Editing Tesseract (section)

{{short description|Four-dimensional analogue of the cube}}
{{about|the geometric shape}}
{{Infobox polychoron
| Name=Tesseract<br />8-cell<br />(4-cube)
| Image_File=8-cell-simple.gif
| Type=[[Convex regular 4-polytope]]
| Family=[[Hypercubes]]
| Last=[[Omnitruncated 5-cell|9]]
| Index=10
| Next=[[Rectified tesseract|11]]
| Schläfli={4,3,3}<br />t<sub>0,3</sub>{4,3,2} or {4,3}×{&nbsp;}<br />t<sub>0,2</sub>{4,2,4} or {4}×{4}<br />t<sub>0,2,3</sub>{4,2,2} or {4}×{&nbsp;}×{&nbsp;}<br />t<sub>0,1,2,3</sub>{2,2,2} or {&nbsp;}×{&nbsp;}×{&nbsp;}×{&nbsp;}
| 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}}
| Cell_List=8 [[cube|{4,3}]] [[File:Hexahedron.png|20px]]
| Face_List=24 [[Square (geometry)|{4}]]
| Edge_Count=32
| Vertex_Count=16
| Petrie_Polygon=[[octagon]]
| Coxeter_Group=B<sub>4</sub>, [3,3,4]
| Vertex_Figure=[[File:8-cell verf.svg|80px]]<br />[[Tetrahedron]]
| Dual=[[16-cell]]
| Property_List=[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[isotoxal figure|isotoxal]], [[isohedral figure|isohedral]], [[Hanner polytope]]
}}
{{wikt | tesseract}}
[[File:8-cell net.png|thumb|The [[Dali cross|Dalí cross]], a [[Net (polyhedron)|net]] of a tesseract]]
[[File:Net of tesseract.gif|thumb|The tesseract can be unfolded into eight cubes into 3D space, just as the cube can be unfolded into six squares into 2D space.]]

In [[geometry]], a '''tesseract''' or '''4-cube''' is a [[four-dimensional space|four-dimensional]] [[hypercube]], analogous to a two-[[dimension]]al [[square (geometry)|square]] and a three-dimensional [[cube]].<ref>{{Cite web|title= The Tesseract - a 4-dimensional cube|url= https://www.cut-the-knot.org/ctk/Tesseract.shtml|access-date= 2020-11-09|website= www.cut-the-knot.org}}</ref> Just as the perimeter of the square consists of four edges and the surface of the cube consists of six square [[Face (geometry) |faces]], the [[hypersurface]] of the tesseract consists of eight cubical [[cell (geometry) |cells]], meeting at [[right angle]]s. The tesseract is one of the six [[convex regular 4-polytope]]s.

The tesseract is also called an '''8-cell''', '''C<sub>8</sub>''', (regular) '''octachoron''', or '''cubic prism'''. It is the four-dimensional '''measure polytope''', taken as a unit for hypervolume.<ref>{{Cite book |last=Elte |first=E. L. |author-link=Emanuel Lodewijk Elte |title=The Semiregular Polytopes of the Hyperspaces |date=2005 |publisher=University of Groningen |isbn=1-4181-7968-X |location=Groningen }}</ref> [[Harold Scott MacDonald Coxeter| Coxeter]] labels it the {{math|''γ''<sub>4</sub>}} polytope.{{Sfn|Coxeter|1973|pp=122-123|loc=§7.2. illustration Fig 7.2<small>C</small>}}  The term ''hypercube'' without a dimension reference is frequently treated as a synonym for this specific [[polytope]].

The ''[[Oxford English Dictionary]]'' traces the word ''tesseract'' to [[Charles Howard Hinton]]'s 1888 book ''[[A New Era of Thought]]''. The term derives from the [[Ancient Greek| Greek]] {{lang|grc-Latn|téssara}} ({{wikt-lang|grc|τέσσαρα}} 'four') and {{lang|grc-Latn|aktís}} ({{wikt-lang|grc|ἀκτίς}} 'ray'), referring to the four edges from each vertex to other vertices. Hinton originally spelled the word as ''tessaract''.<ref>
{{cite OED|term=tesseract|ID=199669}}</ref>