Difference between revisions of "Icosahedron"
From Encyclopedia of Mathematics
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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 | + | {{TEX|done}} |
| + | 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 | ||
| − | + | $$V=\frac{5}{12}a^3(3+\sqrt5)\cong2.1817a^3.$$ | |
<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=15308
Icosahedron. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Icosahedron&oldid=15308