M. Beck and S. Robins,

*Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra*, Undergraduate Texts in Mathematics, Vol. 154, Springer, 2007. Theorem 5.2 uses the integer point enumeration method to prove Euler's formula for rational polytopes in any dimension.Bishop and Goldberg,

*Tensor Analysis on Manifolds*, Dover, 1980. They prove the formula \(p_1-p_2+p_3=2\) where the values \(p_i\) denote the numbers of local minima, saddles, and local maxima on a spherical surface. I have adapted their proof, based on rainfall and flooding, to Euler's formula.J. A. Bondy and U. S. R. Murty,

*Graph Theory with Applications*, Elsevier, 1976. Uses the induction on faces.L. Euler, "Elementa doctrinae solidorum",

*Novi Commentarii academiae scientiarum Petropolitanae*, Vol. 4, pp. 109–140.M. Friedman,

*A History of Folding in Mathematics: Mathematizing the Margins*, Science Networks: Historical Studies, Vol. 59. Birkhäuser, 2018. For Maurolico's 1537 statement of Euler's formula, see p. 71.J. Hadamard,

*Elementary Geometry*. According to Alex Bogomolny, a 1958 Russian translation contains the divide and conquer proof. I haven't checked whether other editions do as well.M. Henle,

*A Combinatorial Approach to Topology*, W. H. Freeman, 1979. This book was used in the undergraduate topology course I once took, so it may have been the first place I saw a proof of Euler's formula. Now I'm not so sure about it. It gives some history of the Euler formula, which follows common folklore (erroneously, according to Malkevitch) in attributing the formula to Descartes, and muffs the proof – Henle gives the triangle removal proof without any mention of what is required of the removal sequence or how to find such a sequence.P. Hilton and J. Pederson, "The Euler Characteristic and Pólya's Dream",

*The American Mathematical Monthly*, Vol. 103, 1996, pp. 121–131. This article provides some speculation on how Euler might have found his formula, relating it to Pólya's theories of mathematical discovery. It also gives a proof of equivalence of the Euler characteristic and total angular defect closely related to the sum of angles proof presented here.P. Hliněný, "A short proof of Euler–Poincaré formula", EuroComb 2021. The proof by projection of Schlegel diagrams.

D. A. Klain and G.-C. Rota,

*Introduction to Geometric Probability*, Lezioni Lincee, Cambridge Univ. Press, 1997. Shows that the Euler characteristic can be viewed (along with e.g. volume and surface area) as one of the fundamental translation-invariant functions on unions of convex sets. The valuation-based proof is from section 5.2; the other key sections related to the Euler characteristic are 3.2 (showing that the characteristic is well defined for simplicial complexes) and 7.3 (generalizing Euler's formula to other intrinsic volumes).I. Lakatos,

*Proofs and Refutations: The Logic of Mathematical Discovery*, Cambridge University Press, 1976. Uses the triangle removal proof of Euler's formula as a key example for an investigation of what mathematical proof means. He also describes a proof based on binary homology theory.J. Lawrence, "A short proof of Euler's relation for convex polytopes", Canadian Mathematical Bulletin 40(4):471-474, 1997. The source of the hyperplane arrangement proof.

A. M. Legendre,

*Élements de géométrie*, Paris, 1794. Credited by Sommerville as source of the proof by spherical sum of angles.S. A. J. Lhuilier, "Mémoir sur la polyédrométrie, contenant une démonstration directe du théorème d'Euler sur les polyèdres, et un examen de diverses exceptions auxquelles ce théorème est assujetti",

*Gergonne Ann. Math.*, Vol. 3, 1812, p. 169. Credited by Sommerville as co-source (with Steiner) of the proof by sum of angles.J. H. van Lint and R. M. Wilson,

*A Course in Combinatorics*, Cambridge University Press, 1992. Proves Euler's formula via induction on edges. They also mention but do not prove a generalization to higher dimensional polytopes in their section on Möbius inversion.J. Malkevitch, "The first proof of Euler's formula",

*Mitteilungen aus dem Mathem. Seminar Giessen*, Heft 165, Teil III (Coxeter Festschrift), pp. 77-82. Describes the early history of the formula, including its discovery by Euler and proof by Legendre, rebutting the folklore theory that the formula was instead discovered by Descartes and proven by Hirsch.J. Propp, "Euler measure as generalized cardinality", arXiv:math/0203289, 2002.

D. M. Y. Sommerville,

*An Introduction to the Geometry of N Dimensions*, Dover 1958. Includes a chapter on Euler's formula including proofs by shelling, sum of angles, spherical sum of angles, and interdigitating trees.J. Steiner, "Leichter Beweis eines stereometrischen Satzes von Euler",

*Crelle J.*, Vol. 1, 1826, pp. 364-367. Credited by Sommerville as co-source (with Lhuilier) of the proof by sum of angles.W. P. Thurston,

*The Geometry and Topology of Three-Dimensional Manifolds*. These unpublished lecture notes (distributed as samizdat within the topology community) include a proof of Euler's formula based on electrical charge.G. K. C. Von Staudt,

*Geometrie der Lage*, Nürnberg, 1847. Credited by Sommerville as source of the proof by interdigitating trees.H. Tverberg, "How to cut a convex polytope into simplices",

*Geometriae Dedicata*, Vol. 3, No. 2, 1974, pp. 239–240. Proves that any polytope can be cut into simplices by a binary space partition and uses this as the basis for a proof of Euler's formula.J. Weeks,

*The Shape of Space: How to Visualize Surfaces and Three-Dimensional Manifolds*, Dekker, 1985. Proves the excess=area formula for spheres and uses it in the spherical sum of angles proof of the Euler formula.D. Wells,

*The Penguin Dictionary of Curious and Interesting Geometry*, Penguin, 1991. Includes a description of the excess=area property of spherical trigonometry used in the spherical sum of angles proof, as well as a quick mention that Pick's Theorem is equivalent to Euler's formula. Curiously, there is no entry for Euler's formula itself.G. Ziegler,

*Lectures on Polytopes*, Springer, 1995, gives a shelling based proof of Euler's formula that avoids case analysis and extends without change to any higher dimensional convex polytope.

Chen, "On the Euler characteristic of finite unions of convex sets", Disc. Comput. Geom. 10, 1993, 79-93.

Abstract: The Euler characteristic plays an important role in many subjects of discrete and continuous mathematics. For noncompact spaces, its homological definition, being a homotopy invariant, seems not as important as its role for compact spaces. However, its combinatorial definition, as a finitely additive measure, proves to be more applicable in the study of singular spaces such as semialgebraic sets finitely subanalytic sets, etc. The author introduces an interesting integral by means of which the combinatorial Euler characteristic can be defined without the necessity of decomposition and extension as in the traditional treatment for polyhedra and finite unions of compact convex sets. Since finite unions of closed convex sets cannot be obtained by cutting convex sets as in the polyhedral case, a separate treatment of the Euler characteristic for functions generated by indicator functions of closed convex sets and relatively open convex sets is necessary, and this forms the content of this paper.

Levitt, "The Euler characteristic is the unique locally determined numerical homotopy invariant of finite complexes", Disc. Comput. Geom. 7, 1992, 59-67.

Abstract: The author shows that if a numerical homotopy invariant of finite simplicial complexes has a local formula, then, up to multiplication by an obvious constant, the invariant is the Euler characteristic. Moreover, the Euler characteristic itself has a unique local formula.

Rocek and Williams, "On the Euler characteristic of piecewise linear manifolds", Phys. Lett. B, 273, 1991, 95-99.

Abstract: The Dehn-Sommerville relations and the corresponding equations for the angle sums are used to derive two expressions for the Euler characteristic of a simplicial manifold, firstly in terms of the numbers of even dimensional subsimplices, and secondly in terms of even-dimensional deficit angles. In each case the coefficients involved are related to the Bernoulli numbers.

Peter Hilton and Jean Pedersen. "Euler's theorem for polyhedra: a topologist and geometer respond". Comment to: "A new look at Euler's theorem for polyhedra" [Amer. Math. Monthly 101 (1994), no. 2, 109-128; MR 95c:52032] by B. Grunbaum and G. C. Shephard. With a response by Grunbaum and Shephard. Amer. Math. Monthly 101 (1994), no. 10, 959-962.

Krzysztof Przeslawski. "Linear algebra of convex sets and the Euler characteristic". Linear and Multilinear Algebra 31 (1992), no. 1-4, 153-191.

G. Thomas Sallee. "Euler's relation and where it led". Convexity and related combinatorial geometry (Norman, Okla., 1980), pp. 45--55, Lecture Notes in Pure and Appl. Math., 76, Dekker, New York, 1982.

Proofs of Euler's Formula.

From the Geometry Junkyard,
computational
and recreational geometry pointers.

David Eppstein,
Theory Group,
ICS,
UC Irvine.