# Definition: Def. 11.25: Cube

### (Definition 25 from Book 11 of Euclid's “Elements”)

A cube is a solid figure contained by six equal squares.

### Modern Formulation

A cube is (the only)1 regular prism (hexahedron) .

### Cartesian Coordinates of the Vertices and Faces of a Cube

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

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