Abstract
A graph is planar if it can be drawn in the plane ℝ2 without crossing edges (or, equivalently, on the 2-dimensional sphere S 2). We talk of a plane graph if such a drawing is already given and fixed. Any such drawing decomposes the plane or sphere into a finite number of connected regions, including the outer (unbounded) region, which are referred to as faces. Euler’s formula exhibits a beautiful relation between the number of vertices, edges and faces that is valid for any plane graph. Euler mentioned this result for the first time in a letter to his friend Goldbach in 1750, but he did not have a complete proof at the time. Among the many proofs of Euler’s formula, we present a pretty and “self-dual” one that gets by without induction. It can be traced back to von Staudt’s book “Geometrie der Lage” from 1847.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. D. Chakerian: Sylvester’s problem on collinear points and a relative Amer. Math. Monthly 77 (1970), 164–167.
G. Pick: Geometrisches zur Zahlenlehre, Sitzungsberichte Lotos (Prag) Natur-med. Verein für Böhmen 19 (1899), 311–319.
K. G. C. von Staudt: Geometrie der Lage, Verlag der Fr. Korn’schei Buchhandlung, Nürnberg 1847.
N. E. Steenrod: Solution 4065/Editorial Note, Amer. Math. Monthly 51 (1944), 170–171.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Aigner, M., Ziegler, G.M. (2001). Three applications of Euler’s formula. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04315-8_10
Download citation
DOI: https://doi.org/10.1007/978-3-662-04315-8_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-04317-2
Online ISBN: 978-3-662-04315-8
eBook Packages: Springer Book Archive