An octahedron is a solid figure contained by eight equal and equilateral triangles.
An octahedron is (the only)^{1} regular three-dimensional polyhedron with \(8\) faces, \(12\) edges, and \(6\) vertices.
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
This will be proven in the Prop. 18 of Book 13, thus the octahedron is well-defined. ↩