In geometry, the Andreini tessellations are the complete set of 28 uniform (space-filling) honeycombs of 3-space. They are the three dimensional equivalent to the uniform tilings of the plane. They are named in honor of A. Andreini who studied and enumerated these tessellation forms around 1905. (See references below) Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ... A tessellated plane. ... A tessellation of space fills space with solids, e. ... This table shows the 11 uniform tilings of the plane, and their dual tilings. ...

A uniform honeycomb is constructed by identical sets of convex uniform polyhedral cells around each vertex. Uniform cells can include the 5 Platonic, 13 Archimedean solids, and infinite sets of uniform prisms and antiprisms. In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting at each of its vertices. ... 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, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ...

The 28 tessellations can be divided into 4 groups by the existence of parallel uniform polygon-tiled planes:

1. 10 are tessellations with no planar face-tilings.
2. 12 are formed by stacked prisms of the 11 planar face-tilings.
3. 1 has alternate planes of vertices, but only has one planar face-tiling.
4. 5 have alternate planes of vertices with both planar face-tiled.

These groups are listed largely in order of least to most polyhedra per vertex. The first group has 4 to 6 polyhedra per vertex, while the last has 12 to 14!

The listing below follows the four groups, and orders them by [n/m] where n is the number of cells per vertex, and m is the number of types of cells.

Bitruncated cubic honeycomb

• No face-tiling planes (with cubic honeycomb naming) (4 to 6 polyhedra per vertex)
  1. [4/1] 4 truncated octahedra
     • Bitruncated cubic honeycomb
     • Dual Disphenoid tetrahedral honeycomb
  2. [4/2] 2 truncated cuboctahedra, 2 octagonal prisms
     • Omnitruncated cubic honeycomb
  3. [4/3] 2 truncated cuboctahedra, 1 truncated octahedron, 1 cube
     • Cantitruncated cubic honeycomb
  4. [4/3] 2 truncated cuboctahedra, 1 truncated cube, 1 truncated tetrahedron
     • Cantitruncated alternated cubic honeycomb
  5. [5/2] 4 truncated cubes, 1 octahedron
     • Truncated cubic honeycomb
  6. [5/3] 2 rhombicuboctahedra, 1 cuboctahedron, 2 cubes
     • Cantellated cubic honeycomb
  7. [5/3] 3 rhombicuboctahedra, 1 tetrahedron, 1 cube
     • Runcinated alternated cubic honeycomb
  8. [5/3] 2 truncated octahedra, 2 truncated tetrahedra, 1 cuboctahedron
     • Truncated alternated cubic honeycomb
  9. [5/4] 1 truncated cube, 1 rhombicuboctahedron, 2 octagonal prisms, 1 cube
     • Runcitruncated cubic honeycomb
  10. [6/2] 4 cuboctahedra, 2 octahedra
      • Rectified cubic honeycomb

Cubic honeycomb lattice

• Identical faced-tiling planes (6 to 12 cells per vertex)
  1. [6/1] 6 hexagonal prisms (hexagonal tiling planes),
     • Hexagonal prismatic honeycomb,
     • dual Triangular prismatic honeycomb
  2. [6/2] 4 dodecagonal prisms, 2 triangular prisms (3.12.12 tiling planes)
     • Truncated hexagonal prismatic honeycomb
  3. [6/2] 4 octagonal prisms, 2 cubes (4.8.8 tiling planes)
     • Truncated square prismatic honeycomb
  4. [6/3] 2 dodecagonal prisms, 2 hexagonal prisms, 2 cubes (4.6.12 tiling planes)
     • Omnitruncated triangular-hexagonal prismatic honeycomb
  5. [8/1] 8 cubes (square tiling planes)
     • Cubic honeycomb
     • Self-dual
  6. [8/2] 4 hexagonal prisms, 4 triangular prisms, (3.6.3.6 tiling planes)
     • Triangular-hexagonal prismatic honeycomb
  7. [8/3] 2 hexagonal prisms, 2 triangular prisms, 4 cubes (3.4.6.4 tiling planes)
     • Rhombitriangular-hexagonal prismatic honeycomb
  8. [10/2] 2 hexagonal prisms, 8 triangular prisms (3.3.3.3.6 tiling planes)
     • Snub triangular-hexagonal prismatic honeycomb
  9. [10/2] (I) 6 triangular prisms, 4 cubes (3.3.3.4.4 tiling planes),
     • Elongated triangular prismatic honeycomb
  10. [10/2] (II) 6 triangular prisms, 4 cubes, (4.4.4.4 tiling planes) Gyrated layers
      • Gyroelongated triangular prismatic honeycomb
  11. [10/2] 6 triangular prisms, 4 cubes (3.3.4.3.4 tiling planes)
      • Snub square prismatic honeycomb
  12. [12/1] (I) 12 triangular prisms (triangular tiling planes)
      • Triangular prismatic honeycomb,
      • dual Hexagonal prismatic honeycomb

• Alternating planes, one face-tiled (8 polyhedra per vertex)
  1. [8/2] 6 truncated tetrahedra, 2 tetrahedra (3.6.3.6 tiling planes)
     • Bitruncated alternated cubic honeycomb

Tetra-Octa honeycomb

• Alternating planes, both face-tiled (12 to 14 polyhedra per vertex)
  1. [12/1] (II) 12 triangular prisms (square tiling planes) Gyrated layers
     • Gyrated triangular prismatic honeycomb
  2. [13/3] (I) 3 octahedra, 4 tetrahedra, 6 triangular prisms (triangular tiling planes)
     • Elongated cubic honeycomb
  3. [13/3] (II) 3 octahedra, 4 tetrahedra, 6 triangular prisms (triangular tiling planes) Gyrated layers
     • Gyroelongated cubic honeycomb
  4. [14/2] (I) 8 tetrahedra, 6 octahedra (triangular tiling planes)
     • Tetrahedral-octahedral honeycomb
     • Dual Rhombic dodecahedral honeycomb
  5. [14/2] (II) 8 tetrahedra, 6 octahedra (triangular tiling planes) Gyrated layers
     • Gyrated tetrahedral-octahedral honeycomb
     • Dual trapezo-rhombic dodecahedral honeycomb

The (I) and (II) forms have the same vertex polyhedra, but repeat differently. The (II) forms have a rotation symmetry group. Tesselation of space using truncated octahedra. ... Tesselation of space using truncated octahedra. ... The bitruncated cubic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra. ... The bitruncated cubic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra. ... The Disphenoid tetrahedral honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of identical nonregular tetrahedral cells. ... Image File history File links Download high resolution version (902x902, 1221 KB) Summary View of cubic honeycomb generated by software: [1] Curved Spaces v1. ... Image File history File links Download high resolution version (902x902, 1221 KB) Summary View of cubic honeycomb generated by software: [1] Curved Spaces v1. ... The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. ... In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. ... In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. ... In geometry, the Great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. ... In geometry, the Small rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. ... In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. ... In geometry, the prismatic trisquare tiling is a semiregular tiling of the Euclidean plane. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. ... In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ... In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. ... The Bitruncated alternated cubic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of tetrahedra, and truncated tetrahedra. ... Image File history File links Tetrahedral-octahedral_honeycomb. ... Image File history File links Tetrahedral-octahedral_honeycomb. ... In geometry, the Square tiling is a regular tiling of the Euclidean plane. ... The Gyrated triangular prismatic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of triangular prisms. ... In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ... The Elongated cubic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of tetrahedra, octahedra, and triangular prisms. ... In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ... The Gyroelongated cubic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of tetrahedra, octahedra, and triangular prisms. ... In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ... The tetrahedral-octahedral honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of alternating tetrahedra and octahedra. ... The rhombic dodecahedra honeycomb is a tessellation (or honeycomb) in Euclidean 3-space. ... In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ... The Gyrated tetrahedral-octahedral honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of tetrahedra and octahedra. ... The trapezo-rhombic dodecahedral honeycomb is a tessellation (or honeycomb) in Euclidean 3-space. ...

All 28 Andreini tessellations are found in crystal arrangements. Quartz crystal A crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. ...

The Tetrahedral-octahedral honeycomb is of special importance since its vertices form a cubic close-packing of spheres. The space-filling trusses 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) 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 (March 3, 1847 – August 2, 1922) was a Scottish scientist and inventor. ... In the U.S. postage stamp commemorating Buckminster Fuller and his contributions to architecture and science, some of his inventions are visible. ... 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. ...

[1] [2] [3] [4]. Octet trusses are now one of the most common type of truss used in construction.

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

References

• Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
• A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative, Mem. Societa Italiana della Scienze, Ser.3, 14 (1905) 75–129.
  • Italian-to-English: "On the regular and semiregular nets of polyhedra and on the correspondents correlative nets, Mem. Italian Societa of Sciences"