| 释义 |
- Topological types
- Symmetries of 5 types
- Examples
- Parallelotope
- See also
- References
In geometry a parallelohedron is a polyhedron that can tessellate 3-dimensional spaces with face-to-face contacts via translations. This requires all opposite faces be congruent. Parallelohedra can only have parallelogonal faces, either parallelograms or hexagons with parallel opposite edges. There are 5 types, first identified by Evgraf Fedorov in his studies of crystallographic systems. Topological typesThe vertices of parallelohedra can be computed by linear combinations of 3 to 6 vectors. Each vector can have any length greater than zero, with zero length becoming degenerate, or becoming a smaller parallelohedra. The greatest parallelohedron is a truncated octahedron which is also called a 4-permutahedron and can be represented with in a 4D in a hyperplane coordinates as all permutations of the counting numbers (1,2,3,4). A belt mn means n directional vectors, each containing m coparallel congruent edges. Every type has order 2 Ci central inversion symmetry in general, but each has higher symmetry geometries as well. Parallelohedra with edges colored by direction| Name | Cube
(parallelepiped) | Hexagonal prism
Elongated cube | Rhombic dodecahedron | Elongated dodecahedron | Truncated octahedron |
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| Images |
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Edge
types | 3 edge-lengths | 3+1 edge-lengths | 4 edge-lengths | 4+1 edge-lengths | 6 edge-lengths |
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| Belts | 43 | 43, 61 | 64 | 64, 41 | 66 |
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Symmetries of 5 types There are 5 types of parallelohedra, although each has forms of varied symmetry. | # | Polyhedron | Symmetry
(order) | Image | Vertices | Edges | Faces | Belts | | 1 | Rhombohedron | Ci (2) | 8 | 12 | 6 | 43 |
|---|
| Trigonal trapezohedron | D3d (12) | | Parallelepiped | Ci (2) | | Rectangular cuboid | D2h (8) | | Cube | Oh (24) | | 2 | Hexagonal prism | Ci (2) | 12 | 18 | 8 | 61, 43 |
|---|
| D6h (24) | | 3 | Rhombic dodecahedron | D4h (16) | 14 | 24 | 12 | 64 |
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| D2h (8) | | Oh (24) | | 4 | Elongated dodecahedron | D4h (16) | 18 | 28 | 12 | 64, 41 |
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| D2h (8) | | | 5 | Truncated octahedron | Oh (24) | 24 | 36 | 14 | 66 |
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Examples High symmetric examples| Pm3m (221) | Im3m (229) | Fm3m (225) | |
{{CDD>node_1|4|node|3|node|4|node
{{CDD|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node | {{CDD>node_1|6|node|3|node|2|node_1|infin|node | {{CDD>node_f1|3|node|split1-43|nodes | Elongated dodecahedral | {{CDD>node|4|node_1|3|node_1|4|node |
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| General symmetry examples |
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Parallelotope In higher dimensions a parallelohedron is called a parallelotope. There are 52 variations for 4-dimensional parallelotopes.[1][2] See also- Parallelogon – analogous space-filling polygons in 2D, with parallelograms and hexagons
- Plesiohedron – a broader class of isohedral space-filling polyhedra
References1. ^Crystal Symmetries: Shubnikov Centennial Papers, edited by B. K. Vainshtein, I. Hargittai [https://books.google.com/books?id=UdnSBQAAQBAJ&lpg=PA435&dq=52%20FOUR-DIMENSIONAL%20PARALLELOTOPES&pg=PA435#v=onepage&q=52%20FOUR-DIMENSIONAL%20PARALLELOTOPES&f=false]
2. ^Once more about the 52 four-dimensional parallelotopes, Michel Deza, Viacheslav Grishukhin (2003) [https://arxiv.org/abs/math/0307171]
- The facts on file: Geometry handbook, Catherine A. Gorini, 2003, {{isbn|0-8160-4875-4}}, p. 117
- Coxeter, H. S. M. Regular polytopes (book), 3rd ed. New York: Dover, pp. 29–30, p. 257, 1973.
- Tutton, A. E. H. Crystallography and Practical Crystal Measurement, 2nd ed. London: Lubrecht & Cramer, 1964.
- {{mathworld | title = Primary parallelohedron | urlname = PrimaryParallelohedron}}
- {{mathworld | title = Space-filling polyhedron | urlname = Space-FillingPolyhedron}}
- E. S. Fedorov, Nachala Ucheniya o Figurah. [In Russian] (Elements of the theory of figures) Notices Imper. Petersburg Mineralog. Soc., 2nd ser.,24(1885), 1 – 279. Republished by the Acad. Sci. USSR, Moscow 1953.
- [https://web.archive.org/web/20160304000157/http://www.matha.mathematik.uni-dortmund.de/~thilo/contents/fedorov.htm Fedorov's five parallelohedra in R³]
- Fedorov's Five Parallelohedra
1 : Space-filling polyhedra |