The six (or seven) bridges of Kaliningrad: A personal Eulerian Walk, 2006

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Volumn 58, Issue 3, 2007, Pages 529-556

  Mallion, R B ^a  

^a UNIVERSITY OF KENT   (United Kingdom)

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EID: 35348962655     PISSN: 03406253     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Review

References (48)

1. th anniversary of Euler's birth thus fell on 15 April this year. This article is intended to be a small tribute to Euler on this auspicious occasion.
2. Expressed according to the Gregorian Calendar, already then current in Basle and Berlin, places where Euler had worked; however, as Euler died in St. Petersburg his death would have been reported locally on the Julian Calendar as having occurred on 7 September. Some reference-sources and web-sites report the date as such (frequently omitting to point out that this is a Julian date).
3. Other parts of the former East Prussia (some two-thirds of it) were assigned at Yalta and Potsdam to Poland; the Memelgebiet (Memelland, the most easterly part of East Prussia that had been retrieved by Germany in 1939) was annexed by Lithuania after 1945. Before the Second World War, the population of East Prussia was 2.4 million, 99 per cent of them ethnic Germans. By 1953, almost no Germans remained. By 1950, 1.35 million expellees lived in the German Federal Republic, with another 48,000 from the Memelgebiet. (See W. Benz [Editor], Die Vertreibung der Deutschen aus dem Osten, Fischer Taschenbuch, Frankfurt-am-Main, 1995).
4. Others presumably lived in the German Democratic Republic, or emigrated abroad. It is estimated (A. M. de Zayas, The German Expellees in War and Peace, Macmillan, New York and London, 1993) that at least 20 per cent of the population perished during the last few months of the war and in the years immediately afterwards. (These data and the references cited were kindly brought to my attention by Dr. Michael Glasby and Professor Ian Whittle.)
5. The Kneiphof (= a 'pub', 'students' drinking place', or 'hotel where students got drunk'), once a bustling little municipality in its own right, is now a verdant park, entirely bereft of buildings save for the Cathedral, currently in course of restoration. It does not appear even to have an official name in Russian: many people still seem to denote it by its former German name, whilst others refer to it as 'Pregel Island' or 'Kant Island', the latter name arising because the tomb of Königsberg's most famous son, Immanuel Kant (1724-1804), is in the Cathedral, on the island, and his university (the Albertina University [1544-1944]) was also there. Although the epitome of the former German culture in Königsberg, Kant, being a philosopher whom some orthodox communists thought to have influenced Marx, was, exceptionally, not considered persona non grata by the Soviets, and so 'Kant Island' was probably quite an acceptable name to them, not as redolent of the Prussian past as the Germanic 'Kneiphof' was. The other (much larger) island, landmass D - the Lomse Insel - is now called Oktyabrskiy (October Island).
6. See H. Sachs, M. Stiebitz, & R. J. Wilson, An historical note: Euler's Königsberg letters, Journal of Graph Theory 1988, 12, 133-139.
7. For some unknown reason, Ehler drew his map with south at the top and, in Figure 4, in order to make north appear at the top, Ehler's sketch has been rotated 180°. Of course, the consequence of this is that Ehler's hand-written labellings - in any case, almost illegible - are now upside-down! There are two mysteries about Figure 4 (which is extracted from Sachs, Stiebitz & Wilson [see Note 5]). One minor puzzle is that the sketch was sent by Ehler to Euler (with a letter dated 9 March 1736 - see Note 5) some six or seven months after the latter had presented the problem, and its solution, to the St. Petersburg Academy. When writing that letter to Euler, Ehler knew that Euler had solved the problem, and was writing to ask him for the solution. The second - and major - enigma is the appearance of dotted lines across the river, in the south-west corner of the map (as it is displayed in Figure 4), cryptically labelled 'holländischer Baum'. None of the authors of the paper mentioned in Note 5 could throw light on either of these matters when I raised them at Ilmenau on 27 March 2007, Professor Horst Sachs's eightieth birthday. This second mystery is perpetuated (by Euler himself) when, in his letter to G. J. Marinoni of 13 March 1736 (which, if Euler, based in St. Petersburg, were using Julian dates, was 24 March on the Gregorian Calendar, and so Euler would have been writing 15 days after Ehler had written his 9 March letter to Euler), Euler sketches a map of contemporary Königsberg and includes a single line across the river (definitely not indicating a bridge - he clearly and unambiguously depicts bridges by means of two, parallel, lines with shading between them); this line (a) is in the same position as the holländischer Baum of Ehler's map (approximately the location of the present-day railway-bridge), and (b) is somewhat tantalisingly labelled 'h'. Now, whether this is 'h' as an abbreviation for holländischer Baum or whether 'h' is used because that is the eighth letter of the alphabet, and was thus the next letter due to be adopted after Euler had previously labelled his seven bridges 'a' to 'g', is again not clear. This sketch by Euler, sent to Marinoni, is illustrated as Figure 3 of the paper by Sachs, Stiebitz & Wilson, cited in Note 5. (See also later, Note 19.)
8. L. Euler, Solutio problematis ad geometriam situs pertinents, (a) originally published by the St. Petersburg Academy in Commentarii Academiae Scientiarum Imperialis Petropolitanae 1736, 8, 128-140;
9. (b) reprinted in ibid., ed. nova, Bononiae 1752 8, 116-126;
10. (c) also reprinted in Opera Omnia, Series Prima, Opera Mathematica (29 vols), Vol. 7, pp. 1-10,
11. Sub Auspiciis Societatis Scientiarum Naturalium Helveticae, 1911-1956.
12. English translations of Euler's paper are to be found in the following three places: (i) J. R. Newman, Leonhard Euler and the Koenigsberg bridges, Scientific American 1953, 189, 66-70;
13. (ii) J. R. Newman (Editor), The World of Mathematics, Vol. 1, Simon & Schuster, New York, 1956, pp. 573-580;
14. (iii) N. L. Biggs, E. K. Lloyd & R. J. Wilson Graph Theory 1736-1936, Clarendon Press, Oxford, 1976, Chapter 1, pp. 3-8.
15. Robin Wilson has pointed out that close examination of this paper shows that, despite its well-known 1736 date, it was in fact not actually published until 1741; see the facsimile shown in Figure 2 of R. J. Wilson, An Eulerian trail through Königsberg, Journal of Graph Theory 1986, 10, 265-275. The facsimile of the first page of Euler's paper illustrated by Wilson is, however, from a later reprint of the paper (probably that in the the Omnia Opera, (c), cited above); the first page of the true original of Euler's paper is illustrated in Figure 5, here. Note that, on the frontispiece, illustrated in Figure 6, the actual publication date of 1741 - along with the purported, notional date of 1736 - is explicitly acknowledged. It may be remarked in passing that Euler had no fewer than ten other papers 'In Classe Mathematica' of this volume (8) of the Commentarii, in addition to the one on the Königsberg Bridges Problem. The only other author to publish - in his case, two papers - 'In Classe Mathematica' of that volume was Daniel Bernoulli; hence, both Euler and Bernoulli were in august company! Also emphatically worth reading is the fascinating, three-page preface to Volume 8 of the Commentarii, written on 1 December 1740 (Julian date) by Christian Goldbach (himself, as it happens, a native of Königsberg), directly and effusively praising the infant Tsar Ivan III (usually styled Ivan VI who, by then, had reigned for only six weeks), and, indirectly, through him, his mother, the Regent Anna Leopoldovna, expressing the hope that royal support for the Academy would continue. Goldbach poignantly addresses his dedication to the infant Emperor 'in the first year of Your age and Reign.' Declared successor on 5 October - at the age of less than two months - to his great-aunt Empress Anna, the baby Ivan became Emperor on her death only 12 days later, under the regency initially (until he was usurped, just three weeks after that) of a former éminence grise, E. J. von Biron, Duke of Courland, and then of Ivan's mother. (This was a regency which, however, as a result of a coup d'état on 6 December 1741 that overthrew the 15-month-old Tsar, was itself destined to last for little more than one further year). It was largely because of all this chaos, and the xenophobic backlash that followed Empress Anna's policy, during the preceding decade, of promoting Baltic Germans over Russians into prominent positions, that, in 1741 (the year of actual publication of his 'Königsberg' paper), Euler accepted Frederick the Great's invitation to move to the Berlin Academy.
16. Robin Wilson has, indeed, also emphasised that, despite common belief and apparently long-established 'folk-lore', Euler himself did not use the language of Graph Theory, and that, specifically, the graph of the Königsberg Bridges Problem depicted in Figure 8 was not illustrated in any publication until W. W. Rouse Ball's Mathematical Recreations and Problems of Past and Present Times, Macmillan, London, 1892.
17. (See (a) R. J. Wilson, An Eulerian trail through Königsberg, Journal of Graph Theory 1986, 10, 265-275;
18. (b) B. Hopkins & R. J. Wilson, The truth about Königsberg, The College Mathematical Journal [The Mathematical Association of America] 2004, 35, 198-207.)
19. J. J. Sylvester, Chemistry and algebra, Nature 1877-1878, 17, 284-284.
20. This account follows Euler's practice of using upper-case letters (A - D) for the landmasses, and lower-case letters for the bridges (a - g, initially; later, for two post-eighteenth-century bridges, the letters h and i will be used, and a twentieth-century motorway-bridge [known as the Estacada Bridge] will subsequently be denoted by the 'compound' symbol '(a,c)').
21. C. Hierholzer, Über die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren, Mathematische Annalen 1873, 6, 30-32.
22. An English translation of this paper is to be found in N. L. Biggs, E. K. Lloyd & R. J. Wilson, Graph Theory 1736-1936, Clarendon Press, Oxford, 1976, Chapter 1, pp. 11-12.
23. For some helpful insight into Hierholzer's proof, see also J. H. Barnett, Early writings on graph theory: Euler circuits and the Königsberg bridge problem, in J. Lodder, J. Barnett, G. Bezhanishvili, D. Pengelley, D. Ranjan, & H. Leung, Historical Projects in Discrete Mathematics and Computer Science, in B. Hopkins (Editor) Discrete Mathematics Resource Guide, The Mathematical Association of America, Washington D.C., in press. The book just cited is currently available on the following internet web-site at the New Mexico State University at Las Cruces: www.math.nmsu.eduhist_projects (note the 'underscore' separating 't' from 'p' in this web-site address); the article by J. H. Barnett referred to is on pp. 74-86 (with the references on pp. 104-107) of this web-site.
24. Sporadic attention was, however, paid to Euler's solution to this problem, in the succeeding years: more than a century later, for example, in the 1851 Nouvelles Annales de Mathématiques (1851, 10, 106-119), E. Coupy published a French translation of Euler's paper, entitled: Solution d'un problème appartenant à la géométrie de situation, par Euler, and he applied Euler's algorithm to the analogous problem of the Bridges of Paris, connecting the Rive Droite, the Rive Gauche and the islands of the River Seine.
25. 7 was also translated into German, in: A. Speiser, Klassische Stücke der Mathematik, Orell Füssli, Zürich & Leipzig, 1925, pp. 127-138.
26. P. A. Fowler, The Königsberg Bridges - 250 years on, American Mathematical Monthly 1988, 95, 42-43.
27. th Edition, Karl Baedeker, Coblenz, 1870, map between p. 338 and p. 339.
28. L. Saalschütz [untitled] Schriften der Physikalisch- Ökonomischen Gesellschaft zu Königsberg in Preußen 1876, 16, 23-24.
29. Robin Wilson has reported (R. J. Wilson, An Eulerian trail through Königsberg, Journal of Graph Theory 1986, 10, 265-275) that Saalschütz even went so far as to enumerate no fewer than 48 different possible Eulerian Walks, starting in the Kneiphof (A) and ending in the Lomse Insel (D).
30. It is perhaps stretching things somewhat, but Robin Wilson (R. J. Wilson, An Eulerian trail through Königsberg, Journal of Graph Theory 1986, 10, 265-275) has drawn attention to the fact that it has been observed
31. (R. Cooke, Letter to the Editor, American Mathematical Monthly 1984, 91, 662-664) that a map of Königsberg ca. 1800 indicates the existence of a ferry service that joined landmasses A and C. If this ferry service were counted as effectively being a 'bridge', this would have then made the vertex degrees A(6), B(3), C(4), D(3), thereby theoretically enabling an Eulerian 'Walk' that went from B to D (or vice-versa).
32. Baedeker's Germany. A Handbook for Railway Travellers and Motorists, Karl Baedeker, Leipzig, 1936, the map between p. 124 and p. 125.
33. According to an article entitled The Bridges and 'the Pregel's Odour' by Ivan Chechot of St. Petersburg, which is to be found on the Kaliningrad internet web-site http://ncca-kaliningrad.ru/art-guide/?by= p&lang=eng&au=022chechot, it seems that the present-day Reichsbahnbrücke is essentially the same as the one that stood during Third-Reich times, as Chechot reports that it was constructed in the period 1913-1926, the long building-time being attributed to a delay due to World War I. Chechot specifically confirms that cars and trams could pass on the lower tier of the bridge, and railway traffic on the upper tier. He claims, too, that there was also at one time a device that enabled the middle part of the bridge to be turned in order to allow ships to pass. This facility seems, though, now to have disappeared and Chechot states that the bridge is, these days, apparently static. However, in the web-site http://commons.wikimedia.org/ wiki/Image:Most.jpg this claim has been contradicted: it is stated there that the '... middle section of the bridge can be moved vertically to let ships pass'. Be that as it may, it is, however, almost certain that, as now, the Reichsbahnbrücke would have been available to any pedestrian who wished to undertake an Eulerian Walk in the mid-to-late 1930s.
34. 6 for the benefit of Euler. (Note: these 1930s maps say 'Holländer' NOT 'holländischer', as written by Ehler on his 1736 sketch [Figure 4].)
35. th anniversary of the city's foundation, [1255-2005]: MOCTbl KëHnrc6epra [The Bridges of Königsberg], Album IX, 12 postcards, Reklamno-informatsionniy Kholding "39 Region", Kaliningrad, 2005. [Web-site: www.39.ru; e-mail: koenigsberg@39.ru]).
36. The original Soviet intention had apparently been to incorporate the Kaliningrad Oblast into the Soviet Socialist Republic of Lithuania - which would have made sense because of previous historical connections between the two and, more obviously, because, physically, the Oblast is immediately adjacent to Lithuania. However, such was the ruined state of the area after World War II that the Soviet Socialist Republic of Lithuania apparently did not want it. The Kaliningrad Oblast was, accordingly, absorbed into the Russian Soviet Socialist Republic, as an exclave of that Republic.
37. 17,19 this was not determinable solely from an examination of the modern roadmap. The web-site http://commons.wikimedia. org/wiki/Image:Most.jpg did, though, display a photograph by Volkov Vitaly of the present-day railway-bridge and further claimed this railway bridge to be a two-tier one, capable of carrying pedestrians (as well as, incidentally, road vehicles) on its lower level, in addition to railway traffic on its upper level - a claim that we subsequently verified personally, in Kaliningrad itself, by direct inspection, 'on the ground'.
38. Neither was it shown on a large, folding city-map that we bought in Kaliningrad; however, closer inspection revealed that it was on a miniature tourist-map, given to us by the hotel reception.
39. Founded in 1255 by the Teutonic Knights and named 'Königsberg' in honour of King Přemysl Otakar II of Bohemia.
40. The Schloss was thought, in Soviet times, to be too representative of Prussian imperialism. Now, however, there are (somewhat ambitious and possibly unrealistic) plans to re-build it.
41. Wikipedia internet web-site on Kaliningrad: http://en.wikipedia.org/wiki/ Kaliningrad.
42. Usually, in English, spelt 'Philip' and (in any language) 'Melanchthon'.
43. C. J. Scriba, in Dictionary of Scientific Biography, American Council of Learned Societies, New York, 1970-1990, quoted in J. J. J. O'Connor & E. F. Robertson, Mac Biography of Carl Gustav Jacob Jacobi on the internet web-site http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Jacobi.html (University of St. Andrews).
44. Studiosus Kirchhoff, Ueber den Durchgang eines elektrischen Stromes durch eine Ebene, insbesondere durch eine kreisförmige, Annalen der Physik u. Chemie ('Poggendorf's Annalen') 1845, 64, 497-514. In the by-line to this paper, Kirchhoff uses his undergraduate denomination, i.e., Studiosus, in preference to the initial letter of his Christian name and - in what is presumably a reference to the research seminars initiated by Bessel, Jacobi and Neumann - Kirchhoff describes his affiliation as 'Mitglied des physikalischen Seminars zu Königsberg.'
45. G. Kirchhoff, Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird, Annalen der Physik u. Chemie ('Poggendorf's Annalen') 1847, 72, 497-508.
46. There is now a Russian (formerly, Soviet) University of Kaliningrad (founded 1967), situated in landmass C, the former Altstadt, which is entirely unconnected with the Albertina University of Königsberg (1544-1944). Even the Albertina University, however, so terminally physically devastated in 1944, seems to have survived, at least in memory, and perhaps in spirit: in note 91, on page 702 of C. Clarke, Iron Kingdom: the Rise and Downfall of Prussia, 1600-1947, Allen Lane, London, 2006, there is a reference to Jahrbuch der Albertus-Universität zu Königsberg/ Preußen, Vol 6, which, remarkably, is dated 1955. I am indebted to Mr. C. W. Haigh for very kindly drawing my attention to this reference, and to Mr. John Pidoux for further information/speculation on the matter.
47. However, my colleague Mr. John Pidoux, who has visited Kaliningrad more recently, informs me that trams no longer run over these bridges.
48. P. Taylor, What ever happened to those bridges? Mathematics Competition (Journal of the World Federation of National Mathematics Competitions) 2000, 13, 87-97. This article is also available on the following web-site of the Australian Mathematics Trust: http://www.amt.edu.au/koenigs.htm.