Honeycomb (geometry)

Space filling and tessellation of the three-dimensional space based on the filling (typically ) three -dimensional Euclidean spaces with structures having more than the same dimension as the space. For the two-dimensional case, the tiling, tiling see. Space fillings can trivially be complete, that is, the entire volume is occupied (as in a completely filled glass ), or partially, leading to the interesting problem of spatially close packing of spheres. In many practical applications it is of interest to optimize the density of the filling, for example in the packaging industry. Space fillings mathematically abstracted can be found in the domains of space-filling curves, where fractal structures with a fractional dimension n smaller the space dimension and used greater than n - 1 for the filling. Natural quasi- fractals of this kind are often referred to as supply networks in biological organisms (blood vessel system, tracheal ).

Space filling polyhedra

A complete space filling by polyhedra is also called tilings of three-dimensional space. If you try to fill the space with polyhedra of a kind, there are among the convex polyhedra, which are bounded by regular polygons, just five that fill the room alone: cubes, triangular and hexagonal prism, truncated octahedron and the twisted double wedge (Johnson - body J26, also Gyrobifastigium ). Among the so-called the Catalan bodies is only the rhombic dodecahedron space filling.

Hexagonal prism

Triangular prism

Twisted double wedge

Truncated octahedron

Rhombic dodecahedron

Crystallographic restriction

For periodic tilings occurs an interesting phenomenon: Their symmetry groups can only rotate 360 ​​°, 180 °, 120 °, 90 ° and / or 60 ° included ( that is, elements of orders 1, 2, 3, 4 and 6 ), but no rotations about other angles (ie no elements of orders 5, 7 or higher). This situation, by the way, also applies to "real" crystals are referred to as " crystallographic restriction ". However, the order of 5 is possible for quasicrystals that have an "almost" periodic graduation.

Types of 3D tiling

The following examples show how the three-dimensional space can be completely filled with regular or semiregular polyhedra same edge length. Indicated are the number of the polyhedron, which is necessary to form a full solid angle of 4π.

  • 8 tetrahedron 6 octahedra (Figure)
  • 8 cubes ( picture)
  • 6 tetrahedron stumps 2 tetrahedra (Figure)
  • 6 rhombic dodecahedron (Figure)
  • 4 cuboctahedron octahedron 2 ( image )
  • 4 Hexaederstümpfe 1 octahedron ( see right )
  • 4 truncated octahedrons (Figure)
  • 4 rhombic dodecahedron (Figure)
  • Rhombicuboctahedron 3 1 1 cube tetrahedron (Figure)
  • Rhombicuboctahedron 2 2 1 cube cuboctahedron (Figure)
  • Two truncated octahedrons 2 tetrahedra stumps 1 cuboctahedron (Figure)
  • Kuboktaederstümpfe 2 1 1 cube truncated octahedron (Figure)
Sphere packing
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