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| | Hello from Australia. I'm glad to came here. My first name is Clifford. <br>I live in a small city called Weymouth in south Australia.<br>I was also born in Weymouth 30 years ago. Married in November year 2005. I'm working at the the office.<br><br>Also visit my homepage :: [http://gallery.Ershadonline.org/profile/iesymonds FIFA coin generator] |
| {{Infobox polychoron
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| | Name=Tesseract<BR>8-cell<BR>4-cube
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| | Image_File=Schlegel wireframe 8-cell.png
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| | Image_Caption=[[Schlegel diagram]]
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| | Type=[[Convex regular 4-polytope]]
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| | Family=[[Hypercube]]
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| | Last=[[Omnitruncated 5-cell|9]]
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| | Index=10
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| | Next=[[Rectified tesseract|11]]
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| | Schläfli={4,3,3}<BR>t<sub>0,3</sub>{4,3,2} or {4,3}×{ }<BR>t<sub>0,2</sub>{4,2,4} or {4}×{4}<BR>t<sub>0,2,3</sub>{4,2,2} or {4}×{ }×{ }<BR>t<sub>0,1,2,3</sub>{2,2,2} or { }×{ }×{ }×{ }|
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| CD={{CDD|node_1|4|node|3|node|3|node}}<BR>{{CDD|node_1|4|node|3|node|2|node_1}}<BR>{{CDD|node_1|4|node|2|node_1|4|node}}<BR>{{CDD|node_1|4|node|2|node_1|2|node_1}}<BR>{{CDD|node_1|2|node_1|2|node_1|2|node_1}}|
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| Cell_List=8 ([[cube|4.4.4]]) [[File:Hexahedron.png|20px]] |
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| Face_List=24 [[square (geometry)|{4}]] |
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| Edge_Count=32 |
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| Vertex_Count=16 |
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| Petrie_Polygon=[[octagon]]|
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| Coxeter_Group=C<sub>4</sub>, [3,3,4] |
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| Vertex_Figure=[[File:8-cell verf.png|80px]]<BR>[[Tetrahedron]]|
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| Dual=[[16-cell]]|
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| Property_List=[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[isotoxal figure|isotoxal]], [[isohedral figure|isohedral]]
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| }}
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| In [[geometry]], the '''tesseract''', also called an '''8-cell''' or '''regular octachoron''' or '''cubic prism''', is the [[four-dimensional space|four-dimensional]] analog of the [[cube]]; the tesseract is to the cube as the cube is to the [[square (geometry)|square]]. Just as the surface of the cube consists of 6 square [[face (geometry)|faces]], the hypersurface of the tesseract consists of 8 cubical [[cell (geometry)|cells]]. The tesseract is one of the six [[convex regular 4-polytope]]s.
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| A generalization of the cube to dimensions greater than three is called a "[[hypercube]]", "''n''-cube" or "measure [[polytope]]".<ref>[[E. L. Elte]], ''The Semiregular Polytopes of the Hyperspaces'', (1912)</ref> The tesseract is the '''four-dimensional hypercube''', or '''4-cube'''.
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| 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 [[Ancient Greek|Greek]] {{lang|grc|τέσσερεις ακτίνες}} ("four rays"), referring to the four lines from each vertex to other vertices.<ref>http://www.oed.com/view/Entry/199669?redirectedFrom=tesseract#eid</ref> In this publication, as well as some of Hinton's later work, the word was occasionally spelled "tessaract." Some people{{Citation needed|date=July 2013}} have called the same figure a '''tetracube''', and also simply a '''hypercube''' (although a ''tetracube'' can also mean a [[polycube]] made of four cubes, and the term ''hypercube'' is also used with dimensions greater than 4).
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| == Geometry ==
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| The tesseract can be constructed in a number of ways. As a [[regular polytope]] with three [[cube]]s folded together around every edge, it has [[Schläfli symbol]] {4,3,3} with [[Hyperoctahedral_group#By_dimension|hyperoctahedral symmetry]] of order 384. Constructed as a 4D [[hyperprism]] made of two parallel cubes, it can be named as a composite Schläfli symbol {4,3} × { }, with symmetry order 96. As a [[duoprism]], a [[Cartesian product]] of two [[Square (geometry)|squares]], it can be named by a composite Schläfli symbol {4}×{4}, with symmetry order 64. As an [[orthotope]] it can be represented by composite Schläfli symbol { } × { } × { } × { } or { }<sup>4</sup>, with symmetry order 16.
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| 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}.
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| The standard tesseract in [[Euclidean space|Euclidean 4-space]] is given as the [[convex hull]] of the points (±1, ±1, ±1, ±1). That is, it consists of the points:
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| :<math>\{(x_1,x_2,x_3,x_4) \in \mathbb R^4 \,:\, -1 \leq x_i \leq 1 \}</math>
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| A tesseract is bounded by eight [[hyperplane]]s (''x''<sub>i</sub> = ±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.
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| ===Projections to 2 dimensions===
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| [[File:Dimension levels.svg|thumb|left|480px|center|A diagram showing how to create a tesseract from a point]]
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| The construction of a hypercube can be imagined the following way:
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| * '''1-dimensional:''' Two points A and B can be connected to a line, giving a new line segment AB.
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| * '''2-dimensional:''' Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD.
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| * '''3-dimensional:''' Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH.
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| * '''4-dimensional:''' Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP.
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| [[File:Hypercubecubes.svg|thumb||160px]]
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| 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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| 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. 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.
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| Tesseracts are also [[bipartite graph]]s, just as a path, square, cube and tree are.
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| ===Parallel projections to 3 dimensions===
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| [[File:Hypercubeorder binary.svg|thumb|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 fourth row in [[Pascal's triangle]].]]
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| {| class=wikitable
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| |[[File:Orthogonal projection envelopes tesseract.png|thumb|left|Parallel projection envelopes of the tesseract (each cell is drawn with different color faces, inverted cells are undrawn)]]
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| The ''cell-first'' parallel [[graphical projection|projection]] of the tesseract into 3-dimensional space has a [[cube|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.
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| The ''face-first'' parallel projection of the tesseract into 3-dimensional space has a [[cuboid]]al 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.
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| 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.
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| The ''vertex-first'' parallel projection of the tesseract into 3-dimensional space has a [[rhombic dodecahedron|rhombic dodecahedral]] envelope. There are exactly two ways of decomposing a rhombic dodecahedron into 4 congruent [[parallelepiped]]s, 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.
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| |}
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| == Image gallery ==
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| {| class=wikitable width=720
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| |- align=left valign=top
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| | [[File:Tesseract2.svg|150px|right|3-D net of a tesseract]]
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| The tesseract can be unfolded into eight cubes into 3D space, just as the cube can be unfolded into six squares into 2D space ([[:File:Hcube fold.gif|view animation]]). An unfolding of a polytope is called a [[Net (polyhedron)|net]]. There are 261 distinct nets of the tesseract.<ref>{{cite web|url=http://unfolding.apperceptual.com/|title=Unfolding an 8-cell}}</ref> The unfoldings of the tesseract can be counted by mapping the nets to ''paired trees'' (a [[Tree (graph theory)|tree]] together with a [[perfect matching]] in its [[Complement graph|complement]]).
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| | [[File:3D stereographic projection tesseract.PNG|360px]]<BR> [[stereogram|Stereoscopic]] 3D projection of a tesseract (parallel view [[File:Stereogram guide parallel.png|10px]])
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| |}
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| ===Perspective projections===
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| {| class=wikitable width=640
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| |- align=center valign=top
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| |[[File:8-cell-simple.gif|200px]]<BR>A 3D projection of an 8-cell performing a [[SO(4)#Geometry_of_4D_rotations|simple rotation]] about a plane which bisects the figure from front-left to back-right and top to bottom
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| |[[File:Tesseract-perspective-vertex-first-PSPclarify.png|200px]]<BR>Perspective with hidden volume elimination. The red corner is the nearest in [[Four-dimensional space|4D]] and has 4 cubical cells meeting around it.
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| |}
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| {| class=wikitable width=640
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| |- align=center valign=top
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| |[[File:Tesseract tetrahedron shadow matrices.svg|200px|right]]
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| 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.
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| |[[File:Stereographic polytope 8cell.png|200px]]<BR>[[Stereographic projection]]<BR>
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| (Edges are projected onto the [[3-sphere]])
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| |}
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| === 2D orthographic projections===
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| {| class=wikitable
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| |+ [[orthographic projection]]s
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| |- align=center
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| ![[Coxeter plane]]
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| !B<sub>4</sub>
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| !B<sub>3</sub> / D<sub>4</sub> / A<sub>2</sub>
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| !B<sub>2</sub> / D<sub>3</sub>
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| |- align=center
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| !Graph
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| |[[File:4-cube t0.svg|150px]]
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| |[[File:4-cube t0 B3.svg|150px]]
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| |[[File:4-cube t0 B2.svg|150px]]
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| |- align=center
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| ![[Dihedral symmetry]]
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| |[8]
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| |[6]
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| |[4]
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| |- align=center
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| !Coxeter plane
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| !Other
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| !F<sub>4</sub>
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| !A<sub>3</sub>
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| |- align=center
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| !Graph
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| |[[File:4-cube column graph.svg|150px]]
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| |[[File:4-cube t0 F4.svg|150px]]
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| |[[File:4-cube t0 A3.svg|150px]]
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| |- align=center
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| !Dihedral symmetry
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| |[2]
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| |[12/3]
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| |[4]
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| |}
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| == Related uniform polytopes ==
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| {{Convex prismatic prisms}}
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| {{Tesseract family}}
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| It in a sequence of [[regular polychora]] and honeycombs with [[tetrahedron|tetrahedral]] [[vertex figure]]s.
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| {{Tetrahedral vertex figure tessellations}}
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| It in a sequence of [[regular polychora]] and honeycombs with [[cube|cubic]] [[cell (geometry)|cells]].
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| {{Cubic cell tessellations}}
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| ==See also==
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| * [[3-sphere]]
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| * [[Four-dimensional space]]
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| ** [[List of regular polytopes]]
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| * [[Grande Arche]] - a monument and building in the business district of [[La Défense]]
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| * [[Ludwig Schläfli]] - [[Ludwig Schläfli#Polytopes|Polytopes]]
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| * [[List of four-dimensional games]]
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| * Uses in fiction:
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| ** "[[And He Built a Crooked House]]" - a science fiction story featuring a building in the form of a tesseract
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| ** ''[[A Wrinkle in Time]]'' - a science fantasy novel using the word "tesseract" (without reference to its geometrical meaning)
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| ** In the [[Marvel Cinematic Universe]], the [[Cosmic Cube#Film|Cosmic Cube]] is referred to as a tesseract
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| ** In the film ''[[Cube_2#Film|Cube 2:Hypercube]]'' the hypercube is described as a tesseract
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| * Uses in art:
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| ** ''[[Crucifixion (Corpus Hypercubus)]]'' - oil painting by Salvador Dalí
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900
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| * [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
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| ** Coxeter, ''[[Regular Polytopes (book)|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 (n≥5)
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| ** 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 (n≥5)
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| ** '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
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| *** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
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| *** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
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| *** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
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| * [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1<sub>n1</sub>)
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| * [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
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| ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
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| == External links ==
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| * {{MathWorld|title=Tesseract|urlname=Tesseract}}
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| * {{GlossaryForHyperspace | anchor=Tesseract | title=Tesseract}}
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| ** {{PolyCell | urlname = section2.html| title = 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 10}}
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| * {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|x4o3o3o - tes}}
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| * [http://eusebeia.dyndns.org/4d/8-cell.html 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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| * [http://www.polytope.de/c8.html Der 8-Zeller (8-cell)] Marco Möller's Regular polytopes in R<sup>4</sup> (German)
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| * [http://www.polychora.de/wiki/index.php?title=TES WikiChoron: Tesseract]
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| * [http://uoregon.edu/~koch/hypersolids/hypersolids.html 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.
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| * [http://www.cs.sjsu.edu/~rucker/hypercube.htm Hypercube 98] A [[Microsoft Windows|Windows]] program that displays animated hypercubes, by [[Rudy Rucker]]
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| * [http://mrl.nyu.edu/~perlin/demox/Hyper.html ken perlin's home page] A way to visualize hypercubes, by [[Ken Perlin]]
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| * [http://www.math.union.edu/~dpvc/math/4D/ Some Notes on the Fourth Dimension] includes very good animated tutorials on several different aspects of the tesseract, by [http://www.math.union.edu/~dpvc/ Davide P. Cervone]
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| * [http://www.fano.co.uk/hypermodel/tesseract.html Tesseract animation with hidden volume elimination]
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| {{4D regular polytopes}}
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| {{Polytopes}}
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| [[Category:Algebraic topology]]
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| [[Category:Four-dimensional geometry]]
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| [[Category:Polychora| 008]]
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