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In geometry, the Andreini tessellations are the complete set of 28 uniform patterns that fill three-dimensional Euclidean space. They are the three-dimensional analogue to the uniform tilings of the plane. They are named in honor of A. Andreini who studied and enumerated these patterns around 1905 (see References below).
A uniform honeycomb is constructed by identical sets of convex uniform polyhedral cells around each vertex. The cells represented in Andreini honeycombs include three of the five Platonic solids, six of the thirteen Archimedean solids, and five of the infinite family of prisms.
Note: Some of the terms used below are defined in Uniform polychoron#Geometric derivations.
Cubic forms
The cubic honeycomb offers seven derived uniform honeycombs via truncation operations.
| name | cell types (# at each vertex) | # families of continuous face planes | Solids (Partial) | Frames (Perspective) | vertex figure |
|---|---|---|
| cubic | cube (8) | 3 square |
| truncated cubic | truncated cube (4) octahedron (1) | 0 |
| rectified cubic | cuboctahedron (4) octahedron (2) | 0 |
| cantellated cubic | rhombicuboctahedron (2) cube (2) cuboctahedron (1) | 0 |
| bitruncated cubic | truncated octahedron (4) | 0 |
| runcitruncated cubic | truncated cube (1) octagonal prism (2) rhombicuboctahedron (1) cube (1) | 0 |
| cantitruncated cubic | truncated cuboctahedron (2) truncated octahedron (1) cube (1) | 0 |
| omnitruncated cubic | truncated cuboctahedron (2) octagonal prism (2) | 0 |
Alternated cubic forms
The tet-oct (alternated cubic) honeycomb offers four derived uniform honeycombs via truncation operations.
| name | cell types (# at each vertex) | # families of continuous face planes | Solids (Partial) | Frames (Perspective) | vertex figure |
|---|---|---|
| alternated cubic | tetrahedron (8) octahedron (6) | 4 triangle |
| truncated alternated cubic | truncated octahedron (2) truncated tetrahedron (2) cuboctahedron (1) | 0 |
| runcinated alternated cubic | rhombicuboctahedron (3) cube (1) tetrahedron (1) | 0 |
| cantitruncated alternated cubic | truncated cuboctahedron (2) truncated cube (1) truncated tetrahedron (1) | 0 |
| bitruncated alternated cubic | truncated tetrahedron (6) tetrahedron (2) | 4 tri-hex |
Gyrated and elongated forms
There are 5 uniform honeycombs generated by gyrating (rotating 90 degrees) or elongating (stacking alternatingly) of prismatic slab layers.
| name | cell types (# at each vertex) | # families of continuous face planes | Solids (Partial) | Frames (Perspective) | vertex figure |
|---|---|---|---|
| gyrated triangular prismatic | triangular prism (12) | 1 square | |
| gyroelongated triangular prismatic | triangular prism (6) cube (4) | 1 square | |
| gyrated alternated cubic | tetrahedron (8) octahedron (6) | 1 triangle | |
| elongated alternated cubic | triangular prism (6) tetrahedron (4) octahedron (3) | 1 triangle | |
| gyroelongated alternated cubic | triangular prism (6) tetrahedron (4) octahedron (3) | 1 triangle |
Stacked prismatic forms
Omitted from the above tables are ten of the eleven prismatic tilings obtained by stacking the eleven uniform plane tilings in parallel layers. The eleventh is the cubic tiling. The vertex figure of each is an irregular bipyramid whose faces are isosceles triangles.
The gyroelongated triangular prismatic tiling has the same vertex figure as one of the plain prismatic tilings, but their orientations in space differ; likewise the elongated and gyroelongated alternated cubic tilings.
Examples
All 28 Andreini 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 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). * * * *. Octet trusses are now among the most common types 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.
"Andreini tessellation"