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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] [2] General cuboids have many different types.
In the case of the body cuboid, the body (space) diagonal g is irrational. For the edge cuboid, one of the edges a, b, c is irrational. The face cuboid has one of the face diagonals d, e, f irrational. The body cuboid is commonly referred to as the Euler cuboid in honor of Leonhard Euler, who discussed this type of cuboid. [15]
A space-filling tetrahedral disphenoid inside a cube. Two edges have dihedral angles of 90°, and four edges have dihedral angles of 60°. A disphenoid is a tetrahedron with four congruent triangles as faces; the triangles necessarily have all angles acute. The regular tetrahedron is a special case of a disphenoid.
In the case of the coordinates of the six vertices where four faces meet at their acute angles, they are (±2, 0, 0), (0, ±2, 0) and (0, 0, ±2). The rhombic dodecahedron can be seen as a degenerate limiting case of a pyritohedron , with permutation of coordinates (±1, ±1, ±1) and (0, 1 + h , 1 − h 2 ) with parameter h = 1.
For example, with a cube, this is done by first looking straight towards one face. Next, the cube is rotated ±45° about the vertical axis, followed by a rotation of approximately 35.264° (precisely arcsin 1 ⁄ √ 3 or arctan 1 ⁄ √ 2, which is related to the Magic angle) about the horizontal axis. Note that with the cube (see image) the ...
A rhombohedron has two opposite apices at which all face angles are equal; a prolate rhombohedron has this common angle acute, and an oblate rhombohedron has an obtuse angle at these vertices. A cube is a special case of a rhombohedron with all sides square.
A rectangular cuboid with integer edges, as well as integer face diagonals, is called an Euler brick; for example with sides 44, 117, and 240. A perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists. [7] The number of different nets for a simple cube is 11 ...
An example is the rhombicuboctahedron, constructed by separating the cube or octahedron's faces from the centroid and filling them with squares. [8] Snub is a construction process of polyhedra by separating the polyhedron faces, twisting their faces in certain angles, and filling them up with equilateral triangles .