Difference between revisions of "Euler theorem"
From Encyclopedia of Mathematics
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| − | For every polyhedron the number | + | {{TEX|done}} |
| + | For every polyhedron the number $V$ of its vertices plus the number $F$ of its faces minus the number $E$ of its edges is equal to 2: | ||
| − | + | $$V+F-E=2.\tag{*}$$ | |
| − | Euler's theorem hold for polyhedrons of genus | + | Euler's theorem hold for polyhedrons of genus $0$; for polyhedrons of genus $p$ the relation |
| − | + | $$V+F-E=2-2p$$ | |
| − | holds. This theorem was proved by L. Euler (1758); the relation | + | holds. This theorem was proved by L. Euler (1758); the relation \ref{*} was known to R. Descartes (1620). |
Revision as of 17:15, 30 April 2014
For every polyhedron the number $V$ of its vertices plus the number $F$ of its faces minus the number $E$ of its edges is equal to 2:
$$V+F-E=2.\tag{*}$$
Euler's theorem hold for polyhedrons of genus $0$; for polyhedrons of genus $p$ the relation
$$V+F-E=2-2p$$
holds. This theorem was proved by L. Euler (1758); the relation \ref{*} was known to R. Descartes (1620).
How to Cite This Entry:
Euler theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_theorem&oldid=31997
Euler theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_theorem&oldid=31997
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article