Cuboid
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In geometry, a cuboid is a hexahedron with quadrilateral faces, meaning it is a polyhedron with six faces; it has eight vertices and twelve edges. A rectangular cuboid (sometimes also called a "cuboid") has all right angles and equal opposite rectangular faces. Etymologically, "cuboid" means "like a cube", in the sense of a convex solid which can be transformed into a cube (by adjusting the lengths of its edges and the angles between its adjacent faces). A cuboid is a convex polyhedron whose polyhedral graph is the same as that of a cube.[1]Template:R/superscript[2]Template:R/superscript
General cuboids have many different types. When all of the rectangular cuboid's edges are equal in length, it results in a cube, with six square faces and adjacent faces meeting at right angles.[1]Template:R/superscript[3]Template:R/superscript Along with the rectangular cuboids, a parallelepiped is a cuboid with six parallelogram faces. A rhombohedron is a cuboid with six rhombus faces. A square frustum is a frustum with a square base, but the rest of its faces are quadrilaterals; the square frustum is formed by truncating the apex of a square pyramid. In attempting to classify cuboids by their symmetries, Lua error in package.lua at line 80: module 'Module:Footnotes/anchor_id_list' not found. found that there were at least 22 different cases, "of which only about half are familiar in the shapes of everyday objects".[4]Template:R/superscript
There exist quadrilateral-faced hexahedra which are non-convex.
| Image | Name | Faces | Symmetry group |
|---|---|---|---|
 |
Cube | 6 congruent squares | Oh, [4,3], (*432) order 48 |
 |
Trigonal trapezohedron | 6 congruent rhombi | D3d, [2+,6], (2*3) order 12 |
 |
Rectangular cuboid | 3 pairs of rectangles | D2h, [2,2], (*222) order 8 |
 |
Right rhombic prism | 1 pair of rhombi, 4 congruent squares | |
 |
Right square frustum | 2 non-congruent squares, 4 congruent isosceles trapezoids |
C4v, [4], (*44) order 8 |
 |
Twisted trigonal trapezohedron | 6 congruent quadrilaterals | D3, [2,3]+, (223) order 6 |
 |
Right isosceles-trapezoidal prism | 1 pair of isosceles trapezoids; 1, 2 or 3 (congruent) square(s) |
?, ?, ? order 4 |
 |
Rhombohedron | 3 pairs of rhombi | Ci, [2+,2+], (×) order 2 |
 |
Parallelepiped | 3 pairs of parallelograms |
See also
References
- ^ a b Page Module:Citation/CS1/styles.css has no content.Robertson, Stewart A. (1984). Polytopes and Symmetry. Cambridge University Press. p. 75. ISBN 9780521277396.
- ^ Branko Grünbaum has also used the word "cuboid" to describe a more general class of convex polytopes in three or more dimensions, obtained by gluing together polytopes combinatorially equivalent to hypercubes. See: Page Module:Citation/CS1/styles.css has no content.Grünbaum, Branko (2003). Convex Polytopes. Graduate Texts in Mathematics. Vol. 221 (2nd ed.). New York: Springer-Verlag. p. 59. doi:10.1007/978-1-4613-0019-9. ISBN 978-0-387-00424-2. MR 1976856.
- ^ Page Module:Citation/CS1/styles.css has no content.Dupuis, Nathan F. (1893). Elements of Synthetic Solid Geometry. Macmillan. p. 53. Retrieved December 1, 2018.
- ^ Page Module:Citation/CS1/styles.css has no content.Robertson, S. A. (1983). "Polyhedra and symmetry". The Mathematical Intelligencer. 5 (4): 57–60. doi:10.1007/BF03026511. MR 0746897.
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