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Tesseract
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== Related polytopes and honeycombs == The tesseract is 4th in a series of [[hypercube]]: {{Hypercube polytopes}} The tesseract (8-cell) is the third in the sequence of 6 convex regular 4-polytopes (in order of size and complexity). {{Regular convex 4-polytopes}} As a uniform [[duoprism]], the tesseract exists in a [[Uniform 4-polytope#Polygonal prismatic prisms: .5Bp.5D .C3.97 .5B .5D .C3.97 .5B .5D|sequence of uniform duoprisms]]: {''p''}Γ{4}. The regular tesseract, along with the [[16-cell]], exists in a set of 15 [[Truncated tesseract#Related uniform polytopes in tesseract symmetry|uniform 4-polytopes with the same symmetry]]. The tesseract {4,3,3} exists in a [[Hexagonal tiling honeycomb#Polytopes and honeycombs with tetrahedral vertex figures|sequence of regular 4-polytopes and honeycombs]], {''p'',3,3} with [[tetrahedron|tetrahedral]] [[vertex figure]]s, {3,3}. The tesseract is also in a [[Order-5 cubic honeycomb#Related polytopes and honeycombs with cubic cells|sequence of regular 4-polytope and honeycombs]], {4,3,''p''} with [[cube|cubic]] [[cell (geometry)|cells]]. {| class=wikitable style="float:right;" width=320 !Orthogonal||Perspective |- |[[File:4-generalized-2-cube.svg|160px]] |[[File:Complex polygon 4-4-2-stereographic3.svg|160px]] |- |colspan=2|<sub>4</sub>{4}<sub>2</sub>, with 16 vertices and 8 4-edges, with the 8 4-edges shown here as 4 red and 4 blue squares |} The [[regular complex polytope]] <sub>4</sub>{4}<sub>2</sub>, {{CDD|4node_1|4|node}}, in <math>\mathbb{C}^2</math> has a real representation as a tesseract or 4-4 [[duoprism]] in 4-dimensional space. <sub>4</sub>{4}<sub>2</sub> has 16 vertices, and 8 4-edges. Its symmetry is <sub>4</sub>[4]<sub>2</sub>, order 32. It also has a lower symmetry construction, {{CDD|4node_1|2|4node_1}}, or <sub>4</sub>{}Γ<sub>4</sub>{}, with symmetry <sub>4</sub>[2]<sub>4</sub>, order 16. This is the symmetry if the red and blue 4-edges are considered distinct.<ref>Coxeter, H. S. M., ''Regular Complex Polytopes'', second edition, Cambridge University Press, (1991).</ref> {{Clear}}
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