Tetragonal disphenoid honeycomb

{| class="wikitable" align="right" style="margin-left:10px" width="250" !bgcolor=#e7dcc3 colspan=2|Tetragonal disphenoid tetrahedral honeycomb |- |colspan=2 align=center|160px |- |bgcolor=#e7dcc3|Type||convex uniform honeycomb dual |- |bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{CDD|node|4|node_f1|3|node_f1|4|node}} |- |bgcolor=#e7dcc3|Cell type||100px Tetragonal disphenoid |- |bgcolor=#e7dcc3|Face types||isosceles triangle {3} |- |bgcolor=#e7dcc3|Vertex figure||80px tetrakis hexahedron {{CDD|node|4|node_f1|3|node_f1}} |- |bgcolor=#e7dcc3|Space group||Im{{overline|3}}m (229) |- |bgcolor=#e7dcc3|Symmetry||{{brackets|4, 3, 4}} |- |bgcolor=#e7dcc3|Coxeter group||<math>{\tilde{C}}_3</math>, [4, 3, 4] |- |bgcolor=#e7dcc3|Dual||Bitruncated cubic honeycomb |- |bgcolor=#e7dcc3|Properties||cell-transitive, face-transitive, vertex-transitive |} The '''tetragonal disphenoid tetrahedral honeycomb''' is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of identical tetragonal disphenoidal cells. Cells are face-transitive with 4 identical isosceles triangle faces. John Horton Conway calls it an ''oblate tetrahedrille'' or shortened to ''obtetrahedrille''.<ref name=sos295>Symmetry of Things, Table 21.1. Prime Architectonic and Catopric tilings of space, p. 293, 295.</ref>

A cell can be seen as 1/12 of a translational cube, with its vertices centered on two faces and two edges. Four of its edges belong to 6 cells, and two edges belong to 4 cells. : 160px

The tetrahedral disphenoid honeycomb is the dual of the uniform bitruncated cubic honeycomb.

Its vertices form the A{{sup sub|*|3}} / D{{sup sub|*|3}} lattice, which is also known as the body-centered cubic lattice.

== Geometry == 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.

: 300px

An orientation of the tetragonal disphenoid honeycomb can be obtained by starting with a cubic honeycomb, subdividing it at the planes <math>x=y</math>, <math>x=z</math>, and <math>y=z</math> (i.e. subdividing each cube into path-tetrahedra), then squashing it along the main diagonal until the distance between the points (0, 0, 0) and (1, 1, 1) becomes the same as the distance between the points (0, 0, 0) and (0, 0, 1).

== Hexakis cubic honeycomb == {| class="wikitable" align="right" style="margin-left:10px" width="250" !bgcolor=#e7dcc3 colspan=2|Hexakis cubic honeycomb ''Pyramidille''<ref name=sos296>Symmetry of Things, Table 21.1. Prime Architectonic and Catopric tilings of space, p. 293, 296.</ref> |- |bgcolor=#ffffff align=center colspan=2|250px |- |bgcolor=#e7dcc3|Type||Dual uniform honeycomb |- |bgcolor=#e7dcc3|Coxeter–Dynkin diagrams||{{CDD|node_f1|4|node_f1|3|node|4|node}} |- |bgcolor=#e7dcc3|Cell |Isosceles square pyramid 30px |- |bgcolor=#e7dcc3|Faces |Triangle square |- |bgcolor=#e7dcc3|Space group Fibrifold notation||Pm{{overline|3}}m (221) 4−:2 |- |bgcolor=#e7dcc3|Coxeter group||<math>{\tilde{C}}_3</math>, [4, 3, 4] |- |bgcolor=#e7dcc3|vertex figures |30px30px {{CDD|node_f1|3|node|4|node}}, {{CDD|node|4|node_f1|3|node}} |- |bgcolor=#e7dcc3|Dual||Truncated cubic honeycomb |- |bgcolor=#e7dcc3|Properties||Cell-transitive |} The '''hexakis cubic honeycomb''' is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. John Horton Conway calls it a ''pyramidille''.<ref name=sos296/>

Cells can be seen in a translational cube, using 4 vertices on one face, and the cube center. Edges are colored by how many cells are around each of them. :160px

It can be seen as a cubic honeycomb with each cube subdivided by a center point into 6 square pyramid cells.

There are two types of planes of faces: one as a square tiling, and flattened triangular tiling with half of the triangles removed as ''holes''. {| class=wikitable |- align=center !Tiling plane |140px |150px |- align=center !Symmetry |p4m, [4,4] (*442) |pmm, [∞,2,∞] (*2222) |}

=== Related honeycombs === It is dual to the truncated cubic honeycomb with octahedral and truncated cubic cells: :160px

If the square pyramids of the '''pyramidille''' are joined on their bases, another honeycomb is created with identical vertices and edges, called a square bipyramidal honeycomb, or the dual of the rectified cubic honeycomb.

It is analogous to the 2-dimensional tetrakis square tiling: :160px

== Square bipyramidal honeycomb == {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Square bipyramidal honeycomb ''Oblate octahedrille''<ref name=sos296/> |- |bgcolor=#ffffff align=center colspan=2|280px |- |bgcolor=#e7dcc3|Type||Dual uniform honeycomb |- |bgcolor=#e7dcc3|Coxeter–Dynkin diagrams||{{CDD|node|4|node_f1|3|node|4|node}} |- |bgcolor=#e7dcc3|Cell |Square bipyramid 80px |- |bgcolor=#e7dcc3|Faces |Triangles |- |bgcolor=#e7dcc3|Space group Fibrifold notation||Pm{{overline|3}}m (221) 4−:2 |- |bgcolor=#e7dcc3|Coxeter group||<math>{\tilde{C}}_3</math>, [4,3,4] |- |bgcolor=#e7dcc3|vertex figures |30px30px {{CDD|node_f1|3|node|4|node}}, {{CDD|node|4|node_f1|3|node}} |- |bgcolor=#e7dcc3|Dual||Rectified cubic honeycomb |- |bgcolor=#e7dcc3|Properties||Cell-transitive, Face-transitive |} The '''square bipyramidal honeycomb''' is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. John Horton Conway calls it an ''oblate octahedrille'' or shortened to ''oboctahedrille''.<ref name=sos295/>

A cell can be seen positioned within a translational cube, with 4 vertices mid-edge and 2 vertices in opposite faces. Edges are colored and labeled by the number of cells around the edge. :160px

It can be seen as a cubic honeycomb with each cube subdivided by a center point into 6 square pyramid cells. The original cubic honeycomb walls are removed, joining pairs of square pyramids into square bipyramids (octahedra). Its vertex and edge framework is identical to the hexakis cubic honeycomb.

There is one type of plane with faces: a flattened triangular tiling with half of the triangles as ''holes''. These cut face-diagonally through the original cubes. There are also square tiling plane that exist as nonface ''holes'' passing through the centers of the octahedral cells. {| class=wikitable |- align=center !Tiling plane |140px Square tiling "holes" |150px flattened triangular tiling |- align=center !Symmetry |p4m, [4,4] (*442) |pmm, [∞,2,∞] (*2222) |}

=== Related honeycombs === It is dual to the rectified cubic honeycomb with octahedral and cuboctahedral cells: :160px

== Phyllic disphenoidal honeycomb == {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|''Phyllic disphenoidal honeycomb'' ''Eighth pyramidille''<ref name=sos298>Symmetry of Things, Table 21.1. Prime Architectonic and Catopric tilings of space, p. 293, 298.</ref> |- |bgcolor=#ffffff align=center colspan=2|(No image) |- |bgcolor=#e7dcc3|Type||Dual uniform honeycomb |- |bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||{{CDD|node_f1|4|node_f1|3|node_f1|4|node_f1|}} |- |bgcolor=#e7dcc3|Cell |80px Phyllic disphenoid |- |bgcolor=#e7dcc3|Faces |Rhombus Triangle |- |bgcolor=#e7dcc3|Space group Fibrifold notation Coxeter notation||Im{{overline|3}}m (229) 8o:2 <nowiki></nowiki>4,3,4 |- |bgcolor=#e7dcc3|Coxeter group||[4,3,4], <math>{\tilde{C}}_3</math> |- |bgcolor=#e7dcc3|vertex figures |30px30px {{CDD|node_f1|3|node_f1|4|node_f1}}, {{CDD|node_f1|2x|node_f1|4|node_f1}} |- |bgcolor=#e7dcc3|Dual||Omnitruncated cubic honeycomb |- |bgcolor=#e7dcc3|Properties||Cell-transitive, face-transitive |} The '''phyllic disphenoidal honeycomb''' is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. John Horton Conway calls this an ''Eighth pyramidille''.<ref name=sos298/>

A cell can be seen as 1/48 of a translational cube with vertices positioned: one corner, one edge center, one face center, and the cube center. The edge colors and labels specify how many cells exist around the edge. It is one 1/6 of a smaller cube, with 6 phyllic disphenoidal cells sharing a common diagonal axis. :160px

=== Related honeycombs === It is dual to the omnitruncated cubic honeycomb: :160px

==References== {{reflist}} *{{citation | last = Gibb | first = William | title = Paper patterns: solid shapes from metric paper | year = 1990 | journal = Mathematics in School | volume = 19 | issue = 3 | pages = 2–4}}, reprinted in {{citation | editor-last = Pritchard | editor-first = Chris | year = 2003 | title = The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching | publisher = Cambridge University Press | isbn = 0-521-53162-4 | pages = 363–366}}. *{{citation | doi = 10.2307/2689983 | last = Senechal | first = Marjorie | authorlink = Marjorie Senechal | title = Which tetrahedra fill space? | year = 1981 | journal = Mathematics Magazine | volume = 54 | issue = 5 | pages = 227–243 | publisher = Mathematical Association of America | jstor = 2689983}}. *{{cite book | last1 = Conway | first1 = John H. | authorlink1 = John Horton Conway | last2 = Burgiel | first2 = Heidi | last3 = Goodman-Strauss | first3 = Chaim | title = The Symmetries of Things | publisher = A K Peters, Ltd. | date = 2008 | chapter = 21. Naming Archimedean and Catalan Polyhedra and Tilings | pages = 292–298 | isbn = 978-1-56881-220-5}}

Category:3-honeycombs Category:Tetrahedra