Convex uniform honeycomb
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In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
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.
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[edit] 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:
- three of the five Platonic solids,
- six of the thirteen Archimedean solids, and
- five of the infinite family of prisms.
[edit] 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.
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.)
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).
[edit] Tessellations listed by category
[edit] Cubic forms {4,3,4}
Image:Polychoron tetrahedral domain.png 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.)
[edit] Alternated cubic forms: h{4,3,4}
The tet-oct (alternated cubic) honeycomb offers four derived uniform honeycombs via truncation operations.
This set is called alternated cubic because they can be derived from the cubic honeycomb by deleting alternate vertices. The cubic cells degenerate into tetrahedral cells, and the deleted vertices form voids in which new octahedral cells appear.
[edit] 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).
| Referenced indices | name | cell types (# at each vertex) | # families of continuous face planes | Solids (Partial) | Frames (Perspective) | vertex figure |
|---|---|---|---|---|---|---|
| J52 A2' G2 | gyrated alternated cubic | tetrahedron (8) octahedron (6) | 1 triangle | Image:Gyrated alternated cubic.jpg | Image:Gyrated alternated cubic.png | Image:VF-gyrated alternated cubic.png |
| J61 A? G3 | gyroelongated alternated cubic | triangular prism (6) tetrahedron (4) octahedron (3) | 1 triangle | Image:Gyroelongated alternated cubic tiling.png | Image:VF-extended alternated cubic.png | |
| J62 A? G4 | elongated alternated cubic | triangular prism (6) tetrahedron (4) octahedron (3) | 1 triangle | Image:Elongated alternated cubic tiling.png | Image:VF-extended alternated cubic.png |
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.
[edit] 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.
[edit] Square prismatics: {}x{4,4}
| Indices | Schläfli symbol | Name | Picture |
|---|---|---|---|
| J11,15, A1, G22 | {} x {4,4} | Cubic honeycomb (square prismatic honeycomb) | Image:Tile 4,4.svg square tiling |
| J45, A6, G24 | {} x t1{4,4} | Truncated square prismatic honeycomb | Image:Tile 488.svg 4.8.8 tiling |
| J44, A11, G14 | {} x s{4,4} | Snub square prismatic honeycomb | Image:Tile 33434.svg 3.3.4.3.4 tiling |
[edit] Trihexagonal prismatics: {}x{3,6}
| Indices | Schläfli symbol | Name | Picture |
|---|---|---|---|
| J41, A4, G11 | {} x {3,6} | Triangular prismatic honeycomb | Image:Tile 3,6.svg triangular tiling |
| J42, A5, G26 | {} x t2{3,6} | Hexagonal prismatic honeycomb | Image:Tile 6,3.svg hexagonal tiling |
| J43, A8, G18 | {} x t1{3,6} | Triangular-hexagonal prismatic honeycomb | Image:Tile 3636.svg 3.6.3.6 tiling |
| J46, A7, G19 | {} x t1,2{3,6} | Truncated hexagonal prismatic honeycomb | Image:Tile 3bb.svg 3.12.12 tiling |
| J47, A9, G16 | {} x t0,2{3,6} | Rhombitriangular-hexagonal prismatic honeycomb | Image:Tile 3464.svg 3.4.6.4 tiling |
| J48, A12, G17 | {} x s{3,6} | Snub triangular-hexagonal prismatic honeycomb | Image:Tile 33336.svg 3.3.3.3.6 tiling |
| J49, A10, G23 | {} x t0,1,2{3,6} | Omnitruncated triangular-hexagonal prismatic honeycomb | Image:Tile 46b.svg 4.6.12 tiling |
| J65, A11', G13 | {} x {3,6}e | Elongated triangular prismatic honeycomb | Image:Tile 33344.svg 3.3.3.4.4 tiling |
[edit] Gyrated prismatic forms
| Referenced indices | name | cell types (# at each vertex) | # families of continuous face planes | Solids (Partial) | Frames (Perspective) | vertex figure |
|---|---|---|---|---|---|---|
| J63 A? G12 | gyrated triangular prismatic | triangular prism (12) | 1 square | Image:Gyrated triangular prismatic tiling.png | Image:VF-gyrated prismatic triangular.png | |
| J64 A? G15 | gyroelongated triangular prismatic | triangular prism (6) cube (4) | 1 square | Image:Gyroelongated triangular prismatic tiling.png | Image:VF-prismatic extended triangular.png |
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.
[edit] Examples
All 28 of these tessellations are found in crystal arrangements.
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.
[edit] External links
- Uniform Honeycombs in 3-Space VRML models
- Tiling space with regular and semiregular polyhedra MS-PowerPoint
- Elementary Honeycombs
- Uniform partitions of 3-space, their relatives and embedding PDF, 1999
- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
[edit] 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.
