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 plesiohedra include such well-known shapes as the cube, hexagonal prism, rhombic dodecahedron, and truncated octahedron.
of points in Euclidean space is a Delone set if there exists a number
fills space, but its points never come too close to each other.
is symmetric (in the sense needed to define a plesiohedron) if, for every two points
, there exists a rigid motion of space that takes
is a Delone set, the Voronoi cell of each point
The faces of this polyhedron lie on the planes that perpendicularly bisect the line segments from
is symmetric as well as being Delone, the Voronoi cells must all be congruent to each other, for the symmetries of
In this case, the Voronoi diagram forms a honeycomb in which there is only a single prototile shape, the shape of these Voronoi cells.
[1] As with any space-filling polyhedron, the Dehn invariant of a plesiohedron is necessarily zero.
Equivalently, they are the Voronoi cells of lattices, as these are the translationally-symmetric Delone sets.
Plesiohedra are a special case of the stereohedra, the prototiles of isohedral tilings more generally.
[1] For this reason (and because Voronoi diagrams are also known as Dirichlet tesselations) they have also been called "Dirichlet stereohedra"[4] There are only finitely many combinatorial types of plesiohedron.
Two different ones with the largest known number of faces, 38, were discovered by crystallographer Peter Engel.
[1][9] For many years the maximum number of faces of a plesiohedron was an open problem,[10][4] but analysis of the possible symmetries of three-dimensional space has shown that this number is at most 38.
[11] The Voronoi cells of points uniformly spaced on a helix fill space, are all congruent to each other, and can be made to have arbitrarily large numbers of faces.
[12] However, the points on a helix are not a Delone set and their Voronoi cells are not bounded polyhedra.