In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. Table of Geometry, from the 1728 Cyclopaedia. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...

Twenty-eight such honeycombs exist:

• the familiar cubic honeycomb and 7 truncations thereof;
• the alternated cubic honeycomb and 4 truncations thereof;
• 10 prismatic forms based on the uniform plane tilings (11 if including the cubic honeycomb);
• 5 modifications of some of the above by elongation and/or gyration.

They can be considered the three-dimensional analogue to the uniform tilings of the plane. In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. ... In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. ... In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. ... This table shows the 11 uniform tilings of the plane, and their dual tilings. ...

History

• 1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions, including one regular cubic honeycomb, and two semiregular forms with tetrahedra and octahedra.
• 1905: Alfredo Andreini enumerated 25 of these tessellations.
• 1991: Norman Johnson's manuscript Uniform Polytopes identified the complete list of 28.
• 1994: Branko Grünbaum, in his paper Uniform tilings of 3-space, also independently enumerated all 28, after discovering errors in Andreini's publication. He found the 1905 paper, which listed 25, had 1 wrong, and 4 being missing. Grünbaum also states that I. Alexeyev of Russia also independently enumerated these forms around the same time.

Only 14 of the convex uniform polyhedra appear in these patterns: In geometry, a Platonic solid is a convex regular polyhedron. ... Alfredo Andreini was an Italian doctor and entomologist, born on July 27, 1870 in Florence and died on December 11, 1943 in Lippiano. ... Norman W. Johnson is a mathematician, previously at Wheaton College, Norton, Massachusetts. ... Branko Grünbaum is a mathematician who works mainly in geometry and is considered a founder of discrete geometry. ... In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. ...

• three of the five Platonic solids,
• six of the thirteen Archimedean solids, and
• five of the infinite family of prisms.

In geometry, a Platonic solid is a convex regular polyhedron. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ...

Names

This set can be called the regular and semiregular honeycombs. It has been called the Archimedean honeycombs by analogy with the convex uniform (non-regular) polyhedra, commonly called Archimedean solids. Recently Conway has suggested naming the set as the Architectonic tessellations and the dual honeycombs as the Catoptric tessellations. In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ...

The individual honeycombs are listed with names given to them by Norman Johnson. (Some of the terms used below are defined in Uniform polychoron#Geometric derivations.) Norman Lloyd Johnson was born in Ilford Essex, England, in 1917. ... In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ...

For cross-referencing, the are give with list indices from [A]ndreini (1-22), [W]illiams(1-2,9-19), [J]ohnson(11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and [G]runbaum(1-28).

Tessellations listed by infinite Coxeter group families

Symmetry domain and points within cubic cell. {4,3,4} R₄ group

The fundamental infinite Coxeter groups for 3-space are: Image File history File links Polychoron_tetrahedral_domain. ... Image File history File links Polychoron_tetrahedral_domain. ...

1. R₄ {4,3,4} regular cubic - (8 unique forms plus one alternation)
2. S₄ branching group - (11 forms, 3 new)
3. P₄ cyclic group - (5 forms, one new)

The total unique honeycombs above are 13. Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CD_ring. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-00. ... Image File history File links CD_3b. ... Image File history File links CD_4. ... Image File history File links CD_dot. ... Image File history File links CD_p4-1000. ...

The prismatic infinite Coxeter groups for 3-space are:

1. R₃xW₂ - {4,4}x{∞} prismatic group - (2 new forms)
2. V₃xW₂ - {6,3}x{∞} prismatic group - (7 unique forms)
3. P₃xW₂ - (3 3 3)x{∞} prismatic group - (No new forms)
4. W₂xW₂xW₂ - {∞}x{∞}x{∞} prismatic group - (These all become a cubic honeycomb)

The total unique prismatic honeycombs above (excluding the cubic counted previously) are 9. Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... Image File history File links CDW_ring. ... Image File history File links CDW_6. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ...

Finally there are 6 uniform honeycombs which don't have pure reflectional symmetry and are generated by elongation and gyration operations.

Combining these counts, 13, 9, and 6 gives us the total 28 uniform honeycombs.

Cubic forms {4,3,4} - R₄ group

The regular cubic honeycomb, represented by Schläfli symbol {4,3,4}, offers seven unque derived uniform honeycombs via truncation operations. (One redundant form, the runcinated cubic honeycomb, is included for completeness though identical to the cubic honeycomb.)

Reference Indices	Coxeter-Dynkin and Schläfli symbols	name	Cell counts/vertex and positions in cubic honeycomb
			Cell (3)	Face (2)	Edge (1)	Vertex (0)	Solids (Partial)	Frames (Perspective)	vertex figure
J11,15 A1 W1 G22	t0{4,3,4}	cubic	8 (4.4.4)						octahedron
J12,32 A15 W14 G7	t1{4,3,4}	rectified cubic	4 (3.4.3.4)			2 (3.3.3.3)			cuboid
J13 A14 W15 G8	t0,1{4,3,4}	truncated cubic	4 (3.8.8)			1 (3.3.3.3)			square pyramid
J14 A17 W12 G9	t0,2{4,3,4}	cantellated cubic	2 (3.4.4.4)		2 (4.4.4)	1 (3.4.3.4)			wedge
J11,15	t0,3{4,3,4}	runcinated cubic (same as regular cubic)	1 (4.4.4)	3 (4.4.4)	3 (4.4.4)	1 (4.4.4)			octahedron
J16 A3 W2 G28	t1,2{4,3,4}	bitruncated cubic	2 (4.6.6)			2 (4.6.6)			isosceles tetrahedron
J17 A18 W13 G25	t0,1,2{4,3,4}	cantitruncated cubic	2 (4.6.8)		1 (4.4.4)	1 (4.6.6)			irregular tetrahedron
J18 A19 W19 G20	t0,1,3{4,3,4}	runcitruncated cubic	1 (3.8.8)	2 (4.4.8)	1 (4.4.4)	1 (3.4.4.4)			oblique trapezoidal pyramid
J19 A22 W18 G27	t0,1,2,3{4,3,4}	omnitruncated cubic	1 (4.6.8)	1 (4.4.8)	1 (4.4.8)	1 (4.6.8)			irregular tetrahedron
J21,31,51 A2 W9 G1		alternated cubic	8 3.3.3			6 3.3.3.3			cuboctahedron

Coxeter groups in the plane with equivalent diagrams. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Download high-resolution version (873x866, 24 KB) I, the creator of this work, hereby release it into the public domain. ... Image File history File links Cubic_tiling. ... Image File history File links VF-cubic. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... The rectified cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Rectified_cubic_honeycomb. ... Image File history File links Rectified_cubic_tiling. ... Image File history File links VF-rectified_cubic. ... In anatomy, the cuboid bone is a bone in the foot. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... The truncated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high-resolution version (1000x1000, 200 KB)Truncated hexahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Truncated_cubic_honeycomb. ... Image File history File links Truncated_cubic_tiling. ... Image File history File links VF-truncated_cubic. ... In geometry, the square pyramid, a pyramid with a square base and equilateral sides, is one of the Johnson solids (J1). ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... The cantellated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high-resolution version (1000x1000, 227 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Cantellated_cubic_honeycomb. ... Image File history File links Cantellated_cubic_tiling. ... Image File history File links VF-cantellated_cubic. ... In geometry, a wedge is a polyhedral solid defined by two triangles and three trapezoid faces. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... A runcinated cubic honeycomb (partial) - The original cells (purple cubes) are reduced in size. ... The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Download high-resolution version (809x854, 23 KB)Runcinated cubic honeycomb: Cubic cells become purple cubes. ... Image File history File links Cubic_tiling. ... Image File history File links VF-cubic. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... The bitruncated cubic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Image File history File links Download high-resolution version (868x848, 43 KB) Image for Bitruncated_cubic_honeycomb - 2 colored truncated octahedron cells for original cells and vertices. ... Image File history File links Bitruncated_cubic_tiling. ... Image File history File links VF-bitruncated_cubic. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... The cantitruncated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high resolution version (638x639, 23 KB) Summary Great rhombicuboctahedron, U17 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Image File history File links Cantitruncated_cubic_honeycomb. ... Image File history File links Cantitruncated_cubic_tiling. ... Image File history File links VF-cantitruncated_cubic. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... The runcitruncated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high-resolution version (1000x1000, 200 KB)Truncated hexahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... Image File history File links Octagonal_prism. ... In geometry, the octagonal prism is the sixth in an infinite set of prisms formed by square sides and two regular polygon caps. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 227 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... Image File history File links Runcitruncated_cubic_honeycomb. ... Image File history File links Runcitruncated_cubic_tiling. ... Image File history File links VF-runcitruncated_cubic. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... The omnitruncated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high resolution version (638x639, 23 KB) Summary Great rhombicuboctahedron, U17 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Image File history File links Octagonal_prism. ... In geometry, the octagonal prism is the sixth in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Octagonal_prism. ... In geometry, the octagonal prism is the sixth in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Download high resolution version (638x639, 23 KB) Summary Great rhombicuboctahedron, U17 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Image File history File links Omnitruncated_cubic_honeycomb. ... Image File history File links Omnitruncated_cubic_tiling. ... Image File history File links VF-omnitruncated_cubic. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links CDW_hole. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... The tetrahedral-octahedral honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of alternating tetrahedra and octahedra. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Tetrahedral-octahedral_honeycomb. ... Image File history File links Alternated_cubic_tiling. ... Image File history File links VF-alternated_cubic. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ...

S₄ group

The S₄ group offers 11 derived forms via truncation operations, four being unique uniform honeycombs.

The honeycombs from this group are called alternated cubic because the first form can be seen as a cubic honeycomb with alternate vertices cubic cells to tetrahedra, and creating octahedron cells in the deleted gaps.

Nodes are indexed left to right as 0,1,0',3 with 0' being below and interchangable with 0. The alternate cubic names given are based on this ordering.

Referenced indices	Coxeter-Dynkin diagram	name	Cells by location (and count around each vertex)				Solids (Partial)	Frames (Perspective)	vertex figure
			(0)	(1)	(0')	(3)
J21,31,51 A2 W9 G1		alternated cubic			(6) 3.3.3.3	(8) 3.3.3			cuboctahedron
J22,34 A21 W17 G10		truncated alternated cubic		(1) 3.4.3.4	(2) 4.6.6	(2) 3.6.6
J12,32 A15 W14 G7		rectified cubic (rectified alternate cubic)	(2) (3.4.3.4)		(2) (3.4.3.4)	(2) (3.3.3.3)			cuboid
J12,32 A15 W14 G7		rectified cubic (cantellated alternate cubic)	(1) (3.3.3.3)		(1) (3.3.3.3)	(4) (3.4.3.4)			cuboid
J16 A3 W2 G28		bitruncated cubic (cantitruncated alternate cubic)	(1) (4.6.6)		(1) (4.6.6)	(2) (4.6.6)			isosceles tetrahedron
J13 A14 W15 G8		truncated cubic (bicantellated alternate cubic)	(2) (3.8.8)		(2) (3.8.8)	(1) (3.3.3.3)			square pyramid
J11,15 A1 W1 G22		cubic (trirectified alternate cubic)	(4) (4.4.4)		(4) (4.4.4)				octahedron
J23 A16 W11 G5		runcinated alternated cubic	(1) cube		(3) 3.4.4.4	(1) 3.3.3
J14 A17 W12 G9		cantellated cubic (runcicantellated alternate cubic)	(1) (3.4.4.4)	(2) (4.4.4)	(1) (3.4.4.4)	(1) (3.4.3.4)			wedge
J24 A20 W16 G21		cantitruncated alternated cubic (or runcitruncated alternate cubic)		(1) 3.8.8	(2) 4.6.8	(1) 3.6.6
J17 A18 W13 G25		cantitruncated cubic (omnitruncated alternated cubic)	(1) (4.6.8)	(1) (4.4.4)	(1) (4.6.8)	(1) (4.6.6)			irregular tetrahedron

Coxeter groups in the plane with equivalent diagrams. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... Image File history File links CD_ring. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-00. ... Image File history File links CD_3b. ... Image File history File links CD_4. ... Image File history File links CD_dot. ... The tetrahedral-octahedral honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of alternating tetrahedra and octahedra. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Tetrahedral-octahedral_honeycomb. ... Image File history File links Alternated_cubic_tiling. ... Image File history File links VF-alternated_cubic. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links CD_ring. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-10. ... Image File history File links CD_3b. ... Image File history File links CD_4. ... Image File history File links CD_dot. ... The truncated alternated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... Image File history File links Truncated_alternated_cubic_honeycomb. ... Image File history File links Truncated_alternated_cubic_tiling. ... Image File history File links VF-Truncated_alternated_cubic. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-10. ... Image File history File links CD_3b. ... Image File history File links CD_4. ... Image File history File links CD_dot. ... The rectified cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Rectified_cubic_honeycomb. ... Image File history File links Rectified_cubic_tiling. ... Image File history File links VF-rectified_cubic. ... In anatomy, the cuboid bone is a bone in the foot. ... Image File history File links CD_ring. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-01. ... Image File history File links CD_3b. ... Image File history File links CD_4. ... Image File history File links CD_dot. ... The rectified cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Rectified_cubic_honeycomb. ... Image File history File links Rectified_cubic_tiling. ... Image File history File links VF-rectified_cubic. ... In anatomy, the cuboid bone is a bone in the foot. ... Image File history File links CD_ring. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-11. ... Image File history File links CD_3b. ... Image File history File links CD_4. ... Image File history File links CD_dot. ... The bitruncated cubic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Image File history File links Download high-resolution version (868x848, 43 KB) Image for Bitruncated_cubic_honeycomb - 2 colored truncated octahedron cells for original cells and vertices. ... Image File history File links Bitruncated_cubic_tiling. ... Image File history File links VF-bitruncated_cubic. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-10. ... Image File history File links CD_3b. ... Image File history File links CD_4. ... Image File history File links CD_ring. ... The truncated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high-resolution version (1000x1000, 200 KB)Truncated hexahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 200 KB)Truncated hexahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Truncated_cubic_honeycomb. ... Image File history File links Truncated_cubic_tiling. ... Image File history File links VF-truncated_cubic. ... In geometry, the square pyramid, a pyramid with a square base and equilateral sides, is one of the Johnson solids (J1). ... Image File history File links CD_dot. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-00. ... Image File history File links CD_3b. ... Image File history File links CD_4. ... Image File history File links CD_ring. ... The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Download high-resolution version (809x800, 21 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Cubic_tiling. ... Image File history File links VF-cubic. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links CD_ring. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-00. ... Image File history File links CD_3b. ... Image File history File links CD_4. ... Image File history File links CD_ring. ... The runcinated alternated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 227 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Runcinated_alternated_cubic_honeycomb. ... Image File history File links Runcinated_alternated_cubic_tiling. ... Image File history File links VF-runcinated_alternated_cubic. ... Image File history File links CD_ring. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-01. ... Image File history File links CD_3b. ... Image File history File links CD_4. ... Image File history File links CD_ring. ... The cantellated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high-resolution version (1000x1000, 227 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 227 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Cantellated_cubic_honeycomb. ... Image File history File links Cantellated_cubic_tiling. ... Image File history File links VF-cantellated_cubic. ... In geometry, a wedge is a polyhedral solid defined by two triangles and three trapezoid faces. ... Image File history File links CD_ring. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-10. ... Image File history File links CD_3b. ... Image File history File links CD_4. ... Image File history File links CD_ring. ... The cantitruncated alternated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high-resolution version (1000x1000, 200 KB)Truncated hexahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... Image File history File links Download high resolution version (638x639, 23 KB) Summary Great rhombicuboctahedron, U17 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... Image File history File links Cantitruncated_alternated_cubic_honeycomb. ... Image File history File links Cantitruncated_alternated_cubic_tiling. ... Image File history File links VF-cantitruncated_alternated_cubic. ... Image File history File links CD_ring. ... Image File history File links CD_3b. ... Image File history File links CD_downbranch-11. ... Image File history File links CD_3b. ... Image File history File links CD_4. ... Image File history File links CD_ring. ... The cantitruncated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high resolution version (638x639, 23 KB) Summary Great rhombicuboctahedron, U17 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Download high resolution version (638x639, 23 KB) Summary Great rhombicuboctahedron, U17 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Image File history File links Cantitruncated_cubic_honeycomb. ... Image File history File links Cantitruncated_cubic_tiling. ... Image File history File links VF-cantitruncated_cubic. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...

P₄ group

There are 5 forms constructed from the P₄ group, only the quarter cubic honeycomb is unique. In mathematics, given a prime number p, a p-group is a group in which each element has a power of p as its order. ...

Referenced indices	Coxeter-Dynkin diagram	name	Cells by location (and count around each vertex)				Solids (Partial)	Frames (Perspective)	vertex figure
			(0)	(1)	(2)	(3)
J21,31,51 A2 W9 G1		alternated cubic		(4) 3.3.3	(6) 3.3.3.3	(4) 3.3.3			cuboctahedron
J12,32 A15 W14 G7		rectified cubic	(2) (3.4.3.4)	(1) (3.3.3.3)	(2) (3.4.3.4)	(1) (3.3.3.3)			cuboid
J25,33 A13 W10 G6		quarter cubic honeycomb	(1) 3.3.3	(1) 3.3.3	(3) 3.6.6	(3) 3.6.6
J22,34 A21 W17 G10		truncated alternated cubic	(1) 3.6.6	(1) 3.4.3.4	(1) 3.6.6	(2) 4.6.6
J16 A3 W2 G28		bitruncated cubic	(1) (4.6.6)	(1) (4.6.6)	(1) (4.6.6)	(1) (4.6.6)			isosceles tetrahedron

Coxeter groups in the plane with equivalent diagrams. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... Image File history File links CD_p4-1000. ... The tetrahedral-octahedral honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of alternating tetrahedra and octahedra. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Tetrahedral-octahedral_honeycomb. ... Image File history File links Alternated_cubic_tiling. ... Image File history File links VF-alternated_cubic. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links CD_p4-1010. ... The rectified cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Rectified_cubic_honeycomb. ... Image File history File links Rectified_cubic_tiling. ... Image File history File links VF-rectified_cubic. ... In anatomy, the cuboid bone is a bone in the foot. ... Image File history File links CD_p4-1100. ... edge framework The Quarter cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of tetrahedra, and truncated tetrahedra. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... Image File history File links Bitruncated_alternated_cubic_honeycomb. ... Image File history File links Bitruncated_alternated_cubic_tiling. ... Image File history File links VF-bitruncated_alternated_cubic. ... Image File history File links CD_p4-1110. ... The truncated alternated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Image File history File links Truncated_alternated_cubic_honeycomb. ... Image File history File links Truncated_alternated_cubic_tiling. ... Image File history File links VF-Truncated_alternated_cubic. ... Image File history File links CD_p4-1111. ... The bitruncated cubic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Image File history File links Download high-resolution version (868x848, 43 KB) Image for Bitruncated_cubic_honeycomb - 2 colored truncated octahedron cells for original cells and vertices. ... Image File history File links Bitruncated_cubic_tiling. ... Image File history File links VF-bitruncated_cubic. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...

Prismatic forms

Eleven prismatic tilings are obtained by stacking the eleven uniform plane tilings, shown below, in parallel layers. (One of these honeycombs is the cubic, shown above.) The vertex figure of each is an irregular bipyramid whose faces are isosceles triangles. Plane tilings by regular polygons have been widely used since antiquity. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... A bipyramid is a polyhedron formed by joining two identical pyramids base-to-base. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

Square prismatics: {4,4}x{∞} - R₃xW₂ group

There's only 3 unique honeycombs from the square tiling, but all 8 tiling truncations are listed below for completeness, and tiling images are shown by colors corresponding to each form.

Indices	Coxeter-Dynkin and Schläfli symbols	Name	Plane tiling	Solids (Partial)	Tiling
J11,15 A1 G22	{4,4} x {∞}	Cubic honeycomb (Square prismatic honeycomb)	(4.4.4.4)
J45 A6 G24	t0,1{4,4} x {∞}	Truncated square prismatic honeycomb	(4.8.8)
J11,15 A1 G22	t1{4,4} x {∞}	Cubic honeycomb (Rectified square prismatic honeycomb)	(4.4.4.4)
J45 A6 G24	t1,2{4,4} x {∞}	Truncated square prismatic honeycomb (Bitruncated square prismatic honeycomb)	(4.8.8)
J11,15 A1 G22	t2{4,4} x {∞}	Cubic honeycomb (Dual square prismatic honeycomb)	(4.4.4.4)
J11,15 A1 G22	t0,2{4,4} x {∞}	Cubic honeycomb (Cantellated square prismatic honeycomb)	(4.4.4.4)
J45 A6 G24	t0,1,2{4,4} x {∞}	Truncated square prismatic honeycomb (Omnitruncated square prismatic honeycomb)	(4.8.8)
J44 A11 G14	s{4,4}x{∞}	Snub square prismatic honeycomb	(3.3.4.3.4)

Coxeter groups in the plane with equivalent diagrams. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (809x800, 21 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (641x641, 31 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Square tiling I, the creator of this work, hereby release it into the public domain. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The truncated square prismatic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (1049x753, 25 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (640x639, 19 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks [{Truncated square tiling]] I, the creator of this work, hereby release it into the public domain. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (809x800, 21 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (640x639, 9 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Square tiling bicolored as rectified square tiling I, the creator of this work, hereby release it into the public domain. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The truncated square prismatic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (1049x753, 25 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (639x640, 20 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks [{Truncated square tiling]] I, the creator of this work, hereby release it into the public domain. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (809x800, 21 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (640x640, 31 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Square tiling I, the creator of this work, hereby release it into the public domain. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (809x800, 21 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (640x640, 35 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Square tiling tricolored as cantellated square tiling I, the creator of this work, hereby release it into the public domain. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The truncated square prismatic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (1049x753, 25 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (641x640, 26 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Truncated square tiling tricolored as omnitruncated square tiling. ... Image File history File links CDW_hole. ... Image File history File links CDW_4. ... Image File history File links CDW_hole. ... Image File history File links CDW_4. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The snub square prismatic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (1021x795, 26 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (639x640, 31 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Hand-recolored snub version from Image:Nonuniform_tiling_44-t012-snub. ...

Trihexagonal prismatics: {6,3}x{∞} - V₃xW₂ group

Indices	Coxeter-Dynkin and Schläfli symbols	Name	Plane tiling	Solids (Partial)	Tiling
J42 A5 G26	t0{6,3} x {∞}	Hexagonal prismatic honeycomb	(63)
J46 A7 G19	t0,1{6,3} x {∞}	Truncated hexagonal prismatic honeycomb	(3.12.12)
J43 A8 G18	t1{6,3} x {∞}	Triangular-hexagonal prismatic honeycomb	(3.6.3.6)
J42 A5 G26	t1,2{6,3} x {∞}	Truncated triangular prismatic honeycomb (Hexagonal prismatic honeycomb)	(6.6.6)
J41 A4 G11	t2{6,3} x {∞}	Triangular prismatic honeycomb	(36)
J47 A9 G16	t0,2{6,3} x {∞}	Rhombitriangular-hexagonal prismatic honeycomb	(3.4.6.4)
J49 A10 G23	t0,1,2{6,3} x {∞}	Omnitruncated triangular-hexagonal prismatic honeycomb	(4.6.12)
J48 A12 G17	s{6,3} x {∞}	Snub triangular-hexagonal prismatic honeycomb	(3.3.3.3.6)

Coxeter groups in the plane with equivalent diagrams. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... Image File history File links CDW_ring. ... Image File history File links CDW_6. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The hexagonal prismatic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (881x856, 22 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (640x639, 20 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks I, the creator of this work, hereby release it into the public domain. ... Image File history File links CDW_ring. ... Image File history File links CDW_6. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The truncated hexagonal prismatic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (1012x694, 22 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (641x640, 19 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks I, the creator of this work, hereby release it into the public domain. ... Image File history File links CDW_dot. ... Image File history File links CDW_6. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The triangular-hexagonal prismatic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (994x865, 28 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (641x639, 18 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks I, the creator of this work, hereby release it into the public domain. ... Image File history File links CDW_dot. ... Image File history File links CDW_6. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (881x856, 22 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (640x640, 20 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks I, the creator of this work, hereby release it into the public domain. ... Image File history File links CDW_dot. ... Image File history File links CDW_6. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The Triangular prismatic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (743x808, 20 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (641x640, 23 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks I, the creator of this work, hereby release it into the public domain. ... Image File history File links CDW_ring. ... Image File history File links CDW_6. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The rhombitriangular-hexagonal prismatic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the Small rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (990x828, 23 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (640x640, 23 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks I, the creator of this work, hereby release it into the public domain. ... Image File history File links CDW_ring. ... Image File history File links CDW_6. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The omnitruncated triangular-hexagonal prismatic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the Great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (988x757, 23 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (639x640, 22 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks I, the creator of this work, hereby release it into the public domain. ... Image File history File links CDW_hole. ... Image File history File links CDW_6. ... Image File history File links CDW_hole. ... Image File history File links CDW_3. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_infin. ... Image File history File links CDW_dot. ... The Snub triangular-hexagonal prismatic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (952x772, 23 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ... Image File history File links Download high-resolution version (639x640, 34 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Hand-recolored snub version from variation of Image:Uniform tiling 63-t012. ...

Gyrated and elongated forms

Three more uniform honeycombs are generated by breaking one or another of the above honeycombs where its faces form a continuous plane, then rotating alternate layers by 60 or 90 degrees (gyration) and/or inserting a layer of prisms (elongation).

The elongated and gyroelongated alternated cubic tilings have the same vertex figure, but are not alike. In the elongated form, each prism meets a tetrahedron at one triangular end and an octahedron at the other. In the gyroelongated form, prisms that meet tetrahedra at both ends alternate with prisms that meet octahedra at both ends.

The gyroelongated triangular prismatic tiling has the same vertex figure as one of the plain prismatic tilings; the two may be derived from the gyrated and plain triangular prismatic tilings, respectively, by inserting layers of cubes.

Referenced indices	symbol	name	cell types (# at each vertex)	# families of continuous face planes	Solids (Partial)	Frames (Perspective)	vertex figure
J52 A2' G2	h{4,3,4}:g	gyrated alternated cubic	tetrahedron (8) octahedron (6)	1 (3.3.3.3.3.3)
J61 A? G3	h{4,3,4}:ge	gyroelongated alternated cubic	triangular prism (6) tetrahedron (4) octahedron (3)	1 (3.3.3.3.3.3)
J62 A? G4	h{4,3,4}:e	elongated alternated cubic	triangular prism (6) tetrahedron (4) octahedron (3)	1 (3.3.3.3.3.3)
J63 A? G12	{6,3}:g x {∞}	gyrated triangular prismatic	triangular prism (12)	1 (4.4.4.4)
J64 A? G15	{6,3}:ge x {∞}	gyroelongated triangular prismatic	triangular prism (6) cube (4)	1 (4.4.4.4)
J65 A11' G13	{6,3}:e x {∞}	elongated triangular prismatic	triangular prism (6) cube (4)	1 (4.4.4.4) 1 (3.3.3.4.4)

In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... The gyrated alternated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ... Image File history File links Gyrated_alternated_cubic. ... Image File history File links Gyrated_alternated_cubic. ... Image File history File links VF-gyrated_alternated_cubic. ... The Gyroelongated alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of tetrahedra, octahedra, and triangular prisms. ... In geometry, a triangular prism or three-sided prism is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ... Image File history File links Gyroelongated_alternated_cubic_tiling. ... Image File history File links VF-extended_alternated_cubic. ... The Elongated alternated cubic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of tetrahedra, octahedra, and triangular prisms. ... In geometry, a triangular prism or three-sided prism is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ... Image File history File links Elongated_alternated_cubic_tiling. ... Image File history File links VF-extended_alternated_cubic. ... The Gyrated triangular prismatic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of triangular prisms. ... In geometry, a triangular prism or three-sided prism is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... Image File history File links Gyrated_triangular_prismatic_tiling. ... Image File history File links VF-gyrated_prismatic_triangular. ... The gyroelongated triangular prismatic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, a triangular prism or three-sided prism is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... Image File history File links Gyroelongated_triangular_prismatic_tiling. ... Image File history File links VF-prismatic_extended_triangular. ... The elongated triangular prismatic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, a triangular prism or three-sided prism is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. ... Image File history File links Download high-resolution version (863x862, 27 KB)Honeycomb drawn from data in VRML files, created by Stella (software). ...

Examples

All 28 of these tessellations are found in crystal arrangements. Quartz crystal Copper(II) sulfate and iodine crystal Synthetic bismuth crystal Insulin crystals Gallium, a metal that easily forms large single crystals A huge monocrystal of potassium dihydrogen phosphate grown from solution by Saint-Gobain for the megajoule laser of CEA. In chemistry and mineralogy, a crystal is a solid...

The alternated cubic honeycomb is of special importance since its vertices form a cubic close-packing of spheres. The space-filling truss of packed octahedra and tetrahedra was apparently first discovered by Alexander Graham Bell and independently re-discovered by Buckminster Fuller (who called it the octet truss and patented it in the 1940s). [1] [2] [3] [4]. Octet trusses are now among the most common types of truss used in construction. The tetrahedral-octahedral honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of alternating tetrahedra and octahedra. ... Close-packing of spheres refers to arranging an infinite lattice of spheres so that they take up the greatest possible fraction of an infinite 3-dimensional space. ... Truss bridge for a single track railway, converted to pedestrian use and pipeline support. ... Alexander Graham Bell (1847-1922) Alexander Graham Bell (March 3, 1847 – August 2, 1800) was a Scottish scientist and inventor who emigrated to Canada and later the United States. ... Richard Buckminster (Bucky) Fuller (July 12[1], 1895 – July 1, 1983) was an American visionary, designer, architect, poet, author, and inventor. ... Simplified space frame roof with the nearest unit polygon hightlighted in blue A space frame is a truss-like, light weight rigid structure constructed from interlocking struts in a geometric pattern. ...

External links

• Weisstein, Eric W., Honeycomb at MathWorld.
• Uniform Honeycombs in 3-Space VRML models
• Elementary Honeycombs
• Uniform partitions of 3-space, their relatives and embedding PDF, 1999
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra

Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

References

• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
• Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. 
• Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. ISBN 0-500-34033-1. 
• A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.