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Concept
Plesiohedron
Summary
In geometry, a plesiohedron is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set.
Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps. The resulting honeycomb will have symmetries that take any copy of the plesiohedron to any other copy.
The plesiohedra include such well-known shapes as the cube, hexagonal prism, rhombic dodecahedron, and truncated octahedron.
The largest number of faces that a plesiohedron can have is 38.
A set of points in Euclidean space is a Delone set if there exists a number such that every two points of are at least at distance apart from each other and such that every point of space is within distance of at least one point in . So fills space, but its points never come too close to each other. For this to be true, must be infinite.
Additionally, the set is symmetric (in the sense needed to define a plesiohedron) if, for every two points and of , there exists a rigid motion of space that takes to and to . That is, the symmetries of act transitively on .
The Voronoi diagram of any set of points partitions space into regions called Voronoi cells that are nearer to one given point of than to any other. When is a Delone set, the Voronoi cell of each point in is a convex polyhedron. The faces of this polyhedron lie on the planes that perpendicularly bisect the line segments from to other nearby points of .
When is symmetric as well as being Delone, the Voronoi cells must all be congruent to each other, for the symmetries of must also be symmetries of the Voronoi diagram. In this case, the Voronoi diagram forms a honeycomb in which there is only a single prototile shape, the shape of these Voronoi cells. This shape is called a plesiohedron. The tiling generated in this way is isohedral, meaning that it not only has a single prototile ("monohedral") but also that any copy of this tile can be taken to any other copy by a symmetry of the tiling.
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