Configurations Between Geometry and Combinatorics

CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Discrete Applied Mathematics 138 (2004) 79–88 www.elsevier.com/locate/dam Conÿgurations between geometry and combinatorics Harald Gropp Universitat Heidelberg, Muhlingstr. 19, D-69121 Heidelberg, Germany Received 1 November 2000; received in revised form 24 January 2003; accepted 27 February 2003 Abstract The combinatorial structure conÿguration which was already deÿned as early as 1876 is the topic of this paper. In their early years conÿgurations were regarded to be ÿnite geometrical substructures of the plane. The best example is the wrong picture of a conÿguration 103 by Kantor. However, before 1890 conÿgurations were already established as purely combinatorial structures. In the 20th century the problem of realization of conÿgurations over ÿelds and of drawing of conÿgurations in the real plane has been investigated. ? 2003 Elsevier B.V. All rights reserved. Keywords: Conÿguration; Geometry; Realization; Drawing 1. Introduction The word “conÿguration” is quite often used in mathematics, in particular in combinatorics, as a synonym for substructure. However, it is a well-deÿned combinatorial structure like a graph or design. Moreover, it is indeed one of the oldest combinatorial structures, deÿned already in a geometrical context in 1876. A modern deÿnition is as follows. Deÿnition 1.1. A conÿguration (vr;bk ) is a ÿnite 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. E-mail address: [email protected] (H. Gropp). 0166-218X/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0166-218X(03)00271-3 80 H. Gropp / Discrete Applied Mathematics 138 (2004) 79–88 For further information see [9]; there are only two other books which contain information on conÿgurations [16,18], the former was reprinted recently. In this paper the development of conÿgurations which were born in geometry during the last 125 years is described and discussed. Section 2 describes the early years of conÿgurations and the conFict between their geometrical and combinatorial properties, represented by the wrong picture of Kantor. The breakthrough of Martinetti (see Section 3) opened the research of conÿgurations as purely combinatorial objects. However, the old geometrical origin of conÿgurations again occurs in the modern realization and drawing problems of our century (Section 4). The ÿrst part (Sections 2 and 3) contains several citations from original papers of the last century which are not so easily available today in order to enable the reader to follow the development from geometry to combinatorics. For further information see [5,11]. There is also a paper [15] in the German language which more explicitly describes the situation in StraHburg after 1871 as well as the possible connection of conÿgurations to Hilbert’s Grundlagen der Geometrie of 1899. The topic of Section 4 is discussed in more detail in [12,13] which contain further theoretical background, many references, and several drawings of conÿgurations. 2. Conÿgurations born in geometry 2.1. The book “Geometrie der Lage” The word conÿguration in the above sense was used in Theodor Reye’s book Geometrie der Lage for the ÿrst time. In the second edition of his book of 1876 [21] Reye mentioned conÿgurations as mathematical objects in their own right as gen- eralizations of the Desargues conÿguration 103. Von SJatzen uberJ das Dreieck will ich nur den folgenden nennen: Liegen zwei Dreiecke ABC und A1B1C1 so (Fig. 3), daH die Verbindungslinien AA1, BB1 und CC1 gleichnamiger Eckpunkte sich in einem und demselben Punkte S schneiden, so begegnen sich die gleichnamigen Seiten AB und A1B1, BC und B1C1, CA und C1A1 in drei Punkten C2, A2, B2 einer Geraden, und umgekehrt. The above citation in the ÿrst edition of Reye’s book describes the theorem of Desargues. In the second edition of 1876 the following sentence is included. Die den Satz erlJauternde Figur verdient Beachtung als ReprJasentant einer Gattung von merkwurdigen,J durch eine gewisse RegelmJassigkeit ausgezeichneten Conÿgu- rationen. Sie besteht aus 10 Punkten und 10 Geraden; auf jeder der Geraden liegen drei von den 10 Punkten, und durch jeden dieser Punkte gehen drei von den 10 Geraden. Here Reye describes the ÿgure related to the above theorem (which is the conÿgura- tion of Desargues) as the representative of a certain family of conÿgurations consisting H. Gropp / Discrete Applied Mathematics 138 (2004) 79–88 81 of 10 points and 10 lines such that there are three points on each line and three lines through each point. 2.2. Some biographical remarks Since most of the mathematicians who are involved in the research of conÿgurations do not belong to the well-known ones in the following there are some short biographical notes on some of them. Theodor Reye was born in RitzebuttelJ near Cuxhaven in 1838, was a professor of mathematics in Zurich,J Aachen, and (after 1872) in StraHburg, and died in WurzburgJ in 1919. Reye started his research in applied mathematics and physics. In ZurichJ he turned his interests towards geometry. Seligmann Kantor was born in Soborten near Teplitz (now Teplice in the Czech Republic) in 1857, studied in Wien, Roma, StraHburg, and Paris, and became a Pri- vatdocent in Praha in 1881. In 1886 he left academic life and lived in Italy. Vittorio Martinetti was born in Scorzalo near Mantova in 1859, became a professor of mathematics at the university of Messina in 1887, and died in Milano in 1936. Heinrich Eduard SchrJoter was born in KJonigsberg (now Kaliningrad in Russia) in 1829, studied in Berlin and KJonigsberg, became a professor in Breslau (now Wroc law in Poland) in 1858, and died in Breslau in 1892. 2.3. The ÿrst constructions and the “wrong” picture A few years later a formal deÿnition for conÿgurations ni was given in [22]. Eine Conÿguration ni in der Ebene besteht aus n Punkten und n Geraden in solcher Lage, dass jede der n Geraden i von den n Punkten enthJalt und durch jeden der n Punkte i von den n Geraden gehen. Reye still discusses conÿgurations as ÿnite geometrical substructures of the plane. This becomes totally clear when he discusses the existence problem of conÿgurations n3 for small values of n as follows. Reelle Conÿgurationen 83 existiren nicht; dagegen hat Herr S. Kantor drei ver- schiedenartige Conÿgurationen 93 und zehn verschiedene 103 nachgewiesen. In the case of conÿgurations 83 Reye remarks that there are no real conÿgurations. In fact, there is a unique conÿguration 83, however, not realizable over the real numbers. He mentions a result of Kantor that there are three diPerent conÿgurations 93 and ten diPerent conÿgurations 103. In the same paper Reye describes the “problem of conÿgurations” that is, of ÿnding all diPerent conÿgurations ni for given numbers n and i and discussing their properties as follows. Das Problem der Conÿgurationen nun verlangt, daH alle verschiedenartigen, zu den Zahlen n und i gehJorigen Conÿgurationen ni ermittelt und daH ihre wichtigsten Eigenschaften aufgesucht werden. 82 H. Gropp / Discrete Applied Mathematics 138 (2004) 79–88 Fig. 1. This deÿnition corresponds to the situation where v=b=n, i.e. the number of points is equal to the number of lines. In the plane the axiom that through two distinct points there is at most one line is automatically fulÿlled. Indeed, 1 year earlier already Kantor [17] had constructed all 10 conÿgurations 103. These are combinatorial structures having the same parameters as the Desargues conÿguration but are pairwise nonisomorphic. Although the constructions of Kantor are done from a combinatorial point of view and the geometrical aspect is not touched at all in his paper, at the end of the journal 10 drawings of these 10 conÿgurations in the plane are included. However, one of these drawings (see Fig. 1) is wrong (one line consists of two straight segments not forming an angle of 180◦). A few years later it was proved by SchrJoter (see below) for the ÿrst time that exactly one of these 10 conÿgurations cannot be drawn in the plane. However, it seems that to Kantor it was not clear whether there is a diPerence between the combinatorial and the geometrical aspect of a conÿguration. In his paper Kantor announces the construction of all conÿgurations 103 and 113.He does not use Reye’s notation which is now standard; as far as it is known the further paper on conÿgurations 113 of Kantor never appeared. In dieser und einer folgenden Abhandlung werde ich die sJammtlichen mJoglichen Formen der ebenen Conÿgurationen (3,3) aus 10, respective 11 Geraden und Punk- ten geben. ..... Es wird sich im Verlaufe unserer Raisonnements ergeben, daH es 10 essentiell verschiedene Gestalten der (3; 3)10 gibt, von denen bisher uberhauptJ nur eine bekannt war. To summarize these ÿrst years the diPerence between conÿgurations realized or drawn in real geometry and conÿgurations as purely combinatorial structures seems to H. Gropp / Discrete Applied Mathematics 138 (2004) 79–88 83 be unknown. Apparently even Reye thinks that there are 10 diPerent conÿgurations 103 in the real plane. 3. Conÿgurations as combinatorial structures It is not clear whether Reye remained interested in conÿgurations in the years which followed. Kantor left academic life and moved to Italy as mentioned above. The interest in conÿgurations moved to Breslau in Germany and to Italy. 3.1. The breakthrough of Martinetti Already a few years later, at the latest in Martinetti’s paper of 1887 [19], it is clear that conÿgurations are only considered from a combinatorial point of view. Mar- tinetti ÿnds a recursive construction method to construct all conÿgurations n3 if all conÿgurations (n − 1)3 are known and by using his method constructs all 31 con- ÿgurations 113.

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