Plesiohedron

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. Although there are only a finite number of combinatorially distinct types of plesiohedron, the complete list is not known. The largest number of faces that a plesiohedron can have is also unknown, but this maximum number of faces is at least 38 and at most 92.

A set ${\displaystyle S}$ of points in Euclidean space is a Delone set if there exists a number ${\displaystyle \varepsilon >0}$ such that every two points of ${\displaystyle S}$ are at least at distance ${\displaystyle \varepsilon }$ apart from each other and such that every point of space is within distance ${\displaystyle 1/\varepsilon }$ of at least one point in ${\displaystyle S}$. So ${\displaystyle S}$ fills space, but its points never come too close to each other. For this to be true, ${\displaystyle S}$ must be infinite. Additionally, the set ${\displaystyle S}$ is symmetric (in the sense needed to define a plesiohedron) if, for every two points ${\displaystyle p}$ and ${\displaystyle q}$ of ${\displaystyle S}$, there exists a rigid motion of space that takes ${\displaystyle S}$ to ${\displaystyle S}$ and ${\displaystyle p}$ to ${\displaystyle q}$. That is, the symmetries of ${\displaystyle S}$ act transitively on ${\displaystyle S}$.

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