Euler’s Polyhedron Formula in mizar

  • Jesse Alama
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6327)


Euler’s polyhedron formula asserts for a polyhedron p that V − E + F = 2, where V, E, and F are, respectively, the numbers of vertices, edges, and faces of p. Motivated by I. Lakatos’s philosophy of mathematics as presented in his Proofs and Refutations, in which the history of Euler’s formula is used as a case study to illustrate Lakatos’s views, we formalized a proof of Euler’s formula formula in the mizar system. We describe some of the notable features of the proof and sketch an improved formalization in progress that takes a deeper mathematical perspective, using the basic results of algebraic topology, than the initial formalization did.


Algebraic Topology Natural Deduction Proof Check Mizar Mathematical Library Mizar System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Alama, J.: Euler’s polyhedron formula. Formalized Mathematics 16(1), 7–17 (2008), CrossRefMathSciNetGoogle Scholar
  2. 2.
    Euler, L.: Demonstratio nonnullarum insignium proprietatum quibus solida hedris planis inclusa sunt praedita. Novi Commentarii Academiae Scientarum Petropolitanae 4, 94–108 (1758)Google Scholar
  3. 3.
    Euler, L.: Elementa doctrinae solidorum. Novi Commentarii Academiae Scientarum Petropolitanae 4, 109–140 (1758)Google Scholar
  4. 4.
    Juskevich, A.P., Winter, E. (eds.): Leonhard Euler und Christian Goldbach: Briefwechsel 1729-1764. Akademie-Verlag, Berlin (1965)Google Scholar
  5. 5.
    Lakatos, I.: Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press, Cambridge (1976)zbMATHGoogle Scholar
  6. 6.
    mizar, Google Scholar
  7. 7.
    Poincaré, H.: Sur la généralisation d’un théorème d’Euler relatif aux polyèdres. Comptes Rendus de Séances de l’Academie des Sciences 117, 144 (1893)Google Scholar
  8. 8.
    Poincaré, H.: Complément à l’analysis situs. Rendiconti del Circolo Matematico di Palermo 13, 285–343 (1899)zbMATHCrossRefGoogle Scholar
  9. 9.
    Pontryagin, L.S.: Foundations of Combinatorial Topology. Dover Publications, New York (1999)Google Scholar
  10. 10.
    Samelson, H.: In defense of Euler. L’Enseignement Mathématique 42, 377–382 (1996)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Spanier, E.H.: Algebraic Topology. Springer, Heidelberg (1968)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jesse Alama
    • 1
  1. 1.Center for Artificial Intelligence, Department of Computer Science, Faculty of Science and TechnologyNew University of LisbonPortugal

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