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A solid figure having eight triangular faces, twelve edges and six vertices, with 4 faces at each vertex. If all edges have the same length, it is one of the five regular polyhedra (Platonic solids); if the edge length is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068100/o0681001.png" />, then the volume of the octahedron is
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A solid figure having eight triangular faces, twelve edges and six vertices, with 4 faces at each vertex. If all edges have the same length, it is one of the five regular polyhedra (Platonic solids); if the edge length is $a$, then the volume of the octahedron is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068100/o0681002.png" /></td> </tr></table>
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$$v=\frac{a^3\sqrt 2}{3}\approx0.4714a^3.$$
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o068100a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o068100a.gif" />
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====Comments====
 
====Comments====
The Schläfli symbol of an octahedron is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068100/o0681003.png" />. When the edges all have the same length one deals with the regular octahedron, reciprocal to the cube; it can be regarded either as a triangular anti-prism or as a square double-pyramid. As one of the five Platonic polyhedra (cf. [[Regular polyhedra|Regular polyhedra]]; [[Platonic solids|Platonic solids]]) it represents the ancient element air. It occurs in nature as a crystal of chrome alum. In terms of its circumradius as unit of measurement, its six vertices have Cartesian coordinates
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The Schläfli symbol of an octahedron is $\{3,4\}$. When the edges all have the same length one deals with the regular octahedron, reciprocal to the cube; it can be regarded either as a triangular anti-prism or as a square double-pyramid. As one of the five Platonic polyhedra (cf. [[Regular polyhedra|Regular polyhedra]]; [[Platonic solids|Platonic solids]]) it represents the ancient element air. It occurs in nature as a crystal of chrome alum. In terms of its circumradius as unit of measurement, its six vertices have Cartesian coordinates
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068100/o0681004.png" /></td> </tr></table>
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$$(\pm1,0,0),\quad(0,\pm1,0)\quad(0,0,\pm1);$$
  
thus it has edge-length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068100/o0681005.png" />, inradius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068100/o0681006.png" /> and volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068100/o0681007.png" />. Its 4 pairs of opposite faces (or the 4 diameters of the cube) are freely permuted by the octahedral group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068100/o0681008.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068100/o0681009.png" />.
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thus it has edge-length $\sqrt2$, inradius $\sqrt{1/3}$ and volume $4/3$. Its 4 pairs of opposite faces (or the 4 diameters of the cube) are freely permuted by the octahedral group $\mathfrak S_4$ of order $4!=24$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.T. Bennett,  "Deformable octahedra"  ''Proc. London Math. Soc. (2)'' , '''10'''  (1912)  pp. 309–343</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Regular polytopes" , Methuen  (1948)  pp. 5</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.T. Bennett,  "Deformable octahedra"  ''Proc. London Math. Soc. (2)'' , '''10'''  (1912)  pp. 309–343</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Regular polytopes" , Methuen  (1948)  pp. 5</TD></TR></table>

Latest revision as of 14:08, 30 July 2014

A solid figure having eight triangular faces, twelve edges and six vertices, with 4 faces at each vertex. If all edges have the same length, it is one of the five regular polyhedra (Platonic solids); if the edge length is $a$, then the volume of the octahedron is

$$v=\frac{a^3\sqrt 2}{3}\approx0.4714a^3.$$

Figure: o068100a


Comments

The Schläfli symbol of an octahedron is $\{3,4\}$. When the edges all have the same length one deals with the regular octahedron, reciprocal to the cube; it can be regarded either as a triangular anti-prism or as a square double-pyramid. As one of the five Platonic polyhedra (cf. Regular polyhedra; Platonic solids) it represents the ancient element air. It occurs in nature as a crystal of chrome alum. In terms of its circumradius as unit of measurement, its six vertices have Cartesian coordinates

$$(\pm1,0,0),\quad(0,\pm1,0)\quad(0,0,\pm1);$$

thus it has edge-length $\sqrt2$, inradius $\sqrt{1/3}$ and volume $4/3$. Its 4 pairs of opposite faces (or the 4 diameters of the cube) are freely permuted by the octahedral group $\mathfrak S_4$ of order $4!=24$.

References

[a1] G.T. Bennett, "Deformable octahedra" Proc. London Math. Soc. (2) , 10 (1912) pp. 309–343
[a2] H.S.M. Coxeter, "Regular polytopes" , Methuen (1948) pp. 5
How to Cite This Entry:
Octahedron. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Octahedron&oldid=32562
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article