Andreini tessellation

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Image:Cubic honeycomb.png Image:Bitruncated cubic tiling.png Image:Truncated octahedra.jpg Image:Cantellated cubic tiling.png Image:Cantitruncated cubic tiling.png Image:Runcitruncated cubic tiling.png Image:Omnitruncated cubic tiling.png Image:Cantitruncated alternated cubic tiling.png Image:Runcinated alternated cubic tiling.png Image:Truncated alternated cubic tiling.png Image:Tetrahedral-octahedral honeycomb.png

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)

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.

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.

Note: See Uniform polychoron#Geometric derivations for some terminology for some of these names below.

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.

• 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

• Identical faced-tiling planes (6 to 12 cells per vertex)
  1. [6/1] 6 hexagonal prisms (hexagonal tiling planes),
     • Hexagonal 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

• 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

• 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 alternated cubic honeycomb
  3. [13/3] (II) 3 octahedra, 4 tetrahedra, 6 triangular prisms (triangular tiling planes) Gyrated layers
     • Gyroelongated alternated 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.

All 28 Andreini tessellations are found in crystal arrangements.

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)

[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 (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.