Cuboid
{{short description|Convex polyhedron with six faces with four edges each}}
{{other uses}}
File:Generic quadrilateral hexahedron.svgIn 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.{{r|alexander84|grunbaum}}
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.{{r|alexander84|dupius}} Along with the rectangular cuboids, parallelepiped is a cuboid with six parallelogram. 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, {{harvtxt|Robertson|1983}} found that there were at least 22 different cases, "of which only about half are familiar in the shapes of everyday objects".{{r|robertson}}
There exist quadrilateral-faced hexahedra which are non-convex.
| class="wikitable center"
|+ style="text-align:center;"|Some notable cuboids | |||
| Image||Name||Faces||Symmetry group | |||
|---|---|---|---|
| 110px | Cube | {{math|6}} congruent squares | {{math|Oh, [4,3], (*432)}} order {{math|48}} |
| 50px | Trigonal trapezohedron | {{math|6}} congruent rhombi | {{math|D3d, [2+,6], (2*3)}} order {{math|12}} |
| 110px | Rectangular cuboid | {{math|3}} pairs of rectangles | rowspan=2|{{math|D2h, [2,2], (*222)}} order {{math|8}} |
| 110px | Right rhombic prism | {{math|1}} pair of rhombi, {{math|4}} congruent squares | |
| 110px | Right square frustum | {{math|2}} non-congruent squares, {{nowrap|{{math|4}} congruent isosceles trapezoids}} | {{math|C4v, [4], (*44)}} order {{math|8}} |
| 110px | Twisted trigonal trapezohedron | {{math|6}} congruent quadrilaterals | {{math|D3, [2,3]+, (223)}} order {{math|6}} |
| 70px | Right isosceles-trapezoidal prism | {{math|1}} pair of isosceles trapezoids; {{nowrap|{{math|1}}, {{math|2}} or {{math|3}} (congruent) square(s)}} | {{math|?, ?, ?}} order {{math|4}} |
| 110px | Rhombohedron | {{math|3}} pairs of rhombi | rowspan=2|{{math|Ci, [2+,2+], (×)}} order {{math|2}} |
| 110px | Parallelepiped | {{math|3}} pairs of parallelograms |
See also
References
{{reflist|refs=
| title = Polytopes and Symmetry
| url = https://archive.org/details/polytopessymmetr0000robe
| url-access = registration
| last = Robertson | first = Stewart A.
| publisher = Cambridge University Press
| year = 1984
| isbn = 9780521277396
| page = [https://archive.org/details/polytopessymmetr0000robe/page/75 75]
}}
| last = Dupuis | first = Nathan F.
| url = https://archive.org/details/elementssynthet01dupugoog/page/n68
| title = Elements of Synthetic Solid Geometry
| publisher = Macmillan
| year = 1893
| page = 53
| access-date = December 1, 2018
}}
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: {{cite book
| last = Grünbaum | first = Branko | author-link = Branko Grünbaum
| doi = 10.1007/978-1-4613-0019-9
| edition = 2nd
| isbn = 978-0-387-00424-2
| location = New York
| mr = 1976856
| page = 59
| publisher = Springer-Verlag
| series = Graduate Texts in Mathematics
| title = Convex Polytopes
| title-link = Convex Polytopes
| volume = 221
| year = 2003
}}
| last = Robertson | first = S. A.
| doi = 10.1007/BF03026511
| issue = 4
| journal = The Mathematical Intelligencer
| mr = 746897
| pages = 57–60
| title = Polyhedra and symmetry
| volume = 5
| year = 1983
}}
}}
{{Commons category|Hexahedra with cube topology}}
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