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This equation, stated by Euler in 1758, [2] is known as Euler's polyhedron formula. [3] It corresponds to the Euler characteristic of the sphere (i.e. χ = 2 {\displaystyle \ \chi =2\ } ), and applies identically to spherical polyhedra .
In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case.
In the mathematical field of Galois cohomology, the local Euler characteristic formula is a result due to John Tate that computes the Euler characteristic of the group cohomology of the absolute Galois group G K of a non-archimedean local field K.
The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k , where k is the non-orientable genus.
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characteristic.
Euler's Gem: The Polyhedron Formula and the Birth of Topology is a book on the formula + = for the Euler characteristic of convex polyhedra and its connections to the history of topology. It was written by David Richeson and published in 2008 by the Princeton University Press , with a paperback edition in 2012.
A tabulation of the numbers in a triangular array is called the Euler triangle or Euler's triangle. It shares some common characteristics with Pascal's triangle . Values of A ( n , k ) {\textstyle A(n,k)} (sequence A008292 in the OEIS ) for 0 ≤ n ≤ 9 {\textstyle 0\leq n\leq 9} are:
The formula is still valid if x is a complex number, and is also called Euler's formula in this more general case. [1] Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". [2]