In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a plesiohedron is a special kind of
space-filling polyhedron
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 const ...
, defined as the
Voronoi cell
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed t ...
of a symmetric
Delone set
In the mathematical theory of metric spaces, ε-nets, ε-packings, ε-coverings, uniformly discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points, a ...
.
Three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
can be completely filled by copies of any one of these shapes, with no overlaps. The resulting
honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic Beeswax, wax cells built by honey bees in their beehive, nests to contain their larvae and stores of honey and pollen.
beekeeping, Beekee ...
will have symmetries that take any copy of the plesiohedron to any other copy.
The plesiohedra include such well-known shapes as the
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
,
hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices..
Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used ...
,
rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
Properties
The rhombic dodecahedro ...
, and
truncated octahedron
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
.
The largest number of faces that a plesiohedron can have is 38.
Definition
A set
of points in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
is a
Delone set
In the mathematical theory of metric spaces, ε-nets, ε-packings, ε-coverings, uniformly discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points, a ...
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
Rigid or rigidity may refer to:
Mathematics and physics
*Stiffness, the property of a solid body to resist deformation, which is sometimes referred to as rigidity
*Structural rigidity, a mathematical theory of the stiffness of ensembles of rig ...
of space that takes
to
and
to
. That is, the symmetries of
act transitively on
.
The
Voronoi diagram
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed ...
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
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
. 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
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
to each other, for the symmetries of
must also be symmetries of the Voronoi diagram. In this case, the Voronoi diagram forms a
honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic Beeswax, wax cells built by honey bees in their beehive, nests to contain their larvae and stores of honey and pollen.
beekeeping, Beekee ...
in which there is only a single
prototile
In the mathematical theory of tessellations, a prototile is one of the shapes of a tile in a tessellation.
Definition
A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint in ...
shape, the shape of these Voronoi cells. This shape is called a plesiohedron. The tiling generated in this way is
isohedral
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
, 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.
[.]
As with any space-filling polyhedron, the
Dehn invariant
In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled ("dissection problem, dissected") into another, and whether a polyhedron or its dissections can Honeycomb (geometry), tile s ...
of a plesiohedron is necessarily zero.
Examples
The plesiohedra include the five
parallelohedra. These are polyhedra that can tile space in such a way that every tile is symmetric to every other tile by a translational symmetry, without rotation. 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.
For this reason (and because Voronoi diagrams are also known as Dirichlet tesselations) they have also been called "Dirichlet stereohedra"
There are only finitely many combinatorial types of plesiohedron. Notable individual plesiohedra include:
*The five parallelohedra: the
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
(or more generally the
parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ...
),
hexagonal prism
In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices..
Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used ...
,
rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
Properties
The rhombic dodecahedro ...
,
elongated dodecahedron
In geometry, the elongated dodecahedron, extended rhombic dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedron is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular de ...
, and
truncated octahedron
In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
.
*The
triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A unif ...
, the prototile of the
triangular prismatic honeycomb
The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 3-space. It is composed entirely of triangular prisms.
It is constructed from a triangular ti ...
. More generally, each of the 11 types of
Laves tiling
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.
There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their dua ...
of the plane by congruent convex polygons (and each of the subtypes of these tilings with different symmetry groups) can be realized as the Voronoi cells of a symmetric Delone set in the plane. It follows that the prisms over each of these shapes are plesiohedra. As well as the triangular prisms, these include prisms over certain quadrilaterals, pentagons, and hexagons.
*The
gyrobifastigium
In geometry, the gyrobifastigium is the 26th Johnson solid (). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile ...
is a stereohedron but not a plesiohedron, because the points at the centers of the cells of its face-to-face tiling (where they are forced to go by symmetry) have differently-shaped Voronoi cells. However, a flattened version of the gyrobifastigium, with faces made of
isosceles right triangle
A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45° ...
s and
silver rectangle
Silver is a chemical element with the symbol Ag (from the Latin ', derived from the Proto-Indo-European ''h₂erǵ'': "shiny" or "white") and atomic number 47. A soft, white, lustrous transition metal, it exhibits the highest electrical cond ...
s, is a plesiohedron.
*The
triakis truncated tetrahedron, the prototile of the
triakis truncated tetrahedral honeycomb
The triakis truncated tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of triakis truncated tetrahedra. It was discovered in 1914.
Voronoi tessellation
It is the Voronoi tessellation of the carbo ...
and the plesiohedron generated by the
diamond lattice
The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was diamond, other elements in group 14 also adopt this structure, including α-tin, the sem ...
*The
trapezo-rhombic dodecahedron
In geometry, the trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is a convex dodecahedron with 6 rhombic and 6 trapezoidal faces. It has symmetry. A concave form can be constructed with an identical net, seen as excavating trig ...
, the prototile of the
trapezo-rhombic dodecahedral honeycomb
The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equal ...
and the plesiohedron generated by the
hexagonal close-packing
*The 17-sided Voronoi cells of the
Laves graph
In geometry and crystallography, the Laves graph is an infinite and highly symmetric system of points and line segments in three-dimensional Euclidean space, forming a periodic graph. Three equal-length segments meet at 120° angles at each po ...
Many other plesiohedra are known. Two different ones with the largest known number of faces, 38, were discovered by crystallographer Peter Engel.
For many years the maximum number of faces of a plesiohedron was an
open problem In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is know ...
,
[.]
but analysis of the possible symmetries of three-dimensional space has shown that this number is at most 38.
[
The Voronoi cells of points uniformly spaced on a ]helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, ...
fill space, are all congruent to each other, and can be made to have arbitrarily large numbers of faces.[.] However, the points on a helix are not a Delone set and their Voronoi cells are not bounded polyhedra.
A modern survey is given by Schmitt.[.]
References
External links
*{{MathWorld, title=Space-Filling Polyhedron, id=Space-FillingPolyhedron, mode=cs2
Space-filling polyhedra