HISTORICAL ASPECTS OF THE DISCOVERY OF THE EULER CHARACTERISTIC AND SOME OF ITS DEVELOPMENTS IN MODERN TOPOLOGY
DOI:
10.47976/RBHM2009v9n1765-75Keywords:
Euler characteristic, topology, characteristic classesAbstract
We begin by describing where and when Euler obtained the famous formula V + F = E + 2, which relates the number of vertices, edges and faces of a polyhedron that satisfies certain conditions. A few considerations are made about the relation of this formula with other problems and some difficulties of the original proof given by Euler. Then we move to the end of the 19th and beginning of the 20th century when the Euler haracteristic and its generalization were linked to new topics in topology. Finally we present some of the generalizations of Euler characteristic which are used in recent (in the past 50 years) developments of topology.
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References
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Borsari, L. D., Cardona, F. and Wong, P. 2009. “Equivariant path fields on topological manifolds”. Top. Methods on Nonlinear Analysis, vol. 33, no. 1, 1-16.
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Euler, L. 1758a. “Elementa doctrinae solidorum”. Novi commentarii academiae scientiarum Imperialis petropolitanae, vol. 4, 109-140, reprinted in Opera Omnia, Series I, Volume 26, 71-93 (Eneström Index E230).
Euler, L. 1758b. “Demonstratio nonnullatum insifnium proprietatum, quibus solidahedris planis inclusa sunt praedita”. Novi commentarii academiae scientiarum Imperialis petropolitanae, vol. 4, 140-160, reprinted in Opera Omnia Series I, Volume 26, 94-108 (Eneström Index E231).
Grünbaum, B. and Shephard, G. C. 1994. “A new look at Euler’s theorem for polyhedra”. Amer. Math. Monthly, vol. 101, no. 2, 109-128.
Borsari, L. D., Cardona, F. and Wong, P. 2009. “Equivariant path fields on topological manifolds”. Top. Methods on Nonlinear Analysis, vol. 33, no. 1, 1-16.
Brown, R. 1965. “Path fields on manifolds”. Trans. Amer. Math. Soc., vol. 118, 180-191.
Brown, R. 1971. The Lefschetz fixed point theorem. Glenview and London: Scott, Foresman and Co.
Cauchy, A. L. 1813. “Recherche sur les polyèdres – première mémoire”. Journal de l'École Polytechnique, vol. 9 (Cahier 16), 66-86.
Euler, L. 1758a. “Elementa doctrinae solidorum”. Novi commentarii academiae scientiarum Imperialis petropolitanae, vol. 4, 109-140, reprinted in Opera Omnia, Series I, Volume 26, 71-93 (Eneström Index E230).
Euler, L. 1758b. “Demonstratio nonnullatum insifnium proprietatum, quibus solidahedris planis inclusa sunt praedita”. Novi commentarii academiae scientiarum Imperialis petropolitanae, vol. 4, 140-160, reprinted in Opera Omnia Series I, Volume 26, 94-108 (Eneström Index E231).
Grünbaum, B. and Shephard, G. C. 1994. “A new look at Euler’s theorem for polyhedra”. Amer. Math. Monthly, vol. 101, no. 2, 109-128.
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2020-11-03
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GONÇALVES, Daciberg Lima. HISTORICAL ASPECTS OF THE DISCOVERY OF THE EULER CHARACTERISTIC AND SOME OF ITS DEVELOPMENTS IN MODERN TOPOLOGY. Brazilian Journal on the History of Mathematics, São Paulo, vol. 9, no. 17, p. 65–75, 2020. DOI: 10.47976/RBHM2009v9n1765-75. Disponível em: https://mail.rbhm.org.br/index.php/RBHM/article/view/170. Acesso em: 25 nov. 2024.
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