown. The

16th vertex is projected to infinity and the four edges to it are not

shown.

Stereographic projection

(Edges are projected onto the 3-

sphere)

2D orthographic projections[edit]

orthographic projections

Coxeter plane B4 B3 / D4 / A2 B2 / D3


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Graph

Dihedral

symmetry[8] [6] [4]

Coxeter plane Other F4 A3

Graph

Dihedral

symmetry[2] [12/3] [4]

Related uniform polytopes[edit]

Name tesseract

rectifie

dtessera

ct

truncat

edtessera

ct

cantell

atedtessera

ct

runcin

atedtessera

ct

bitrunc

atedtessera

ct

cantitrun

catedtesseract

runcitru

ncatedtesseract

omnitrun

catedtesseract

Coxe

ter-

Dynk

in

diagr

am

Schl

fli

symbol

{4,3,3}t1{4,3,3

}

t0,1{4,3,

3}

t0,2{4,3,

3}

t0,3{4,3,

3}

t1,2{4,3,

3}

t0,1,2{4,3,3

}

t0,1,3{4,3,3

}

t0,1,2,3{4,3,

3}

Schle

gel

diagr

am


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B4

Coxe

ter

plane

grap

h

Nam

e16-cell

rectifie

d

16-cell

truncat

ed

16-cell

cantell

ated

16-cell

runcin

ated

16-cell

bitrunc

ated

16-cell

cantitrun

cated

16-cell

runcitru

ncated

16-cell

omnitrun

cated

16-cell

Coxe

ter-

Dynk

in

diagr

am

Schl

fli

symb

ol

{3,3,4}t1{3,3,4

}

t0,1{3,3,

4}

t0,2{3,3,

4}

t0,3{3,3,

4}

t1,2{3,3,

4}

t0,1,2{3,3,4

}

t0,1,3{3,3,4

}

t0,1,2,3{3,3,

4}

Schle

gel

diagr

am

B4

Coxe

ter

plane

grap

h

See also[edit]

3-sphere Four-dimensional space

o List of regular polytopes Grande Arche - a monument and building in the business district ofLa Dfense Ludwig Schlfli - Polytopes List of four-dimensional games Uses in fiction:

o "And He Built a Crooked House" - a science fiction story featuring abuilding in the form of a tesseract

o A Wrinkle in Time - a science fantasy novel using the word tesseract


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o In the Marvel Cinematic Universe, the Cosmic Cube is referred to as atesseract

Uses in art:o Crucifixion (Corpus Hypercubus) - oil painting by Salvador Dal

Notes[edit]1. ^http://www.oed.com/view/Entry/199669?redirectedFrom=tesseract#eid2. ^"Unfolding an 8-cell".

References[edit]

T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions,Messenger of Mathematics, Macmillan, 1900

H.S.M. Coxeter:o Coxeter,Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-

486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular

polytopes in n-dimensions (n5)

o H.S.M. Coxeter,Regular Polytopes, 3rd Edition, Dover New York, 1973,p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-

dimensions (n5)

o Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. ArthurSherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-

Interscience Publication, 1995, ISBN 978-0-471-01003-6[1]

(Paper 22) H.S.M. Coxeter,Regular and Semi Regular Polytopes I,[Math. Zeit. 46 (1940) 380-407, MR 2,10]

(Paper 23) H.S.M. Coxeter,Regular and Semi-Regular Polytopes II,[Math. Zeit. 188 (1985) 559-591]

(Paper 24) H.S.M. Coxeter,Regular and Semi-Regular Polytopes III,[Math. Zeit. 200 (1988) 3-45]

John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)

Norman JohnsonUniform Polytopes, Manuscript (1991)o N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.

(1966)

External links[edit]

Weisstein, Eric W., "Tesseract",MathWorld. Olshevsky, George, Tesseractat Glossary for Hyperspace.

o 2. Convex uniform polychora based on the tesseract (8-cell) andhexadecachoron (16-cell) - Model 10, George Olshevsky.

Richard Klitzing, 4D uniform polytopes (polychora), x4o3o3o - tes The Tesseract Ray traced images with hidden surface elimination. This site provides

a good description of methods of visualizing 4D solids.


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Der 8-Zeller (8-cell) Marco Mller'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 thesix regular hypersolids of four-dimensional space.

Hypercube 98 A Windows program that displays animated hypercubes, by RudyRucker

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

v t e

Convex regular polychora

5-cell8-

cell16-cell 24-cell 120-cell 600-cell

{3,3,3}pentachoron

{4,3,3}tesseract

{3,3,4}hexadecachoron

{3,4,3}icositetrachoron

{5,3,3}hecatonicosachoron

{3,3,5}hexacosichoron

v t e

Fundamental convex regular and uniform polytopes in dimensions 210

Family An BCn DnE6 / E7 / E8

/ F4 / G2Hn

Regular polygon Triangle Square Hexagon Pentagon

Uniform polyhedron TetrahedronOctahedron

CubeDemicube

Dodecahedron

Icosahedron

Uniform polychoron 5-cell16-cell

TesseractDemitesseract 24-cell

120-cell 600-

cell

Uniform 5-polytope 5-simplex 5-orthoplex 5-cube5-demicube

Uniform 6-polytope 6-simplex6-orthoplex

6-cube6-demicube 122 221


7-cube7-demicube

132 231

321

Uniform 8-polytope 8-simplex 8-orthoplex 8-demicube 142 241


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8-cube 421


9-cube9-demicube


10-cube10-demicube

Uniform n-polytope n-simplexn-orthoplex

n-cuben-demicube

1k2 2k1

k21

n-pentagonal

polytope

Topics: Polytope families Regular polytope List of regular polytopes

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