Difference between revisions of "Euler theorem"
From Encyclopedia of Mathematics
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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: | 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: | ||
| − | $$V+F-E=2.\tag{*}$$ | + | $$V+F-E=2.\label{*}\tag{*}$$ |
Euler's theorem hold for polyhedrons of genus $0$; for polyhedrons of genus $p$ the relation | Euler's theorem hold for polyhedrons of genus $0$; for polyhedrons of genus $p$ the relation | ||
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$$V+F-E=2-2p$$ | $$V+F-E=2-2p$$ | ||
| − | holds. This theorem was proved by L. Euler (1758); the relation \ | + | holds. This theorem was proved by L. Euler (1758); the relation \eqref{*} was known to R. Descartes (1620). |
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| + | [[Category:Geometry]] | ||
Latest revision as of 17:34, 14 February 2020
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:
$$V+F-E=2.\label{*}\tag{*}$$
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 \eqref{*} 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