A cube is a solid figure contained by six equal squares.
A cube is (the only)^{1} regular prism (hexahedron) .
The Cartesian coordinates \((x,y,z)\) of all \(8\) vertices of a cube centered at the origin are given by:
\[\begin{array}{lrrr} \text{Vertex}&x&y&z\\ v_{1}&-1&-1&1\\ v_{2}&-1&-1&-1\\ v_{3}&1&-1&-1\\ v_{4}&1&-1&1\\ v_{5}&1&1&1\\ v_{6}&1&1&-1\\ v_{7}&-1&1&-1\\ v_{8}&-1&1&1\\ \end{array}\]
The \(6\) faces of the octahedron are equilateral triangles with the following vertices:
\[\begin{array}{lccccc} \text{Face}\\ f_{1}&v_{1}&v_{2}&v_{3}&v_{4}\\ f_{2}&v_{4}&v_{3}&v_{6}&v_{5}\\ f_{3}&v_{5}&v_{6}&v_{7}&v_{8}\\ f_{4}&v_{8}&v_{7}&v_{2}&v_{1}\\ f_{5}&v_{8}&v_{1}&v_{4}&v_{5}\\ f_{6}&v_{2}&v_{7}&v_{6}&v_{3}\\ \end{array}\]
Corollaries: 1
Definitions: 2
Proofs: 3 4 5 6
Propositions: 7 8 9 10
Sections: 11
This will be proven in the Prop. 18 of Book 13, thus the cube is well-defined. ↩