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Difference between revisions of "Icosahedron"

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One of the five regular polytopes. An icosahedron has 20 (triangular) faces, 30 edges and 12 vertices (at each of which 5 edges meet). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050020/i0500201.png" /> is the length of an edge of the icosahedron, then its volume is given by
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One of the five regular polytopes. An icosahedron has 20 (triangular) faces, 30 edges and 12 vertices (at each of which 5 edges meet). If $a$ is the length of an edge of the icosahedron, then its volume is given by
  
<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/i/i050/i050020/i0500202.png" /></td> </tr></table>
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$$V=\frac{5}{12}a^3(3+\sqrt5)\cong2.1817a^3.$$
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i050020a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i050020a.gif" />

Latest revision as of 16:13, 11 April 2014

One of the five regular polytopes. An icosahedron has 20 (triangular) faces, 30 edges and 12 vertices (at each of which 5 edges meet). If $a$ is the length of an edge of the icosahedron, then its volume is given by

$$V=\frac{5}{12}a^3(3+\sqrt5)\cong2.1817a^3.$$

Figure: i050020a

Comments

The regular polytopes are also called the Platonic solids.

The symmetry group of the icosahedron plays a role in various branches of mathematics, and led F. Klein to his famous book [a2].

References

[a1] H.S.M. Coxeter, "Regular polytopes" , Dover, reprint (1973)
[a2] F. Klein, "Lectures on the icosahedron and the solution of equations of the fifth degree" , Dover, reprint (1956) (Translated from German)
How to Cite This Entry:
Icosahedron. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Icosahedron&oldid=31504