80:
786:
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,
794:
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
841:
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
853:
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.
1119:. But as he writes (p. 429), Voronoi's conjecture for dimensions at most four was already proven by Delaunay. For the classification of three-dimensional parallelohedra into these five types, see
732:. 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
740:, 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"
89:
166:
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shape, the shape of these
Voronoi cells. This shape is called a plesiohedron. The tiling generated in this way is
846:, but analysis of the possible symmetries of three-dimensional space has shown that this number is at most 38.
829:
779:
807:
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There are only finitely many combinatorial types of plesiohedron. Notable individual plesiohedra include:
1039:(2011), "On the number of facets of three-dimensional Dirichlet stereohedra IV: quarter cubic groups",
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145:
967:
614:. The faces of this polyhedron lie on the planes that perpendicularly bisect the line segments from
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of points partitions space into regions called
Voronoi cells that are nearer to one given point of
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191:
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are combinatorially equivalent to
Voronoi tilings, and Erdahl proves this in the special case of
887:
736:, as these are the translationally-symmetric Delone sets. Plesiohedra are a special case of the
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43:
can be completely filled by copies of any one of these shapes, with no overlaps. The resulting
1111:. Voronoi conjectured that all tilings of higher dimensional spaces by translates of a single
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1134:
1036:
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1326:(2003), "Arbitrarily large neighborly families of congruent symmetric convex 3-polytopes",
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44:
1163:; Dolbilin, N. P.; Ĺ togrin, M. I. (1978), "Combinatorial and metric theory of planigons",
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must also be symmetries of the
Voronoi diagram. In this case, the Voronoi diagram forms a
8:
1330:, Monogr. Textbooks Pure Appl. Math., vol. 253, Dekker, New York, pp. 267–278,
932:(September 1991), "Voronoi diagrams—a survey of a fundamental geometric data structure",
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Erdahl, R. M. (1999), "Zonotopes, dicings, and
Voronoi's conjecture on parallelohedra",
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fills space, but its points never come too close to each other. For this to be true,
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Zeitschrift fĂĽr
Kristallographie, Kristallgeometrie, Kristallphysik, Kristallchemie
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will have symmetries that take any copy of the plesiohedron to any other copy.
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apart from each other and such that every point of space is within distance
1141:, University of California Press, Berkeley, Calif.-London, pp. 48–50,
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951:. See especially section 1.2.1, "Regularly Placed Sites", pp. 354–355.
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these include prisms over certain quadrilaterals, pentagons, and hexagons.
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is symmetric as well as being Delone, the Voronoi cells must all be
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66:. The largest number of faces that a plesiohedron can have is 38.
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1265:(1985), "69.14 Space Filling with Identical Symmetrical Solids",
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The plesiohedra include such well-known shapes as the
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Trudy Matematicheskogo Instituta Imeni V. A. Steklova
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893:Bulletin of the American Mathematical Society
1200:Notices of the American Mathematical Society
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782:. More generally, each of the 11 types of
716:As with any space-filling polyhedron, the
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964:; Moews, D. (1995), "Polytopes that fill
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890:(1980), "Tilings with congruent tiles",
296:must be infinite. Additionally, the set
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812:triakis truncated tetrahedral honeycomb
720:of a plesiohedron is necessarily zero.
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828:and the plesiohedron generated by the
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814:and the plesiohedron generated by the
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857:A modern survey is given by Schmitt.
681:to each other, for the symmetries of
1132:
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996:Discrete and Computational Geometry
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13:
1190:Schoen, Alan H. (June–July 2008),
1041:Beiträge zur Algebra und Geometrie
834:The 17-sided Voronoi cells of the
14:
1411:
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1086:European Journal of Combinatorics
728:The plesiohedra include the five
161:{\displaystyle \varepsilon >0}
986:{\displaystyle \mathbb {R} ^{n}}
705:in which there is only a single
97:A 17-sided plesiohedron and its
87:
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16:Type of space-filling polyhedron
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907:10.1090/S0273-0979-1980-14827-2
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1121:GrĂĽnbaum & Shephard (1980)
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780:triangular prismatic honeycomb
229:{\displaystyle 1/\varepsilon }
168:such that every two points of
1:
1241:10.1524/zkri.1981.154.3-4.199
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808:triakis truncated tetrahedron
747:The five parallelohedra: the
460:. That is, the symmetries of
69:
1139:Polyhedra: a visual approach
822:trapezo-rhombic dodecahedron
201:{\displaystyle \varepsilon }
7:
723:
10:
1416:
1377:"Space-Filling Polyhedron"
993:and scissors congruence",
634:to other nearby points of
1135:"Close-packing polyhedra"
1063:10.1007/s13366-011-0010-5
797:isosceles right triangles
236:of at least one point in
188:are at least at distance
142:if there exists a number
1305:Schmitt, Moritz (2016),
1268:The Mathematical Gazette
550:than to any other. When
29:space-filling polyhedron
1400:Space-filling polyhedra
1192:"On the graph (10,3)-a"
830:hexagonal close-packing
824:, the prototile of the
810:, the prototile of the
778:, the prototile of the
751:(or more generally the
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60:rhombic dodecahedron
39:. Three-dimensional
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1233:1981ZK....154..199E
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1035:Sabariego, Pilar;
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801:silver rectangles
694:{\displaystyle S}
670:{\displaystyle S}
647:{\displaystyle S}
627:{\displaystyle p}
612:convex polyhedron
603:{\displaystyle S}
583:{\displaystyle p}
563:{\displaystyle S}
543:{\displaystyle S}
523:{\displaystyle S}
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473:{\displaystyle S}
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433:{\displaystyle p}
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376:, there exists a
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33:Voronoi cell
25:plesiohedron
24:
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836:Laves graph
738:stereohedra
510:of any set
107:Laves graph
1324:Kim, Scott
861:References
140:Delone set
70:Definition
37:Delone set
1382:MathWorld
1117:zonotopes
1054:0708.2114
711:isohedral
707:prototile
703:honeycomb
679:congruent
224:ε
196:ε
150:ε
99:honeycomb
45:honeycomb
1394:Category
1207:(6): 663
734:lattices
724:Examples
21:geometry
1354:2034721
1342:Bibcode
1289:3616930
1249:0598811
1229:Bibcode
1177:0558946
1147:0451161
1108:1703597
1071:2842627
1019:1318797
916:0585178
105:of the
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767:, and
114:A set
101:, the
62:, and
1332:arXiv
1285:JSTOR
1195:(PDF)
1049:arXiv
851:helix
657:When
610:is a
256:. So
138:is a
820:The
806:The
799:and
790:The
774:The
749:cube
506:The
420:and
336:and
153:>
52:cube
23:, a
1277:doi
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1225:154
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1094:doi
1059:doi
1005:doi
944:doi
902:doi
755:),
590:in
483:on
440:to
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356:of
19:In
1396::
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979:n
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898:3
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689:S
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622:p
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220:/
216:1
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156:0
122:S
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