# Definition: Def. 11.26: Octahedron

An octahedron is a solid figure contained by eight equal and equilateral triangles.

### Modern Formulation

An octahedron is (the only)1 regular three-dimensional polyhedron with $$8$$ faces, $$12$$ edges, and $$6$$ vertices.

### Cartesian Coordinates of the Vertices and Faces of an Octahedron

The Cartesian coordinates $$(x,y,z)$$ of all $$6$$ vertices of an octahedron centered at the origin are given by

$\begin{array}{lrrr} \text{Vertex}&x&y&z\\ v_{1}&1&0&0\\ v_{2}&0&1&0\\ v_{3}&0&0&1\\ v_{4}&-1&0&0\\ v_{5}&0&-1&0\\ v_{6}&0&0&-1\\ \end{array}$

The $$8$$ faces of the octahedron are equilateral triangles with the following vertices:

$\begin{array}{lccccc} \text{Face}\\ f_{1}&v_{1}&v_{2}&v_{3}\\ f_{2}&v_{1}&v_{2}&v_{6}\\ f_{3}&v_{1}&v_{3}&v_{5}\\ f_{4}&v_{1}&v_{5}&v_{6}\\ f_{5}&v_{2}&v_{3}&v_{4}\\ f_{6}&v_{2}&v_{4}&v_{6}\\ f_{7}&v_{3}&v_{4}&v_{5}\\ f_{8}&v_{4}&v_{5}&v_{6}\\ \end{array}$

Problems: 1
Proofs: 2 3
Propositions: 4 5
Sections: 6

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