Space-filling polyhedron (original) (raw)
In geometry, a space-filling polyhedron is a polyhedron that can be used to fill all of three-dimensional space via translations, rotations and/or reflections, where filling means that, taken together, all the instances of the polyhedron constitute a partition of three-space. Any periodic tiling or honeycomb of three-space can in fact be generated by translating a primitive cell polyhedron. (It is natural to wonder whether bubbles in a foam are made of regular polyhedra. The answer is No.)
| Property | Value |
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| dbo:abstract | In geometry, a space-filling polyhedron is a polyhedron that can be used to fill all of three-dimensional space via translations, rotations and/or reflections, where filling means that, taken together, all the instances of the polyhedron constitute a partition of three-space. Any periodic tiling or honeycomb of three-space can in fact be generated by translating a primitive cell polyhedron. Any parallelepiped tessellates Eucledian 3-space, and more specifically any of five parallelohedra such as the rhombic dodecahedron, which is one of nine edge-transitive and face-transitive solids. Examples of other space-filling polyhedra include the set of five convex polyhedra with regular faces, which include the triangular prism, hexagonal prism, gyrobifastigium, cube, and truncated octahedron; a set that intersects with that of the five parallelohedra. (It is natural to wonder whether bubbles in a foam are made of regular polyhedra. The answer is No.) (en) |
| dbo:wikiPageExternalLink | http://mathworld.wolfram.com/Space-FillingPolyhedron.html https://archive.org/details/spacestructures00loeb https://archive.org/details/spacestructures00loeb/page/n140 |
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| rdfs:comment | In geometry, a space-filling polyhedron is a polyhedron that can be used to fill all of three-dimensional space via translations, rotations and/or reflections, where filling means that, taken together, all the instances of the polyhedron constitute a partition of three-space. Any periodic tiling or honeycomb of three-space can in fact be generated by translating a primitive cell polyhedron. (It is natural to wonder whether bubbles in a foam are made of regular polyhedra. The answer is No.) (en) |
| rdfs:label | Space-filling polyhedron (en) |
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