Self-dual configurations and regular graphs
HTML articles powered by AMS MathViewer
- by H. S. M. Coxeter PDF
- Bull. Amer. Math. Soc. 56 (1950), 413-455
References
-
1. H. F. Baker, Principles of geometry, vol. 1, 2d ed., Cambridge, 1929.
1a. H. F. Baker, ibid, vol. 3, Cambridge, 1934.
- H. F. Baker, Note to the preceding paper by C. V. H. Rao, Proc. Cambridge Philos. Soc. 42 (1946), 226–229. MR 19323, DOI 10.1017/s0305004100022970
- Gaston Benneton, Configurations harmoniques et quaternions, Ann. Sci. École Norm. Sup. (3) 64 (1947), 1–58 (French). MR 0023537, DOI 10.24033/asens.941
- Wilhelm Blaschke, Projektive Geometrie, Wolfenbütteler Verlagsanstalt, Wolfenbüttel-Hannover, 1947 (German). MR 0025741 5. W. Blaschke and G. Bol, Geometrie der Gewebe, Berlin, 1938.
- R. C. Bose, An affine analogue of Singer’s theorem, J. Indian Math. Soc. (N.S.) 6 (1942), 1–15. MR 6735
- H. R. Brahana, Regular Maps and Their Groups, Amer. J. Math. 49 (1927), no. 2, 268–284. MR 1506619, DOI 10.2307/2370756
- W. Burnside, Theory of groups of finite order, Dover Publications, Inc., New York, 1955. 2d ed. MR 0069818 9. A. Cayley, Sur quelques théorèmes de la géométrie de position(1846), Collected Mathematical Papers, vol. 1, 1889, pp. 317-328. 10. H. Cox, Application of Grassmann’s Ausdehnungslehre to properties of circles, Quart. J. Math. vol. 25 (1891) pp. 1-71. 11. H. S. M. Coxeter, Regular skew polyhedra in three and four dimensions, Proc. London Math. Soc. (2) vol. 43 (1937) pp. 33-62. 12. H. S. M. Coxeter, Regular polytopes, London, 1948.
- H. S. M. Coxeter, Configurations and maps, Rep. Math. Colloquium (2) 8 (1949), 18–38. MR 29494
- H. S. M. Coxeter, The Real Projective Plane, McGraw-Hill Book Co., Inc., New York, N.Y., 1949. MR 0030205 15. L. Cremona, Teoremi stereometrici, dei quali si deducono le proprietà dell’esagrammo di Pascal, Memorie della Reale Accademia dei Lincei vol. 1 (1877) pp. 854-874. 16. P. Du Val, On the directrices of a set of points in a plane, Proc. London Math. Soc. (2) vol. 35 (1933) pp. 23-74. 17. G. Fano, Sui postulati fondamentali della geometria proiettiva in uno spazio lineare a un numero qualunque di dimensioni, Giornale di Matematiche vol. 30 (1892) pp. 106-132.
- J. M. Feld, Configurations Inscriptible in a Plane Cubic Curve, Amer. Math. Monthly 43 (1936), no. 9, 549–555. MR 1523764, DOI 10.2307/2301402 18. R. M. Foster, Geometrical circuits of electrical networks, Transactions of the American Institute of Electrical Engineers vol. 51 (1932) pp. 309-317. 19. R. Frucht, Die Gruppe des Petersen’schen Graphen und der Kantensysteme der regulären Polyeder, Comment. Math. Helv. vol. 9 (1937) pp. 217-223. 20. G. Gallucci, Studio della figura delle otto rette e sue applicazioni alla geometria del tetraedro ed alla teoria della configurazioni, Rendiconto dell’Accademia delle Scienze fisiche e matematiche (Sezione della Società reale di Napoli) (3) vol. 12 (1906) pp. 49-79. 21. P. J. Heawood, Map-colour theorem, Quart. J. Math. vol. 24 (1890) pp. 332-338. 22. L. Henneberg, Die graphische Statik der starren Körper, Encyklopädie der Mathematischen Wissenschaften vol. 4.1 (1908) pp. 345-434. 23. E. Hess, Weitere Beiträge zur Theorie der räumlichen Configurationen, Verhandlungen den K. Leopoldinisch-Carolinischen Deutschen Akademie Naturforscher vol. 75 (1899) pp. 1-482.
- W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry. Vol. I, Cambridge, at the University Press; New York, The Macmillan Company, 1947. MR 0028055 25. R. W. H. T. Hudson, Kummer’s quartic surface, Cambridge, 1905. 26. S. Kantor, Über die Configurationen (3, 3) mit den Indices 8, 9 und ihren Zusammenhang mit den Curven dritter Ordnung, Sitzungsberichte der Mathematisch-Naturwissenschaftliche Classe der K. Akademie der Wissenschaften, Wien vol. 84.1 (1882) pp. 915-932. 27. S. Kantor, Die Configurationen (3, 3)10, ibid. pp. 1291-1314. 28. D. König, Theorie der endlichen und unendlichen Graphen, Leipzig, 1936.
- Karl Kommerell, Die Pascalsche Konfiguration $9_3$, Deutsche Math. 6 (1941), 16–32 (German). MR 5630 30. A. Kowalewski, W. R. Hamilton’s Dodekaederaufgabe als Buntordnungsproblem, Sitzungsberichte der Mathematisch-Naturwissenschaftliche Klasse der K. Akademie der Wissenschaften, Wien vol. 126.2a (1917) pp. 67-90. 31. F. W. Levi, Geometrische Konfigurationen, Leipzig, 1929.
- F. W. Levi, Finite Geometrical Systems, University of Calcutta, Calcutta, 1942. MR 0006834 33. V. Martinetti, Sopra alcune configurazioni piane, Annali di Matematica (2) vol. 14 (1887) pp. 161-192. 33a. V. Martinetti, Sulle configurazioni piane μ, Annali di Matematica (2) vol. 15 (1888) pp. 1-26. 34. V. Martinetti, Alcune considerazioni sulla configurazione di Kummer, Rend. Circ. Mat. Palermo vol. 16 (1902) pp. 196-203. 35. G. A. Miller, H. F. Blichfeldt, and L. E. Dickson, Theory and applications of finite groups, New York, 1916. 36. A. F. Möbius, Kann von zwei dreiseitigen Pyramiden eine jede in Bezug auf die andere um- und eingeschrieben zugleich heissen? (1828), Gesammelte Werke, vol. 1, 1886, pp. 439-446. 37. F. Morley and F. V. Morley, Inversive geometry, Boston, 1933. 38. J. Petersen, Les 36 officiers, Annuaire des Mathématiciens (1902) pp. 413-427. 39. H. W. Richmond, On the figure of six points in space of four dimensions, Quart. J. Math. vol. 31 (1900) pp. 125-160.
- Herbert W. Richmond, On a chain of theorems due to Homersham Cox, J. London Math. Soc. 16 (1941), 105–107. MR 4964, DOI 10.1112/jlms/s1-16.2.105 40. G. de B. Robinson, On the orthogonal groups in four dimensions, Proc. Cambridge Philos. Soc. vol. 27 (1931) pp. 37-48.
- A. Schönflies, Ueber die regelmässigen Configurationen $n^3$, Math. Ann. 31 (1888), no. 1, 43–69 (German). MR 1510469, DOI 10.1007/BF01204635
- James Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), no. 3, 377–385. MR 1501951, DOI 10.1090/S0002-9947-1938-1501951-4 43. D. M. Y. Sommerville, An introduction to the geometry of n dimensions, London, 1929. 44. G. K. C. von Staudt, Geometrie der Lage, Nürnberg, 1847. 45. C. Stephanos, Sur les systèmes desmiques de trois tétraèdres, Bull. Sci. Math. (2) vol. 3 (1879) pp. 424-456. 46. W. Threlfall, Gruppenbilder, Abhandlungen der Mathematisch-Physischen Klasse der Sächsischen Akademie der Wissenschaften vol. 41.6 (1932) pp. 1-59.
- W. T. Tutte, A family of cubical graphs, Proc. Cambridge Philos. Soc. 43 (1947), 459–474. MR 21678, DOI 10.1017/s0305004100023720
- Oswald Veblen, Collineations in a finite projective geometry, Trans. Amer. Math. Soc. 8 (1907), no. 3, 366–368. MR 1500790, DOI 10.1090/S0002-9947-1907-1500790-8 49. O. Veblen and J. W. Young, Projective geometry, vol. 1, Boston, 1910. 50. H. Whitney, Planar graphs, Fund. Math. vol. 21 (1933) pp. 73-84.
- Max Zacharias, Untersuchungen über ebene Konfigurationen $(12_4, 16_3)$, Deutsche Math. 6 (1941), 147–170 (German). MR 17930
Additional Information
- Journal: Bull. Amer. Math. Soc. 56 (1950), 413-455
- DOI: https://doi.org/10.1090/S0002-9904-1950-09407-5
- MathSciNet review: 0038078