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Difference between revisions of "Euler theorem"

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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 \ref{*} was known to R. Descartes (1620).
+
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
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article