Configurations between geometry and combinatorics. (English) Zbl 1045.51002
A configuration \((v_r,b_k)\) is a finite incidence structure of \(v\) points and \(b\) lines such that (i) there are \(k\) points on each line and \(r\) lines through each point, and (ii) there is at most one line through two given points.
In this paper, the development of configurations which were born in geometry during the last 125 years is described. The author discusses the early years of configurations and the conflict between their geometrical and combinatorial properties, represented by the wrong picture of a configuration \(10_3\) by Kantor. The breakthrough of Martinetti opened the research of configurations as purely combinatorial objects.
The paper contains several citations from original papers which are not so easily available today in order to enable the reader to follow the development from geometry to combinatorics, and reflects very well some mathematical problems like the drawing and the realization problem of configurations as typical problems of the 20th century with a lot of impact on visualization and computers.
In this paper, the development of configurations which were born in geometry during the last 125 years is described. The author discusses the early years of configurations and the conflict between their geometrical and combinatorial properties, represented by the wrong picture of a configuration \(10_3\) by Kantor. The breakthrough of Martinetti opened the research of configurations as purely combinatorial objects.
The paper contains several citations from original papers which are not so easily available today in order to enable the reader to follow the development from geometry to combinatorics, and reflects very well some mathematical problems like the drawing and the realization problem of configurations as typical problems of the 20th century with a lot of impact on visualization and computers.
Reviewer: Serguey M. Pokas (Odessa)
References:
| [1] | Betten, A.; Betten, D., Tactical decompositions and some configurations \(v_4\), J. Geom., 66, 27-41 (1999) · Zbl 0947.51010 |
| [2] | Betten, A.; Brinkmann, G.; Pisanski, T., Counting symmetric configurations \(v_3\), Discrete Appl. Math., 99, 331-338 (2000) · Zbl 0940.05020 |
| [3] | M. Boben, H. Gropp, T. Pisanski, Computer drawings of non-symmetric configurations, extended abstract, 14th European Workshop on Computational Geometry, Barcelona, 1998.; M. Boben, H. Gropp, T. Pisanski, Computer drawings of non-symmetric configurations, extended abstract, 14th European Workshop on Computational Geometry, Barcelona, 1998. |
| [4] | Coxeter, H. S.M., Desargues configurations and their collineation groups, Math. Proc. Cambridge Philos. Soc., 78, 227-246 (1975) · Zbl 0306.50009 |
| [5] | Gropp, H., On the history of configurations, (Dı́ez, A.; Echeverrı́a, J.; Ibarra, A., International Symposium on Structures in Mathematical Theories (1990), Univ. del Pais Vasco: Univ. del Pais Vasco Bilbao), 263-268 |
| [6] | Gropp, H., The construction of all configurations \((12_4, 16_3)\), Ann. Discrete Math., 51, 85-91 (1992) · Zbl 0767.05035 |
| [7] | Gropp, H., Non-symmetric configurations with deficiencies 1 and 2, Ann. Discrete Math., 52, 227-239 (1992) · Zbl 0767.05034 |
| [8] | Gropp, H., Non-symmetric configurations with natural index, Discrete Math., 124, 87-98 (1994) · Zbl 0791.05012 |
| [9] | H. Gropp, Configurations, in: C.J. Colbourn, J.H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996, pp. 253-255.; H. Gropp, Configurations, in: C.J. Colbourn, J.H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996, pp. 253-255. · Zbl 0836.00010 |
| [10] | Gropp, H., Configurations and \((r,1)\)-designs, Discrete Math., 129, 113-137 (1994) · Zbl 0801.05017 |
| [11] | H. Gropp, On the history of configurations II—Austria and the rest of the world, IV, Österreichisches Symp. zur Geschichte der Mathematik, Neuhofen/Ybbs, 1995, pp. 21-25.; H. Gropp, On the history of configurations II—Austria and the rest of the world, IV, Österreichisches Symp. zur Geschichte der Mathematik, Neuhofen/Ybbs, 1995, pp. 21-25. |
| [12] | H. Gropp, The drawing of configurations, in: F.J. Brandenburg (Ed.), Graph Drawing GD’95 Proceedings, Springer, Berlin, Heidelberg, New York, 1995, pp. 267-276.; H. Gropp, The drawing of configurations, in: F.J. Brandenburg (Ed.), Graph Drawing GD’95 Proceedings, Springer, Berlin, Heidelberg, New York, 1995, pp. 267-276. |
| [13] | Gropp, H., Configurations and their realization, Discrete Math., 174, 137-151 (1997) · Zbl 0892.51001 |
| [14] | Gropp, H., On some infinite series of \((r,1)\)-designs, Discrete Appl. Math., 99, 13-21 (2000) · Zbl 0942.05012 |
| [15] | Gropp, H., Die Configurationen von Theodor Reye in Straßburg nach 1876, (Toepell, M., Mathematik im Wandel, Mathematikgeschichte und Unterricht III (2001), Hildesheim: Hildesheim Berlin), 287-301 |
| [16] | D. Hilbert, S. Cohn-Vossen, Anschauliche Geometrie, Springer, Berlin, 1932, 2. Aufl. 1996.; D. Hilbert, S. Cohn-Vossen, Anschauliche Geometrie, Springer, Berlin, 1932, 2. Aufl. 1996. · JFM 58.0597.01 |
| [17] | Kantor, S., Die Configurationen \((3,3)_{10}\), Sitzungsber. Akad. Wiss. Wien, Math.-Nat. Kl., 84, 1291-1314 (1881) · JFM 13.0460.05 |
| [18] | F.W. Levi, Geometrische Konfigurationen, S. Hirzel, Leipzig, 1929.; F.W. Levi, Geometrische Konfigurationen, S. Hirzel, Leipzig, 1929. · JFM 55.0351.03 |
| [19] | Martinetti, V., Sulle configurazioni piane \(μ_3\), Ann. Mat. Pura Appl., 15, 1-26 (1887) · JFM 19.0587.02 |
| [20] | de Pasquale, V., Sui sistemi ternari di 13 elementi, Rend. Reale Ist. Lombardo Sci. Lett., 2, 32, 213-221 (1899) · JFM 30.0208.03 |
| [21] | Th. Reye, Geometrie der Lage I, C. Rümpler, Hannover, 1867, 2. Aufl. 1876.; Th. Reye, Geometrie der Lage I, C. Rümpler, Hannover, 1867, 2. Aufl. 1876. |
| [22] | Reye, Th., Das Problem der Configurationen, Acta Math., 1, 93-96 (1882) · JFM 14.0546.02 |
| [23] | H. Schröter, Ueber lineare Constructionen zur Herstellung der Configurationen \(n_3\); H. Schröter, Ueber lineare Constructionen zur Herstellung der Configurationen \(n_3\) |
| [24] | H. Schröter, Ueber die Bildungsweise und geometrische Construction der Configurationen \(10_3\); H. Schröter, Ueber die Bildungsweise und geometrische Construction der Configurationen \(10_3\) |
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