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Given a cube with edge length . The face diagonal of a cube is the diagonal of a square , and the space diagonal of a cube is a line connecting two vertices that is not in the same face, formulated as . Both formulas can be determined by using Pythagorean theorem.
Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid, Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues that there are no further convex regular polyhedra.
If the edge lengths of a cuboid are a, b, and c, then the distinct rectangular faces have edges (a, b), (a, c), and (b, c); so the respective face diagonals have lengths +, +, and +. Thus each face diagonal of a cube with side length a is a 2 {\displaystyle a{\sqrt {2}}} .
h is the height of the wedge-shaped "roof" above the faces of that cube with edge length 2. An important case is h = 1 / 2 (a quarter of the cube edge length) for perfect natural pyrite (also the pyritohedron in the Weaire–Phelan structure). Another one is h = 1 / φ = 0.618... for the regular dodecahedron.
A magic square is an arrangement of numbers in a square grid so that the sum of the numbers along every row, column, and diagonal is the same. Similarly, one may define a magic cube to be an arrangement of numbers in a cubical grid so that the sum of the numbers on the four space diagonals must be the same as the sum of the numbers in each row, each column, and each pillar.
3D model of a truncated cube. In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and δ S +1, where δ S is the silver ratio, √ 2 +1.
Doubling the cube is the construction, using only a straightedge and compass, of the edge of a cube that has twice the volume of a cube with a given edge. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots.
The term unit cube or unit hypercube is also used for hypercubes, or "cubes" in n-dimensional spaces, for values of n other than 3 and edge length 1. [1] [2]Sometimes the term "unit cube" refers in specific to the set [0, 1] n of all n-tuples of numbers in the interval [0, 1].