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Platonic solid
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==Cartesian coordinates== For Platonic solids centered at the origin, simple [[Cartesian coordinate system|Cartesian coordinates]] of the vertices are given below. The Greek letter <math>\varphi</math> is used to represent the [[golden ratio]] <math>\frac{1+\sqrt{5}}{2}\approx 1.6180</math>. {| class=wikitable style="text-align:center;" |+ Parameters ! scope="row" | Figure ! colspan=2 | Tetrahedron !! Octahedron !! Cube !! colspan=2 | Icosahedron !! colspan=2 | Dodecahedron |- ! scope="row" | Faces |colspan=2|4||8||6||colspan=2|20||colspan=2|12 |- ! scope="row" | Vertices |colspan=2|4||6 (2 × 3)||8||colspan=2|12 (4 × 3)||colspan=2|20 (8 + 4 × 3) |- ! Position|| 1||2 || |||| 1||2|| 1||2 |- style="vertical-align:top;" ! scope="row" style="vertical-align:middle;" | Vertex <br/>coordinates | {{nowrap|(1, 1, 1)}}<BR/>{{nowrap|(1, −1, −1)}}<BR/>{{nowrap|(−1, 1, −1)}}<BR/>{{nowrap|(−1, −1, {{fsp}}1)}} | {{nowrap|(−1, −1, −1)}}<BR/>{{nowrap|(−1, 1, 1)}}<BR/>{{nowrap|({{fsp}}1, −1, {{fsp}}1)}}<BR/>{{nowrap|({{fsp}}1, {{fsp}}1, −1)}} | {{nowrap|(±1, {{fsp}}0, {{fsp}}0)}}<BR/>{{nowrap|({{fsp}}0, ±1, {{fsp}}0)}}<BR/>{{nowrap|({{fsp}}0, {{fsp}}0, ±1)}} | {{nowrap|(±1, ±1, ±1)}} | {{nowrap|({{fsp}}0, ±1, ±''φ'')}}<BR/>{{nowrap|(±1, ±''φ'', {{fsp}}0)}}<BR/>{{nowrap|(±''φ'', {{fsp}}0, ±1)}}||{{nowrap|({{fsp}}0, ±''φ'', ±1)}}<BR/>{{nowrap|(±''φ'', ±1, {{fsp}}0)}}<BR/>{{nowrap|(±1, {{fsp}}0, ±''φ'')}} | {{nowrap|(±1, ±1, ±1)}}<BR/>{{nowrap|({{fsp}}0, ±{{sfrac|1|''φ''}}, ±''φ'')}}<BR/>{{nowrap|(±{{sfrac|1|''φ''}}, ±''φ'', {{fsp}}0)}}<BR/>{{nowrap|(±''φ'', {{fsp}}0, ±{{sfrac|1|''φ''}})}} | {{nowrap|(±1, ±1, ±1)}}<BR/>{{nowrap|({{fsp}}0, ±''φ'', ±{{sfrac|1|''φ''}})}}<BR/>{{nowrap|(±''φ'', ±{{sfrac|1|''φ''}}, {{fsp}}0)}}<BR/>{{nowrap|(±{{sfrac|1|''φ''}},{{fsp}}}} 0, ±''φ'') |} The coordinates for the tetrahedron, dodecahedron, and icosahedron are given in two positions such that each can be deduced from the other: in the case of the tetrahedron, by changing all coordinates of sign ([[central symmetry]]), or, in the other cases, by exchanging two coordinates ([[reflection (geometry)|reflection]] with respect to any of the three diagonal planes). These coordinates reveal certain relationships between the Platonic solids: the vertices of the tetrahedron represent half of those of the cube, as {4,3} or {{CDD|node_1|4|node|3|node}}, one of two sets of 4 vertices in dual positions, as h{4,3} or {{CDD|node_h|4|node|3|node}}. Both tetrahedral positions make the compound [[stellated octahedron]]. The coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform [[truncated octahedron]], t{3,4} or {{CDD|node_1|3|node_1|4|node}}, also called a ''[[Icosahedron#Pyritohedral symmetry|snub octahedron]]'', as s{3,4} or {{CDD|node_h|3|node_h|4|node}}, and seen in the [[compound of two icosahedra]]. Eight of the vertices of the dodecahedron are shared with the cube. Completing all orientations leads to the [[compound of five cubes]].
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