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 ...

and

crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...

, a stereohedron 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 ...

that fills space isohedrally, meaning that the

symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

of the tiling take any copy of the stereohedron to any other copy. Two-dimensional analogues to the stereohedra are called planigons. Higher dimensional

polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...

s can also be stereohedra, while they would more accurately be called stereotopes.

Plesiohedra

A subset of stereohedra are called

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 ...

s, defined as the Voronoi cells 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 ...

Parallelohedron In geometry, a parallelohedron is a polyhedron that can be translated without rotations in 3-dimensional Euclidean space to fill space with a honeycomb in which all copies of the polyhedron meet face-to-face. There are five types of parallelohedr ...

s are plesiohedra which are space-filling by translation only. Edges here are colored as parallel vectors.

Other periodic stereohedra

The

catoptric tessellation Catoptrics (from grc-gre, κατοπτρικός ''katoptrikós'', "specular", from grc-gre, κάτοπτρον ''katoptron'' "mirror") deals with the phenomena of reflection (physics), reflected light and image, image-forming Optics, optical s ...

contain stereohedra cells.

Dihedral angles A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...

are integer divisors of 180°, and are colored by their order. The first three are the fundamental domains of

_3

_3

, and

_3

symmetry, represented by Coxeter-Dynkin diagrams: , and .

_3

is a half symmetry of

_3

, and

_3

is a quarter symmetry. Any space-filling stereohedra with symmetry elements can be

dissected Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause ...

into smaller identical cells which are also stereohedra. The name modifiers below, half, quarter, and eighth represent such dissections. Other convex polyhedra that are stereohedra but not parallelohedra nor plesiohedra include 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 ...

References

* {{SpringerEOM, title=Stereohedron, first=A. B., last=Ivanov, id=Stereohedron&oldid=31579 * B. N. Delone, N. N. Sandakova, ''Theory of stereohedra'' Trudy Mat. Inst. Steklov., 64 (1961) pp. 28–51 (Russian) * Goldberg, Michael ''Three Infinite Families of Tetrahedral Space-Fillers'' Journal of Combinatorial Theory A, 16, pp. 348–354, 1974. * Goldberg, Michael ''The space-filling pentahedra'', Journal of Combinatorial Theory, Series A Volume 13, Issue 3, November 1972, Pages 437-44
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* Goldberg, Michael ''The Space-filling Pentahedra II'', Journal of Combinatorial Theory 17 (1974), 375–378

* Goldberg, Michael ''On the space-filling hexahedra'' Geom. Dedicata, June 1977, Volume 6, Issue 1, pp 99–10
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* Goldberg, Michael ''On the space-filling heptahedra'' Geometriae Dedicata, June 1978, Volume 7, Issue 2, pp 175–18
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* Goldberg, Michael ''Convex Polyhedral Space-Fillers of More than Twelve Faces.'' Geom. Dedicata 8, 491-500, 1979. * Goldberg, Michael ''On the space-filling octahedra'', Geometriae Dedicata, January 1981, Volume 10, Issue 1, pp 323–33
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* Goldberg, Michael ''On the Space-filling Decahedra''. Structural Topology, 1982, num. Type 10-I
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* Goldberg, Michael ''On the space-filling enneahedra'' Geometriae Dedicata, June 1982, Volume 12, Issue 3, pp 297–30
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Space-filling polyhedra