Editing Octahedron (section)

=== As a Platonic solid ===
{{multiple image
 | image1 = Kepler Octahedron Air.jpg
 | caption1 = Sketch of a regular octahedron by [[Johannes Kepler]]
 | image2 = Mysterium Cosmographicum solar system model.jpg
 | caption2 = [[Johannes Kepler|Kepler's]] Platonic solid model of the [[Solar System]]
 | align = right
 | total_width = 300
}}
The regular octahedron is one of the [[Platonic solid]]s, a set of polyhedrons whose faces are [[Congruence (geometry)|congruent]] [[regular polygons]] and the same number of faces meet at each vertex.{{r|hs}} This ancient set of polyhedrons was named after [[Plato]] who, in his [[Timaeus (dialogue)|''Timaeus'']] dialogue, related these solids to nature. One of them, the regular octahedron, represented the [[classical element]] of [[Wind (classical element)|wind]].{{r|cromwell}}

Following its attribution with nature by Plato, [[Johannes Kepler]] in his ''[[Harmonices Mundi]]'' sketched each of the Platonic solids.{{r|cromwell}} In his ''[[Mysterium Cosmographicum]]'', Kepler also proposed the [[Solar System]] by using the Platonic solids setting into another one and separating them with six spheres resembling the six planets. The ordered solids started from the innermost to the outermost: regular octahedron, [[regular icosahedron]], [[regular dodecahedron]], [[regular tetrahedron]], and [[cube]].{{r|livio}}

Like its dual, the regular octahedron has three properties: any two faces, two vertices, and two edges are transformed by rotation and reflection under the symmetry orbit, such that the appearance remains unchanged; these are [[isohedral]], [[isogonal figure|isogonal]], and [[isotoxal]] respectively. Hence, it is considered a [[regular polyhedron]]. Four triangles surround each vertex, so the regular octahedron is <math> 3.3.3.3 </math> by [[vertex configuration]] or <math> \{3,4\} </math> by [[Schläfli symbol]].{{r|wd}}