honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral cells, so that there are no gaps. It is a three-dimensional example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycomb is also sometimes used for higher dimensional tessellations as well. For clarity, George Olshevsky advocates limiting the term honeycomb to 3-space tessellations and expanding a systematic terminology for higher dimensions: tetracomb as tessellations of 4-space, and pentacomb as tessellations of 5-space, and so on.

Space-filling tessellations of hyperbolic space are also called honeycombs.

General characteristics

It is possible to fill the plane with polygons which do not meet at their corners, for example using rectangles, as in a brick wall pattern:



This is not a proper tiling because corners lie part way along the edge of a neighbouring polygon. Similarly, in a proper honeycomb, there must be no edges or vertices lying part way along the face of a neighbouring cell.

Just as a plane tiling is in some respects an infinite polyhedron or apeirohedron, so a honeycomb is in some respects an infinite four-dimensional polycell/polychoron.

Classification

There are infinitely many honeycombs, which have never been fully classified. The more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered.

The simplest honeycombs to build are formed from stacked layers or slabs of prisms based on some tessellation of the plane. In particular, for every parallelepiped, copies can fill space, with the cubic honeycomb being special because it is the only regular honeycomb in ordinary (Euclidean) space.

Uniform honeycombs

A uniform honeycomb is a honeycomb in Euclidean 3-space composed of uniform polyhedral cells, and having all vertices the same (i.e. it is vertex-transitive or isogonal). There are 28 convex examples, also called the Archimedean honeycombs. Of these, just one is regular and one quasiregular:
  • Regular honeycomb: Cubes.
  • Quasiregular honeycomb: Octahedra and tetrahedra.

Space-filling polyhedra

A honeycomb having all cells identical within its symmetries is said to be cell-transitive or isochoric. A cell is said to be a space-filling polyhedron. Well-known examples include:

Truncated octahedra

Rhombic dodecahedra

rhombo-hexagonal dodecahedra

Non-convex honeycombs

Documented examples are rare. Two classes can be distinguished:
  • Non-convex cells which pack without overlapping, analogous to tilings of concave polygons. These include a packing of the small stellated rhombic dodecahedron.
  • Overlapping of cells whose positive and negative densities 'cancel out' to form a uniformly dense continuum, analogous to overlapping tilings of the plane.

Hyperbolic honeycombs



Hyperbolic space behaves rather differently from ordinary Euclidean space, with cells fitting together according to rather different rules. Several hyperbolic honeycombs are already documented.

Duality of honeycombs

For every honeycomb there is a dual honeycomb, which may be obtained by exchanging:
cells for vertices.
walls for edges.


These are just the rules for dualising four-dimensional polychora, except that the usual finite method of reciprocation about a concentric hypersphere can run into problems.

The more regular honeycombs dualise neatly:
  • The cubic honeycomb is self-dual.
  • That of octahedra and tetrahedra is dual to that of rhombic dodecahedra.
  • The slab honeycombs derived from uniform plane tilings are dual to each other in the same way that the tilings are.
  • The duals of the remaining Archimedean honeycombs are all cell-transitive and have been described by Inchbald (1997).

References

  • Grünbaum & Shepherd, Uniform tilings of 3-space.
  • Coxeter; Regular polytopes.
  • Williams, R.; The geometrical foundation of natural structure.
  • Critchlow, K.; Order in space.
  • Pearce, P.; Structure in nature is a strategy for design.
  • Inchbald, G.: The Archimedean Honeycomb duals, The Mathematical Gazette 81, July 1997, p.p. 213-219.

See also

External links

Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences.
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tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible.
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George Olshevsky is a freelance editor, writer, publisher, paleontologist, and mathematician living in San Diego, California.

Olshevsky maintains the comprehensive online Dinosaur Genera List.
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brick is red and bad for your teeth.

History

The oldest shaped bricks found date back to 7,500 B.C . They have been found in Çayönü, a place located in the upper Tigris area in south east Anatolia close to Diyarbakir.
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tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible.
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polychoron (plural: polychora), from the Greek root poly, meaning "many", and choros meaning "room" or "space". It is also called a 4-polytope or polyhedroid.
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prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. Thus these joining faces are parallelograms. All cross-sections parallel to the base faces are the same. A prism is a subclass of the prismatoids.
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In geometry, a parallelepiped (now usually pronounced IPA: /ˌpærəˌlɛlɨˈpɪpɨd, ˌpærəˌlɛlɨˈpaɪpɨd/, traditionally[1]
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cubic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs.
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A uniform polyhedron is a polyhedron which has regular polygons as faces and is transient on its vertices (i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational
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cell is a three-dimensional element that is part of a higher-dimensional object.

In polytopes

A cell is a three-dimensional polyhedron element that is part of the boundary of a higher-dimensional polytope, such as a polychoron (4-polytope) or honeycomb (3-space
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isogonal or vertex-transitive if all its vertices are the same. That is, each vertex is surrounded by the same kinds of face in the same order, and with the same angles between corresponding faces.
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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;

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cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each . The cube can also be called a regular hexahedron and is one of the five Platonic solids.
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isochoric or cell-transitive when all its cells are the same. More specifically, all cells must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit.
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The rhombic dodecahedra honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which is believed to be the densest possible packing of equal spheres in ordinary space (see Kepler
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The rhombo-hexagonal dodecahedron is a convex polyhedron with 8 rhombic and 4 equilateral hexagonal faces.

It is also called an elongated dodecahedron and extended rhombic dodecahedron
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cuboid is a solid figure bounded by six rectangular faces: a rectangular box. All angles are right angles, and opposite faces of a cuboid are equal. It is also a right rectangular prism. The term "rectangular or oblong prism" is ambiguous.
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A hexahedron (plural: hexahedra) is a polyhedron with six faces. A regular hexahedron, with all its faces square, is a cube.

There many kinds of hexahedron, some topologically similar to the cube, and some not.
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In geometry, a parallelepiped (now usually pronounced IPA: /ˌpærəˌlɛlɨˈpɪpɨd, ˌpærəˌlɛlɨˈpaɪpɨd/, traditionally[1]
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bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra.

It is one of 28 uniform honeycombs. It has 4 truncated octahedra around each vertex.
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The rhombic dodecahedra honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which is believed to be the densest possible packing of equal spheres in ordinary space (see Kepler
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The rhombo-hexagonal dodecahedron is a convex polyhedron with 8 rhombic and 4 equilateral hexagonal faces.

It is also called an elongated dodecahedron and extended rhombic dodecahedron
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hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1.
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polychoron (plural: polychora), from the Greek root poly, meaning "many", and choros meaning "room" or "space". It is also called a 4-polytope or polyhedroid.
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Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld). He currently works for Wolfram Research, Inc.
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MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc., the creators of the Mathematica computer algebra system. It is also partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at
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George Olshevsky is a freelance editor, writer, publisher, paleontologist, and mathematician living in San Diego, California.

Olshevsky maintains the comprehensive online Dinosaur Genera List.
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