Cuboid

In geometry, a cuboid is a quadrilateral-faced convex hexahedron (a polyhedron with six faces). "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 or/and the angles between its adjacent faces. In general mathematical language, a cuboid is a convex polyhedron whose polyhedral graph is the same as that of a cube.^[1]^[2]

A special case of a cuboid is a rectangular cuboid, with six rectangle faces and adjacent faces meeting at right angles. A special case of a rectangular cuboid is a cube, with six square faces and adjacent faces meeting at right angles.^[1]^[3]

Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, as is a right square frustum (the shape formed by truncating the apex of a right square pyramid).

In attempting to classify cuboids by their symmetries, S. A. Robinson found that there were at least 22 different cases, "of which only about half are familiar in the shapes of everyday objects".^[4]

The numbers $F$ of faces, $V$ of vertices, and $E$ of edges of any convex polyhedron are related by Euler's formula: $F + V - E = 2$ .

In the case of a cuboid, this formula gives: $6 + 8 - 12 = 2$ ;
that is, like a cube, a cuboid has six faces, eight vertices, and twelve edges.

Gallery[edit]

Some notable cuboids (quadrilateral-faced convex hexahedra)
Cube	6 congruent squares	$O h, [4,3], (*432)$ order $48$
Trigonal trapezohedron	6 congruent rhombi	$D 3d, [2 +,6], (2*3)$ order $12$
Rectangular cuboid	3 pairs of rectangles	$D 2h, [2,2], (*222)$ order $8$
Right rhombic prism	1 pair of rhombi, 4 congruent squares
Right square frustum	2 non-congruent squares, 4 congruent isosceles trapezoids	$C 4v, [4], (*44)$ order $8$
Twisted trigonal trapezohedron	6 congruent twisted kites	$D 3, [2,3] +, (223)$ order $6$
Right trapezoidal prism	1 pair of isosceles trapezoids; 1, 2, or 3 (congruent) square(s)	$?, ?, ?$ order $4$
Rhombohedron	3 pairs of rhombi	$C i, [2 +,2 +], (\times)$ order $2$
Parallelepiped	3 pairs of parallelograms

Note:
There exist quadrilateral-faced hexahedra which are non-convex.

References[edit]

^ ^a ^b Robertson, Stewart Alexander (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: 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.
^ Dupuis, Nathan Fellowes (1893). Elements of Synthetic Solid Geometry. Macmillan. p. 53. Retrieved December 1, 2018.
^ Robertson, S. A. (1983). "Polyhedra and symmetry". The Mathematical Intelligencer. 5 (4): 57–60. doi:10.1007/BF03026511. MR 0746897.

[alexander84-1] Robertson, Stewart Alexander (1984). Polytopes and Symmetry. Cambridge University Press. p. 75. ISBN 9780521277396.

[2] 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: 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.

[3] Dupuis, Nathan Fellowes (1893). Elements of Synthetic Solid Geometry. Macmillan. p. 53. Retrieved December 1, 2018.

[4] Robertson, S. A. (1983). "Polyhedra and symmetry". The Mathematical Intelligencer. 5 (4): 57–60. doi:10.1007/BF03026511. MR 0746897.

[1]

[2]

[3]

[4]

Gallery[edit]

See also[edit]

References[edit]