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| !bgcolor=#e7dcc3 colspan=2|Tetragonal disphenoid tetrahedral honeycomb
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| |bgcolor=#ffffff align=center colspan=2|[[Image:Disphenoid tetrah hc.png|300px]]
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| |bgcolor=#e7dcc3|Type||[[convex uniform honeycomb]] dual
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node|4|node_f1|3|node_f1|4|node}}
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| |bgcolor=#e7dcc3|Cell type||[[Image:Disphenoid tetrahedron.png|100px]]<BR>[[Tetragonal disphenoid]]
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| |bgcolor=#e7dcc3|Face types||[[isosceles triangle]] {3}
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| |bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Tetrakishexahedron.jpg|80px]]<BR>[[tetrakis hexahedron]]<BR>{{CDD|node|4|node_f1|3|node_f1}}
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| |bgcolor=#e7dcc3|[[Space group]]||Im{{overline|3}}m (229)
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| |bgcolor=#e7dcc3|[[Coxeter notation|Symmetry]]||[<span/>[4,3,4]]
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| |bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{C}}_3</math>, [4,3,4]
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| |bgcolor=#e7dcc3|Dual||[[Bitruncated cubic honeycomb]]
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| |bgcolor=#e7dcc3|Properties||[[cell-transitive]], [[face-transitive]], [[vertex-transitive]]
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| |}
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| The '''tetragonal disphenoid tetrahedral honeycomb''' is a space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in [[Euclidean 3-space]] made up of identical [[tetragonal disphenoid]]al cells. Cells are [[face-transitive]] with 4 identical [[isosceles triangle]] faces.
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| [[John Horton Conway]] calls this honeycomb a '''oblate tetrahedrille'''.
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| This honeycomb's [[vertex figure]] is a [[tetrakis cube]]: 24 disphenoids meet at each vertex. The union of these 24 disphenoids forms a [[rhombic dodecahedron]]. Each edge of the tessellation is surrounded by either four or six disphenoids, according to whether it forms the base or one of the sides of its adjacent isosceles triangle faces respectively. When an edge forms the base of its adjacent isosceles triangles, and is surrounded by four disphenoids, they form an irregular [[octahedron]]. When an edge forms one of the two equal sides of its adjacent isosceles triangle faces, the six disphenoids surrounding the edge form a special type of [[parallelepiped]] called a [[trigonal trapezohedron]].
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| The disphenoid tetrahedral honeycomb is the dual of the uniform [[bitruncated cubic honeycomb]].
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| ==See also==
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| *[[Architectonic and catoptric tessellation]]
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| *[[Cubic honeycomb]]
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| *[[Triakis truncated tetrahedral honeycomb]]
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| ==References==
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| *{{citation
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| | last = Gibb | first = William
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| | title = Paper patterns: solid shapes from metric paper
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| | year = 1990
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| | journal = Mathematics in School
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| | volume = 19
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| | issue = 3
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| | pages = 2–4}}, reprinted in {{citation
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| | editor-last = Pritchard | editor-first = Chris
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| | year = 2003
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| | title = The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching
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| | publisher = Cambridge University Press
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| | isbn = 0-521-53162-4
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| | pages = 363–366}}.
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| *{{citation
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| | doi = 10.2307/2689983
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| | last = Senechal | first = Marjorie | authorlink = Marjorie Senechal
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| | title = Which tetrahedra fill space?
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| | year = 1981
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| | journal = [[Mathematics Magazine]]
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| | volume = 54
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| | issue = 5
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| | pages = 227–243
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| | publisher = Mathematical Association of America
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| | jstor = 2689983}}.
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| [[Category:Honeycombs (geometry)]]
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| {{polychora-stub}}
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