Equidecomposability of Polyhedra: A Solution of Hilbert’s Third Problem in Kraków before ICM 1900

Equidecomposability of Polyhedra: A Solution of Hilbert’s Third Problem in Kraków before ICM 1900 Equidecomposability of Polyhedra: A Solution of Hilbert’s Third Problem in Krakow before ICM DANUTA CIESIELSKA AND KRZYSZTOF CIESIELSKI uring the International Congress of Mathematicians scissors congruent). The notion of rearrangement covers in Paris in 1900, David Hilbert gave a talk on the here only translation or rotation, no part can be flipped over. D problems that, in his opinion, would influence The story started at the beginning of the 19th century, when mathematics in the 20th century. Later, in the printed William Wallace posed this question in 1807. The first proof record of his address [21], twenty-three problems were of the theorem was due to John Lowry [32] in 1814. Inde- presented, now known as Hilbert’s Problems. Hilbert’s pendently, it was proved by Farkas Bolyai in 1832 [8] and by Third Problem concerned polyhedra: given two polyhedra Paul Gerwien in 1833 [17]. Much later, in the 20th century, of equal volumes, is it possible to cut one of them by means Laczkovich proved that any plane polygon can be cut into of planes into finitely many polyhedral pieces that can be finitely many polygonal parts, which can be reassembled reassembled into the other polyhedron? In the sequel, we using only translation into a square equiareal with the orig- will call such polyhedra equidecomposable. Max Dehn inal polygon (see [26], [27]). What about the analogous soon solved the problem. However, this problem had been problem in three-dimensional space? solved in Krako ´ w about twenty years earlier, and neither Hilbert nor Dehn could have known about that. Gerling’s Solution of a Particular Case Let us quote the original formulation of the problem stated The Planar Case by Hilbert in 1900: In the mid-19th century it was common knowledge that any In two letters to Gerling, Gauss expresses his regret two polygons of equal areas could be dissected by lines into that certain theorems of solid geometry depend upon finitely many polygonal pieces that could be rearranged one the method of exhaustion, i.e., in modern phraseol- into the other, that is, they were equidecomposable (or ogy, upon the axiom of continuity (or upon the 2018 The Author(s). This article is an open access publication, Volume 40, Number 2, 2018 55 https://doi.org/10.1007/s00283-017-9748-4 axiom of Archimedes). Gauss mentions in particular the theorem of Euclid, that triangular pyramids of equal altitudes are to each other as their bases. Now the analogous problem in the plane has been solved. Gerling also succeeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent parts. Nevertheless, it seems to me probable that a general proof of this kind for the theorem of Euclid just mentioned is impossible, and it should be our task to give a rigorous proof of its impossibility. This would be obtained as soon as we succeeded in specifying two tetrahedra of equal bases and equal altitudes which can in no way be split up into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split up into congruent tetrahedra. The correspondence between Christian Ludwig Gerling and Carl Friedrich Gauss mentioned previously was pub- lished a long time after ICM 1900, in 1927, by C. Schaeffer [10]. Gerling proved that two symmetric tetrahedra are equidecomposable. As in the planar case, only translations Figure 1. Christian Ludwig Gerling (https://en.wikipedia.org/ and rotations were allowed and no part could be wiki/Christian_Ludwig_Gerling#/media/File:Christian_Ludwig_ transferred by a mirror reflection, which made the problem Gerling.jpg). far from trivial. Gerling sent his solution to Gauss. Let us quote here a letter from Gauss to Gerling, written in Go ¨ttingen on April 17, 1844. The letter is essential for the positively solved. The proof was based on an idea that story, however it is seldom cited in the literature. may be easily illustrated in the planar case. Assume that It was with great pleasure that I read your proof that two triangles are given, such that one of them is the symmetrical but not congruent polyhedra are of reflection image of the other. They can be easily divided equal volume. This is a thorny problem indeed. You into smaller triangles to get six pairs of congruent triangles might say that: (1) any tetrahedron could be dissected (see Fig. 2). The common point of the small triangles in such a way that a certain part is congruent to a part contained in the big triangle is the center of the circle of the second tetrahedron and that two faces of these inscribed in this triangle. Gerling divided a big tetrahedron are perpendicular to a third face and intersect at into twelve small ones with their common point being the equal angles, and (2) it can be shown that any center of the sphere inscribed in the large tetrahedron, and tetrahedron could be dissected into 12 parts using their bases given by a suitable division of the large this construction. tetrahedron’s faces into congruent triangles. Of course, I cannot say that this reasoning is new. Gerling’s method is very clever and, despite the doubts You might consult ‘‘Geometry’’ by Legendre, where of Gauss, original. Unfortunately, as Gauss indicated, the many proofs are given in a more general or easier method does not answer the question of whether every form than before. Unfortunately, I am not in pos- two tetrahedra of equal volumes are equidecomposable. session of this book and I do not have access to it at This question, to be posed again by Hilbert in 1900, was this moment. very natural and well known in the mathematical com- munity in the second half of the 19th century. However, it I must express my regret that your proof does not remained unsolved. lead to a simplification of other theorems in stere- ometry, namely those which depend on the method The Contest in Krako´w of exhaustion, see Book 12 of Euclid, Chapter 5. Against this background, let us now move to the Polish city Perhaps something can still be improved, but unfor- of Krako ´ w, then in the Austro-Hungarian Empire. In 1872, tunately I have no time now to consider this matter the Academy of Arts and Sciences was established there. It further. acted as an institution bringing together Polish scientists Gerling’s solution was correct and the question of whether from the whole of Europe. It stayed in touch with other two irregular tetrahedra such that one of them is a mirror scientific academies in Europe and later admitted as for- reflection of the other can be divided into a finite number eign members many outstanding scientists, among them of pairwise congruent polyhedra (tetrahedra) was The demonstration showing Gerling’s dissection is available at http://demonstrations.wolfram.com/Gerlings12PieceDissectionOfAnIrregularTetrahedronIntoItsMirr/. In 1918, when Poland regained independence, the Academy changed its name to the Polish Academy of Arts and Sciences (Polska Akademia Umieje ˛ tnos ´ ci—PAU). In 1952, under the strong pressure from political authorities, the Academy was forced to cease its activities. The renewal of the Academy took place in 1989. 56 THE MATHEMATICAL INTELLIGENCER Figure 2. Gerling’s solution illustrated in the planar case. mathematicians such as Elie Cartan, Henri Lebesgue, Godfrey Harold Hardy, and Maurice Frechet. On June 12, 1882, the Academy announced a math contest. Władysław Kretkowski (see below for more information about him) presented to the Academy two problems and offered prizes for their solutions. In the ‘‘Report of the Activities of the Mathematics and Natural Sciences Division of the Academy, vol. 11’’ [29], we can read about the result of this competition, as presented at the meeting of the Division on February 20, 1884, by ´ Figure 3. The first page of Birkenmajer’s submission. Scien- Franciszek Karlinski. He announced that nobody had answered the first question, concerning algebra, with the tific Library of PAU and PAN, Ref. ms. 6828 (with permission). prize of 1000 French francs. However, two geometers had sent solutions to the second problem (the prize was 500 francs). In accordance with the rules of the contest, the authors had signed their papers with pseudonyms. The A description of the author’s reasoning follows. Kar- report then moves on to a description of the problem and linski concludes that the jury considered the second the solutions. submission as a paper of high quality that provided a The problem was: solution to the problem and deserved the reward. The Given any two tetrahedra with equal volumes, sub- Division decided to give the prize to the author who had divide one of them by means of planes, if it is sent the paper under the pseudonym ’AEI ‘O HEOR possible, into the smallest possible number of pieces CEXMETPEI. The name of the author of the solution is that can be rearranged so as to form the other tetra- not mentioned in the report. hedron. If this cannot be done at all or can be done When the competition results were announced, it only with certain restrictions, then prove the impos- turned out that the problem had been solved by Ludwik sibility or specify precisely those restrictions. Antoni Birkenmajer, then a 28-year-old teacher of math- As we can see, this is precisely Hilbert’s Third Problem. ematics. The most important question is: was the solution What about the solutions received? correct? In the report, the reasoning is described on two In the report we can read that the first author signed pages. This description indicates that the problem was his submission ‘‘Eureka.’’ The description of the reasoning solved indeed, but this was only a kind of summary. was followed by the final conclusion: ‘‘As we see, the Birkenmajer never published his result. The Science Archives of PAN and PAU only contain a letter from author did not solve the general problem. He described only one case and even this one under some additional Kretkowski in which he suggested announcing the competition. assumptions. So, the paper does not meet the conditions of the competition and cannot be awarded the prize.’’ The Fortunately, we managed to find in the Scientific Library of PAU and PAN the original manuscript by Birkenmajer. report reads further: ‘‘The second paper is completely dif- ferent. It is rigorously scientific, contains 40 pages and 7 Now it is sure that Birkenmajer [6] really presented a correct figures, arranged into three chapters.’’ proof. The Academy comprised then three divisions; the others were concerned with Philology and with History and Philosophy. Franciszek Karlin ´ ski (1830–1906), Polish astronomer, mathematician, and professor of the Jagiellonian University. This is a quotation from Plato and means ‘‘God always geometrizes.’’ Polish Academy of Sciences (Polska Akademia Nauk—PAN), created in 1951. This is no mistake. Indeed, the order of the abbreviations in the official name of the Archives is ‘‘of PAN and PAU’’ and of the Library it is ‘‘of PAU and PAN.’’ 2018 The Author(s). This article is an open access publication, Volume 40, Number 2, 2018 57 characterizing T and T such that four of these Birkenmajer’s Solution 1 2 invariants are independent. If this is satisfied, there Let us describe in brief the idea of Birkenmajer’s forty-page exists just one plane that solves the problem. reasoning. He shows that the required construction is Finally, in the third part of the paper, Birkenmajer connects possible only under some additional assumptions. his previous results with the properties of dihedral angles In the first chapter, he takes a tetrahedron and cuts it by of tetrahedra and gives a partial answer to the question: ‘‘If planes to obtain sections. Such a section may be either a this can be done only with certain restrictions, then specify triangle or a quadrilateral. In the majority of cases, an precisely those restrictions.’’ He proves that there are two obtained quadrilateral is not a parallelogram and we may particular cases when the problem can be solved, that is extend two sides of this quadrilateral and one edge of the tetrahedron to get a new tetrahedron (see Fig. 4). Now, (1) when one face of the tetrahedron is an isosceles Birkenmajer investigates several polyhedra he obtained as triangle and the dihedral angle at the base of the a result of cutting the original tetrahedron by planes and triangle is a right angle, or using the just-mentioned construction. For all of them, he (2) when one face of the tetrahedron is an isosceles uses Euler’s formula on polyhedra. After some calculation triangle and the angle between the bisector of the he comes to interesting formulas connecting the number of vertex angle of this triangle and the edge opposite to its those polyhedra, their faces, edges, and vertices. base is a right angle. In the second chapter, entitled ‘‘The Conditions for the Solvability of the Problem,’’ Birkenmajer’s purpose is to The paper also contains a very interesting appendix. In discover some parameters that will enable him to find the this part Birkenmajer notes that his method of solving the invariants of equidecomposability. Using a formula from problem could in fact be seen as algebraic—that is, the the first chapter, he obtains some conditions wherein there problem may be reduced to a question about a polynomial are as many as 36 parameters! The number of conditions function. Moreover, Birkenmajer writes that this solution that must be satisfied to get tetrahedron T from tetrahe- could be extended to any convex polyhedron. He notices dron T is equal to that the methods of calculus are not suitable for the W ¼ 2nðn  1Þðn  2Þþ 6nðn  1Þþ 9n; investigation of this problem. Unfortunately, his knowledge of algebra was rather poor and he was not able to give the in which n is the number of planes used in cutting T . The necessary conditions for equidecomposability of two author indicates that the existence of a required division polyhedra in polynomial form. depends on the existence of solutions to suitable Dio- We should also note that the construction of an addi- phantine equations. He describes the way to find the tional tetrahedron used by Birkenmajer in Chapter 1 was appropriate construction in a particular case. correct, but not necessary, and it needlessly complicated This leads him to the solution of the main problem, that the solution. It was indicated by the referee on the margin is, that two tetrahedra with equal volumes may not be of the paper, where it was noted that the same result might equidecomposable. He writes: be obtained if the author considered only polyhedra made The task is only possible when there are at least 7 by cutting (see Fig. 5). conditional equations involving 11 invariants Figure 5. A referee’s remark on Birkenmajer’s solution. Figure 4. A part of the fifth page of Birkenmajer’s solution. Scientific Library of PAU and PAN, Ref. ms. 6828 (with Scientific Library of PAU and PAN, Ref. ms. 6828 (with permission). permission). 58 THE MATHEMATICAL INTELLIGENCER Birkenmajer and Kretkowski Who was the author of the solution and who was the person who offered the prize? The names of several Polish mathematicians are well known throughout the world. However, the names of Birkenmajer and Kretkowski are generally not among these. Nevertheless, both mathemati- cians deserve attention. Ludwik Antoni Birkenmajer (1855–1929) was born in Lipsko near Lvov. He studied at the University in Lvov and in 1879 he obtained his Ph.D. from this university. Then he continued studies in Vienna. From 1880 to 1909 Birken- majer taught mathematics and physics at the agricultural ´ ´ gymnasium in Czernichow (near Krakow). Simultane- ously, after habilitation in 1881 at the Jagiellonian University in Krakow, he became a Privatdozent at the Jagiellonian University and he lectured in mathematical physics. In 1897, a Chair of the History of Exact Sciences was created for Birkenmajer at the Jagiellonian University. He was a professor of this university until his death. He was also a member of the Academy of Arts and Sciences in Krako ´ w (since 1893) and a member of the International Astronomical Union in Oxford. Birkenmajer held a very broad spectrum of scientific interests. His Ph.D. dissertation concerned mathematics: it Figure 6. Ludwik Antoni Birkenmajer as a student. Courtesy: was entitled On General Methods of Integration of Differ- Krzysztof Birkenmajer (with permission). entials. His habilitation thesis concerned physics and was devoted to the structure of the Earth. He there compared experimental data to Laplace’s conjectures on the relation between the density and pressure inside the Earth. But Birkenmajer was also an astronomer, geographer, and geo- physicist. He wrote many papers about mathematical models of the shape of the Earth. He made several interesting investigations of the temperature of the water in lakes in the Tatra mountains. Nevertheless, he first was a historian of science and is best known for his achievements in this area. He was particularly interested in the 15th and 16th centuries. His very detailed research on the life and work of Nicolaus Copernicus revealed several facts about Copernicus that were unknown at that time. Birkenmajer published many papers and books about this subject. His book on Copernicus [3] had 736 pages and to date is regarded by historians of science as an extraordinary treatise. More details about Birkenmajer’s work on Copernicus can be found in [18]. In 2011, the Institute for the History of Science of the Polish Academy of Sciences (created in 1954) was named for him and his son, Aleksander Birkenmajer. Concerning his results in mathematics, Birkenmajer was, above all, an applied mathematician. However, he also wrote some papers in pure mathematics, mainly on prime numbers. In particular, he proved that if p is a prime greater 1 1 1 than 3 and the sum þ þ ... þ is presented as a Figure 7. Ludwik Antoni Birkenmajer. PAUart BZS.RKPS. 1 2 p1 fraction, then its numerator is divisible by p ; moreover, 12519.k.8 (with permission). only primes have this property [4]. In another article [5], he analyzed a remainder after division of ð2k  1Þ! by 4k  1 where 4k  1 is a prime and solved a problem stated by area. Unfortunately, all his mathematical papers were Henri Lebesgue. He gave a characterization of Heronian published in Polish, like the report on the solution of the triangles, that is, triangles having integer sides and integer equidecomposability problem [29]. A type of secondary school with a strong vocational focus. Aleksander Birkenmajer (1890–1967), a historian of science and library scientist, professor of the Jagiellonian University and Warsaw University. 2018 The Author(s). This article is an open access publication, Volume 40, Number 2, 2018 59 Figure 8. The grave of Birkenmajer in Rakowicki Cemetery in Figure 9. Władysław Kretkowski. PAUart BZS.RKPS.6818.k.8 Krako ´ w. (Photo by the authors). (with permission). pioneers of modern applications of mathematics in Poland. The name of Władysław Kretkowski (1840–1910) is even Although his achievements were not comparable with the less known, in Poland and elsewhere. results of leading mathematicians of his period, he defi- Kretkowski was born in Wierzbinek near Torun ´ . From nitely was a good mathematician who obtained valuable 1865 to 1867 he studied at the Sorbonne in Paris, from results. which he obtained Diplo ˆme de Licencie e`s Sciences However, Kretkowski also deserves a place in the his- Mathe´matiques. He also graduated from the School for ´ tory of mathematics at the Jagiellonian University for Bridges and Roads (Ecole imperiale des ponts et chausse´es, ´ another reason. He was a very rich Polish nobleman and he now Ecole des Ponts Paris Tech). In Paris, he became devoted considerable means to supporting science, in acquainted with several open mathematical problems. particular mathematics. The contest described earlier with During his studies he published in Nouvelles Annales de the awards financed by Kretkowski was not his only con- Mathematiques solutions of geometrical problems stated by tribution: he supported scientists with many scholarships. Hieronymus Georg Zeuthen and Ludvig Oppernmann [24]. He died in 1910 and in his last will he bequeathed all his In 1879 he applied for a Ph.D. at the University in Lvov and, estate for the development of mathematics. The Academy although he passed the mathematical exam with distinc- of Arts and Sciences was asked to administer scholarships tions and presented eight papers published in Paris and dedicated for studies in leading European mathematical Krako ´ w, he did not obtain a doctorate there (for the story, centers. In his will, he had also included a donation for the see [13]). In 1882 he got a Ph.D. in mathematics from the Jagiellonian University, which made it possible to establish Jagiellonian University on the basis of a dissertation on the a chair in mathematics. And finally, he left his entire valu- applications of functional discriminants in calculus. The able collection of mathematical books—over two thousand paper was refereed by Franciszek Karlinski and Franciszek volumes, most of which were modern—to the mathemati- Mertens (the author of the famous Mertens conjecture; in cal library of the university. 1865–1884 Mertens was a mathematics professor at the Jagiellonian University). Bricard, Hilbert, and Dehn Kretkowski published more than 20 articles in mathe- In 1896, twelve years after Birkenmajer’s result, Raoul Bri- matics. In his papers, he proved several theorems on geometry, analysis, theory of polynomials, and analytical card published the paper [9] in which he proposed a much more general answer for any two polyhedra. The propo- functions. He introduced a very clever method of deter- mining the center and radius of a sphere circumscribed sition crucial for solving the problem (now called Bricard’s condition) states: around an ndimensional simplex [25]. His 57-page-long treatise about determinants published as an appendix in If two polyhedra are equidecomposable, then there exist positive integers n , m , and integer p such that [16] was complimented by Thomas Muir in the fundamental i j monograph on the development of the theory of determi- n a þ n a þ .. .þ n a ¼ m b þ m b þ ...þ m b þ pp; 1 1 2 2 q q 1 1 2 2 l l nants [28]. Kretkowski may also be regarded as one of the 60 THE MATHEMATICAL INTELLIGENCER Figure 11. Max Dehn (https://www.geni.com/people/Max- W-Dehn/6000000000128799755). Courtesy: Mary Proctor Dehn and Christopher Winter (with permission). Figure 10. The caricature of Bricard drawn by Jean-Robert (Public Domain; https://eo.wikipedia.org/wiki/Raoul_Bricard). prisms are zero. However, the Dehn invariant of a regular tetrahedron is not equal to zero. Dehn published his result in two papers. In the first one, where a are measures of dihedral angles of the first published in 1900 [14], he described two polyhedra with polyhedron and b are measures of dihedral angles different Dehn invariants. In the second one, published two of the second polyhedron. years later [15], he proved that two equidecomposable polyhedra must have the same invariants. It is worth noting Unfortunately, Bricard’s reasoning was not correct (see that Dehn’s work is also about Diophantine equations. In [2] and [7]). Soon after, in 1997, Giuseppe Sforza also 1903 Benjamin F. Kagan presented a modified version of published an example of nonequidecomposable polyhedra Dehn’s proof [22]. Kagan’s version was easier to follow by [30]. the reader. For a clear and complete description of Dehn’s Anyway, in 1900 the well-known problem of equide- method and invariants, see [1], [7], and [20]. composability of two tetrahedra was still regarded as Dehn’s method of solution was completely different open. The almost 20-year-old solution by Birkenmajer from that presented by Birkenmajer and from the idea remained unpublished. Birkenmajer was not working as a suggested by Bricard. Moreover, Dehn’s solution to Hil- mathematician and his activities took place far from bert’s Third Problem had one more advantage, from the mathematical centers. Information about his proof was point of view of Bricard’s reasoning: it proved that Bricard’s published only in Polish, in a report of a meeting of the condition was true. Note that Dehn was familiar with the Academy. No wonder that almost nobody knew about it. work of Bricard and Sforza; in [15] he mentioned their Moreover, the problem fitted into an important discussion papers and commented on the results they obtained. of the foundations of geometry that Hilbert participated This could have been the end of the story. However, in. Then Hilbert included the equidecomposability prob- things turned out differently. lem in his famous list. Hilbert’s Third Problem was solved quite soon by Max Dehn who, roughly speaking, found another invariant of Many Years Later equidecomposability which depended on all the dihedral As we know, a special place is held in mathematics by angles of the polyhedron and on the lengths of its edges conditions that characterize some objects in unique form, (for a precise definition see, for example, [1] and [23]). Now that is, necessary and sufficient conditions for some prop- it is called the Dehn invariant. Another obvious invariant erties, as in the Poincare conjecture. For many years, it was was the volume of the polyhedra. Accordingly, a necessary known that the preservation of the volume and Dehn’s condition for equidecomposability of two polyhedra was invariant were necessary for equidecomposability. It was the equality of their volumes and of their Dehn invariants. It great news when it was proved about 60 years later (in an was clear that the Dehn invariants of the cube and all almost forty-page-long paper) by Sydler [31] that these 2018 The Author(s). This article is an open access publication, Volume 40, Number 2, 2018 61 Ludwik and Aleksander Birkenmajer Institute conditions are sufficient. This result is now called Sydler’s theorem. Also, Hilbert’s Third Problem was the basis for for the History of Science further research (see very interesting expository articles [11] Polish Academy of Sciences and [12]). Nowy Swiat 72 In recent years, following the progress of different 00-330 Warszawa branches of mathematics, other equivalent conditions for Poland equidecomposability were found (see [23]). Thus it turned e-mail: smciesie@cyfronet.krakow.pl out that the Third Problem, which seemed to be much simpler than the other Hilbert’s Problems, was very stim- Mathematics Institute, Jagiellonian University ulating for the development of mathematics, even one Łojasiewicza 6 century later. 30-348 Krako ´ w Also one century later, another interesting result con- Poland cerning this topic was obtained. Until 2007 there was no e-mail: krzysztof.ciesielski@im.uj.edu.pl direct proof of Bricard’s condition. In each published proof it was a consequence of the solution of Hilbert’s Third OPEN ACCESS Problem. It was only in 2007 that a paper was published by This article is distributed under the terms of the Creative Benko (see [2]) in which he wrote: Commons Attribution 4.0 International License (http:// In this article we give a short direct proof of Bricard’s creativecommons.org/licenses/by/4.0/), which permits condition that was overlooked for a century. There- unrestricted use, distribution, and reproduction in any fore it provides a new solution to Hilbert’s problem. medium, provided you give appropriate credit to the ori Our proof is completely elementary. Since it uses no ginal author(s) and the source, provide a link to the linear algebra, it could even be presented in a high- Creative Commons license, and indicate if changes were school math club. made. It turned out that, if we suitably modify Bricard’s condition, we can get a different, necessary and sufficient condition for equidecomposability. So, after more than one hundred REFERENCES years, the gap in another method of solving Hilbert’s Third [1] M. Aigner, G. M. Ziegler, Proofs from the Book, Springer-Verlag, Problem was closed! Berlin Heidelberg 2001. It’s time for a conclusion. Speaking about Hilbert’s [2] D. Benko, A New Approach to Hilbert’s Third Problem, The Third Problem, it is definitely Dehn who should be con- American Mathematical Monthly 114(2007), 665–676. sidered the person who solved it. He was the first one to [3] L. A. Birkenmajer, Mikolaj Kopernik. Cz. 1.: Studya nad pracami publish the correct proof. Moreover, his solution was very Kopernika oraz materyaly biograficzne, Skład Gło ´ wny w Ksie ˛ garni stimulating for mathematics and immediately led to the Spo ´ łki Wydawniczej Polskiej, Krako ´ w 1900. answer to another open problem. Nevertheless, it is good [4] L. Birkenmajer, O pewnem twierdzeniu z teoryi liczb, Prace to know that almost twenty years earlier yet another solu- Matematyczno-Fizyczne 7(1896), 12–14. tion of the problem had been given by Ludwik Antoni [5] L. A. Birkenmajer, O zwia ˛ zku twierdzenia Wilsona z teorya ˛ reszt Birkenmajer. Birkenmajer not only showed an example of kwadratowych, Rozprawy Wydzialu Matematyczno-Przyrodnic- nonequidecomposable polyhedra, but also give suit- able invariants. In print, it was only announced and zego Akademii Umieje ˛ tnos ´ ci, 57A(1918), 137–149. summarized (in Polish, a language not widely known by [6] L. Birkenmajer, Zadanie konkursowe z geometryi podane przez p. the mathematical community), but the manuscript exists Dra W. Kretkowskiego, Krako ´ w 1883, Archives of PAN and PAU and shows that Birkenmajer’s reasoning was completely ´ in Krakow. Ref. ms. 6828. different from Dehn’s, Bricard’s, and Sforza’s, and, what is [7] V. G. Boltianskii, Hilbert’s Third Problem, V. H. Winston & Sons most important, correct. (Halsted Press, John Wiley & Sons), Washington DC, 1978 Let us finish with an interesting fact, in some way con- (translated from the Russian version: Tret’ja problema Gil’berta, nected with this story. It is not common knowledge that Nauka, Moscow 1977). Max Dehn was probably the first mathematician to give a [8] W. Bolyai de Bolya, Tentamen. Iuventutem studiosam in ele- correct proof of the Jordan Curve Theorem for polygons menta matheseos purae, elementaris ac sublı`mioris, methodo (see [19]). However, his manuscript containing this result, intuitiva, evidentiaque huic propria, introducendi. Cum appendice dating from 1899, remained unpublished. triplici, Maros Va ´ sa ´ rhelyini; tomus primus 1932, tomus secudus [9] R. Bricard, Sur une question de ge ´ ome ´ trie relative aux polyedres, ACKNOWLEDGMENTS ´ Nouvelles annales de mathe ´ matiques, Ser. 3, 15(1896), s. 331– We thank Krzysztof Kwasniewicz for his help. We also thank the referee for helpful comments. 62 THE MATHEMATICAL INTELLIGENCER ¨ [10] Briefwechsel zwischen Carl Friedrich Gauss und Christian [22] B. Kagan, Uber die Transformation der Polyeder, Math. Ann. Ludwig Gerling Correspondence of Gauss and Gerling), C. 57(1903), 421–424. Schaeffer (ed.), Berlin 1927, Letter No. 343. [23] R. Kellerhals, Old and new about Hilbert’s Third Problem, in: ´ ` [11] P. Cartier, Decomposition des polyedres: Le point sur le European women in mathematics (Loccum, 1999), 179–187, ` ` ´ troisieme probleme de Hilbert, Asterisque 133/134(1986), 261– Hindawi Publ. Corp., Cairo, 2000. 288. [24] L. Kretkowski, Question 769 et 770, Nouvelles Annales de ` ` ´ ´ [12] J.-L. Cathelineau, Quelques aspects du troisieme probleme de Mathematiques ser. 2, 6(1867), 227–231. Hilbert, Gaz. Math., Soc. Math. Fr. 52(1992), 45–71. [25] W. Kretkowski, Rozwia ˛ zanie pewnego zadania z geometryi [13] D. Ciesielska, Sprawa doktoratu Władysława Kretkowskiego, wielowymiarowej, Pamie ˛ tnik Towarzystwa Nauk Scisłych Antiquitates Mathematicae 6(2012), 7–37. 12(1880), 3 pages. [14] M. Dehn, Ueber raumgleiche Polyeder, Nachrichten von der [26] M. Laczkovich, Equidecomposability and discrepancy: a solution Ko ¨ nigl. Gesellschaft der Wissenschaften zu Go ¨ ttingen, Mathe- to Tarski’s circle squaring problem, Journal fu ¨ r die reine und matisch-physikalische Klasse 1900, 345–354. angewandte Mathematik 404(1990), 77–117. [15] M. Dehn, Uber den Rauminhalt, Math. Ann. 55(1902), 465–478. [27] M. Laczkovich, Paradoxical decompositions: a survey of recent [16] W. Folkierski, Zasady rachunku ro ´ z _ niczkowego i całkowego, nakł. results, Proc. First European Congress of Mathematics, Vol. II Biblioteki w Ko ´ rniku, Paris 1870. (Paris, 1992), Progress in Mathematics 120, Birkha ¨ user, Basel [17] P. Gerwien, Zerschneidung jeder beliebigen Anzahl von gleichen 1994. geradlinigen Figuren in dieselben Stu ¨ cke, Journal fu ¨ r die reine [28] T. Muir, The Theory of Determinant in the historical order of und angewandte Mathematik 10(1833), 228–234. development, vol. III, The period 1861 to 1880, Macmillan, [18] A. Goddu, Ludwik Antoni Birkenmajer and Curtis Wilson on the London 1920. Origin of Nicholas Copernicus’s Heliocentrism, Isis 107(2016), [29] Rozprawy i Sprawozdania z Posiedzen ´ Wydzialu Matematyczno- 225–253. Przyrodniczego Akademii Umieje ˛ tnos ´ ci 11(1884), 87–92. [19] H. Guggenheimer, The Jordan Curve Theorem and an unpub- [30] G. Sforza, Un’ osservazione sull’ equivalenza dei poliedri per lished manuscript by Max Dehn, Archive for History of Exact congruenza delle parti, Periodico di matematica 12(1897), 105– Sciences, 17(1977), 193–200. 109. ´ ´ [20] H. Hadwiger, Zum Problem der Zerlegungsgleichheit der Poly- [31] J.-P. Sydler, Conditions necessaires et suffisantes pour l’equiv- eder, Arch. Math, 2(1950), 441–444. alence des polyedres de l’espace euclidean a trois dimensions, [21] D. Hilbert, Mathematical Problems. Lecture delivered before the Comment. Math. Helv. 40(1965), 43–80. International Congress of Mathematicians at Paris 1900 (trans- [32] W. Wallace, J. Lowry, Question 269, New Series of the Mathe- lated by M. W. Newson), Bulletin of American Mathematical matical Repository 3(1814). Thomas Leybourn (ed.), 44–46. Society 19(1902), 437–479. 2018 The Author(s). This article is an open access publication, Volume 40, Number 2, 2018 63 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Mathematical Intelligencer Springer Journals

Equidecomposability of Polyhedra: A Solution of Hilbert’s Third Problem in Kraków before ICM 1900

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Equidecomposability of Polyhedra: A Solution of Hilbert’s Third Problem in Krakow before ICM DANUTA CIESIELSKA AND KRZYSZTOF CIESIELSKI uring the International Congress of Mathematicians scissors congruent). The notion of rearrangement covers in Paris in 1900, David Hilbert gave a talk on the here only translation or rotation, no part can be flipped over. D problems that, in his opinion, would influence The story started at the beginning of the 19th century, when mathematics in the 20th century. Later, in the printed William Wallace posed this question in 1807. The first proof record of his address [21], twenty-three problems were of the theorem was due to John Lowry [32] in 1814. Inde- presented, now known as Hilbert’s Problems. Hilbert’s pendently, it was proved by Farkas Bolyai in 1832 [8] and by Third Problem concerned polyhedra: given two polyhedra Paul Gerwien in 1833 [17]. Much later, in the 20th century, of equal volumes, is it possible to cut one of them by means Laczkovich proved that any plane polygon can be cut into of planes into finitely many polyhedral pieces that can be finitely many polygonal parts, which can be reassembled reassembled into the other polyhedron? In the sequel, we using only translation into a square equiareal with the orig- will call such polyhedra equidecomposable. Max Dehn inal polygon (see [26], [27]). What about the analogous soon solved the problem. However, this problem had been problem in three-dimensional space? solved in Krako ´ w about twenty years earlier, and neither Hilbert nor Dehn could have known about that. Gerling’s Solution of a Particular Case Let us quote the original formulation of the problem stated The Planar Case by Hilbert in 1900: In the mid-19th century it was common knowledge that any In two letters to Gerling, Gauss expresses his regret two polygons of equal areas could be dissected by lines into that certain theorems of solid geometry depend upon finitely many polygonal pieces that could be rearranged one the method of exhaustion, i.e., in modern phraseol- into the other, that is, they were equidecomposable (or ogy, upon the axiom of continuity (or upon the 2018 The Author(s). This article is an open access publication, Volume 40, Number 2, 2018 55 https://doi.org/10.1007/s00283-017-9748-4 axiom of Archimedes). Gauss mentions in particular the theorem of Euclid, that triangular pyramids of equal altitudes are to each other as their bases. Now the analogous problem in the plane has been solved. Gerling also succeeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent parts. Nevertheless, it seems to me probable that a general proof of this kind for the theorem of Euclid just mentioned is impossible, and it should be our task to give a rigorous proof of its impossibility. This would be obtained as soon as we succeeded in specifying two tetrahedra of equal bases and equal altitudes which can in no way be split up into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split up into congruent tetrahedra. The correspondence between Christian Ludwig Gerling and Carl Friedrich Gauss mentioned previously was pub- lished a long time after ICM 1900, in 1927, by C. Schaeffer [10]. Gerling proved that two symmetric tetrahedra are equidecomposable. As in the planar case, only translations Figure 1. Christian Ludwig Gerling (https://en.wikipedia.org/ and rotations were allowed and no part could be wiki/Christian_Ludwig_Gerling#/media/File:Christian_Ludwig_ transferred by a mirror reflection, which made the problem Gerling.jpg). far from trivial. Gerling sent his solution to Gauss. Let us quote here a letter from Gauss to Gerling, written in Go ¨ttingen on April 17, 1844. The letter is essential for the positively solved. The proof was based on an idea that story, however it is seldom cited in the literature. may be easily illustrated in the planar case. Assume that It was with great pleasure that I read your proof that two triangles are given, such that one of them is the symmetrical but not congruent polyhedra are of reflection image of the other. They can be easily divided equal volume. This is a thorny problem indeed. You into smaller triangles to get six pairs of congruent triangles might say that: (1) any tetrahedron could be dissected (see Fig. 2). The common point of the small triangles in such a way that a certain part is congruent to a part contained in the big triangle is the center of the circle of the second tetrahedron and that two faces of these inscribed in this triangle. Gerling divided a big tetrahedron are perpendicular to a third face and intersect at into twelve small ones with their common point being the equal angles, and (2) it can be shown that any center of the sphere inscribed in the large tetrahedron, and tetrahedron could be dissected into 12 parts using their bases given by a suitable division of the large this construction. tetrahedron’s faces into congruent triangles. Of course, I cannot say that this reasoning is new. Gerling’s method is very clever and, despite the doubts You might consult ‘‘Geometry’’ by Legendre, where of Gauss, original. Unfortunately, as Gauss indicated, the many proofs are given in a more general or easier method does not answer the question of whether every form than before. Unfortunately, I am not in pos- two tetrahedra of equal volumes are equidecomposable. session of this book and I do not have access to it at This question, to be posed again by Hilbert in 1900, was this moment. very natural and well known in the mathematical com- munity in the second half of the 19th century. However, it I must express my regret that your proof does not remained unsolved. lead to a simplification of other theorems in stere- ometry, namely those which depend on the method The Contest in Krako´w of exhaustion, see Book 12 of Euclid, Chapter 5. Against this background, let us now move to the Polish city Perhaps something can still be improved, but unfor- of Krako ´ w, then in the Austro-Hungarian Empire. In 1872, tunately I have no time now to consider this matter the Academy of Arts and Sciences was established there. It further. acted as an institution bringing together Polish scientists Gerling’s solution was correct and the question of whether from the whole of Europe. It stayed in touch with other two irregular tetrahedra such that one of them is a mirror scientific academies in Europe and later admitted as for- reflection of the other can be divided into a finite number eign members many outstanding scientists, among them of pairwise congruent polyhedra (tetrahedra) was The demonstration showing Gerling’s dissection is available at http://demonstrations.wolfram.com/Gerlings12PieceDissectionOfAnIrregularTetrahedronIntoItsMirr/. In 1918, when Poland regained independence, the Academy changed its name to the Polish Academy of Arts and Sciences (Polska Akademia Umieje ˛ tnos ´ ci—PAU). In 1952, under the strong pressure from political authorities, the Academy was forced to cease its activities. The renewal of the Academy took place in 1989. 56 THE MATHEMATICAL INTELLIGENCER Figure 2. Gerling’s solution illustrated in the planar case. mathematicians such as Elie Cartan, Henri Lebesgue, Godfrey Harold Hardy, and Maurice Frechet. On June 12, 1882, the Academy announced a math contest. Władysław Kretkowski (see below for more information about him) presented to the Academy two problems and offered prizes for their solutions. In the ‘‘Report of the Activities of the Mathematics and Natural Sciences Division of the Academy, vol. 11’’ [29], we can read about the result of this competition, as presented at the meeting of the Division on February 20, 1884, by ´ Figure 3. The first page of Birkenmajer’s submission. Scien- Franciszek Karlinski. He announced that nobody had answered the first question, concerning algebra, with the tific Library of PAU and PAN, Ref. ms. 6828 (with permission). prize of 1000 French francs. However, two geometers had sent solutions to the second problem (the prize was 500 francs). In accordance with the rules of the contest, the authors had signed their papers with pseudonyms. The A description of the author’s reasoning follows. Kar- report then moves on to a description of the problem and linski concludes that the jury considered the second the solutions. submission as a paper of high quality that provided a The problem was: solution to the problem and deserved the reward. The Given any two tetrahedra with equal volumes, sub- Division decided to give the prize to the author who had divide one of them by means of planes, if it is sent the paper under the pseudonym ’AEI ‘O HEOR possible, into the smallest possible number of pieces CEXMETPEI. The name of the author of the solution is that can be rearranged so as to form the other tetra- not mentioned in the report. hedron. If this cannot be done at all or can be done When the competition results were announced, it only with certain restrictions, then prove the impos- turned out that the problem had been solved by Ludwik sibility or specify precisely those restrictions. Antoni Birkenmajer, then a 28-year-old teacher of math- As we can see, this is precisely Hilbert’s Third Problem. ematics. The most important question is: was the solution What about the solutions received? correct? In the report, the reasoning is described on two In the report we can read that the first author signed pages. This description indicates that the problem was his submission ‘‘Eureka.’’ The description of the reasoning solved indeed, but this was only a kind of summary. was followed by the final conclusion: ‘‘As we see, the Birkenmajer never published his result. The Science Archives of PAN and PAU only contain a letter from author did not solve the general problem. He described only one case and even this one under some additional Kretkowski in which he suggested announcing the competition. assumptions. So, the paper does not meet the conditions of the competition and cannot be awarded the prize.’’ The Fortunately, we managed to find in the Scientific Library of PAU and PAN the original manuscript by Birkenmajer. report reads further: ‘‘The second paper is completely dif- ferent. It is rigorously scientific, contains 40 pages and 7 Now it is sure that Birkenmajer [6] really presented a correct figures, arranged into three chapters.’’ proof. The Academy comprised then three divisions; the others were concerned with Philology and with History and Philosophy. Franciszek Karlin ´ ski (1830–1906), Polish astronomer, mathematician, and professor of the Jagiellonian University. This is a quotation from Plato and means ‘‘God always geometrizes.’’ Polish Academy of Sciences (Polska Akademia Nauk—PAN), created in 1951. This is no mistake. Indeed, the order of the abbreviations in the official name of the Archives is ‘‘of PAN and PAU’’ and of the Library it is ‘‘of PAU and PAN.’’ 2018 The Author(s). This article is an open access publication, Volume 40, Number 2, 2018 57 characterizing T and T such that four of these Birkenmajer’s Solution 1 2 invariants are independent. If this is satisfied, there Let us describe in brief the idea of Birkenmajer’s forty-page exists just one plane that solves the problem. reasoning. He shows that the required construction is Finally, in the third part of the paper, Birkenmajer connects possible only under some additional assumptions. his previous results with the properties of dihedral angles In the first chapter, he takes a tetrahedron and cuts it by of tetrahedra and gives a partial answer to the question: ‘‘If planes to obtain sections. Such a section may be either a this can be done only with certain restrictions, then specify triangle or a quadrilateral. In the majority of cases, an precisely those restrictions.’’ He proves that there are two obtained quadrilateral is not a parallelogram and we may particular cases when the problem can be solved, that is extend two sides of this quadrilateral and one edge of the tetrahedron to get a new tetrahedron (see Fig. 4). Now, (1) when one face of the tetrahedron is an isosceles Birkenmajer investigates several polyhedra he obtained as triangle and the dihedral angle at the base of the a result of cutting the original tetrahedron by planes and triangle is a right angle, or using the just-mentioned construction. For all of them, he (2) when one face of the tetrahedron is an isosceles uses Euler’s formula on polyhedra. After some calculation triangle and the angle between the bisector of the he comes to interesting formulas connecting the number of vertex angle of this triangle and the edge opposite to its those polyhedra, their faces, edges, and vertices. base is a right angle. In the second chapter, entitled ‘‘The Conditions for the Solvability of the Problem,’’ Birkenmajer’s purpose is to The paper also contains a very interesting appendix. In discover some parameters that will enable him to find the this part Birkenmajer notes that his method of solving the invariants of equidecomposability. Using a formula from problem could in fact be seen as algebraic—that is, the the first chapter, he obtains some conditions wherein there problem may be reduced to a question about a polynomial are as many as 36 parameters! The number of conditions function. Moreover, Birkenmajer writes that this solution that must be satisfied to get tetrahedron T from tetrahe- could be extended to any convex polyhedron. He notices dron T is equal to that the methods of calculus are not suitable for the W ¼ 2nðn  1Þðn  2Þþ 6nðn  1Þþ 9n; investigation of this problem. Unfortunately, his knowledge of algebra was rather poor and he was not able to give the in which n is the number of planes used in cutting T . The necessary conditions for equidecomposability of two author indicates that the existence of a required division polyhedra in polynomial form. depends on the existence of solutions to suitable Dio- We should also note that the construction of an addi- phantine equations. He describes the way to find the tional tetrahedron used by Birkenmajer in Chapter 1 was appropriate construction in a particular case. correct, but not necessary, and it needlessly complicated This leads him to the solution of the main problem, that the solution. It was indicated by the referee on the margin is, that two tetrahedra with equal volumes may not be of the paper, where it was noted that the same result might equidecomposable. He writes: be obtained if the author considered only polyhedra made The task is only possible when there are at least 7 by cutting (see Fig. 5). conditional equations involving 11 invariants Figure 5. A referee’s remark on Birkenmajer’s solution. Figure 4. A part of the fifth page of Birkenmajer’s solution. Scientific Library of PAU and PAN, Ref. ms. 6828 (with Scientific Library of PAU and PAN, Ref. ms. 6828 (with permission). permission). 58 THE MATHEMATICAL INTELLIGENCER Birkenmajer and Kretkowski Who was the author of the solution and who was the person who offered the prize? The names of several Polish mathematicians are well known throughout the world. However, the names of Birkenmajer and Kretkowski are generally not among these. Nevertheless, both mathemati- cians deserve attention. Ludwik Antoni Birkenmajer (1855–1929) was born in Lipsko near Lvov. He studied at the University in Lvov and in 1879 he obtained his Ph.D. from this university. Then he continued studies in Vienna. From 1880 to 1909 Birken- majer taught mathematics and physics at the agricultural ´ ´ gymnasium in Czernichow (near Krakow). Simultane- ously, after habilitation in 1881 at the Jagiellonian University in Krakow, he became a Privatdozent at the Jagiellonian University and he lectured in mathematical physics. In 1897, a Chair of the History of Exact Sciences was created for Birkenmajer at the Jagiellonian University. He was a professor of this university until his death. He was also a member of the Academy of Arts and Sciences in Krako ´ w (since 1893) and a member of the International Astronomical Union in Oxford. Birkenmajer held a very broad spectrum of scientific interests. His Ph.D. dissertation concerned mathematics: it Figure 6. Ludwik Antoni Birkenmajer as a student. Courtesy: was entitled On General Methods of Integration of Differ- Krzysztof Birkenmajer (with permission). entials. His habilitation thesis concerned physics and was devoted to the structure of the Earth. He there compared experimental data to Laplace’s conjectures on the relation between the density and pressure inside the Earth. But Birkenmajer was also an astronomer, geographer, and geo- physicist. He wrote many papers about mathematical models of the shape of the Earth. He made several interesting investigations of the temperature of the water in lakes in the Tatra mountains. Nevertheless, he first was a historian of science and is best known for his achievements in this area. He was particularly interested in the 15th and 16th centuries. His very detailed research on the life and work of Nicolaus Copernicus revealed several facts about Copernicus that were unknown at that time. Birkenmajer published many papers and books about this subject. His book on Copernicus [3] had 736 pages and to date is regarded by historians of science as an extraordinary treatise. More details about Birkenmajer’s work on Copernicus can be found in [18]. In 2011, the Institute for the History of Science of the Polish Academy of Sciences (created in 1954) was named for him and his son, Aleksander Birkenmajer. Concerning his results in mathematics, Birkenmajer was, above all, an applied mathematician. However, he also wrote some papers in pure mathematics, mainly on prime numbers. In particular, he proved that if p is a prime greater 1 1 1 than 3 and the sum þ þ ... þ is presented as a Figure 7. Ludwik Antoni Birkenmajer. PAUart BZS.RKPS. 1 2 p1 fraction, then its numerator is divisible by p ; moreover, 12519.k.8 (with permission). only primes have this property [4]. In another article [5], he analyzed a remainder after division of ð2k  1Þ! by 4k  1 where 4k  1 is a prime and solved a problem stated by area. Unfortunately, all his mathematical papers were Henri Lebesgue. He gave a characterization of Heronian published in Polish, like the report on the solution of the triangles, that is, triangles having integer sides and integer equidecomposability problem [29]. A type of secondary school with a strong vocational focus. Aleksander Birkenmajer (1890–1967), a historian of science and library scientist, professor of the Jagiellonian University and Warsaw University. 2018 The Author(s). This article is an open access publication, Volume 40, Number 2, 2018 59 Figure 8. The grave of Birkenmajer in Rakowicki Cemetery in Figure 9. Władysław Kretkowski. PAUart BZS.RKPS.6818.k.8 Krako ´ w. (Photo by the authors). (with permission). pioneers of modern applications of mathematics in Poland. The name of Władysław Kretkowski (1840–1910) is even Although his achievements were not comparable with the less known, in Poland and elsewhere. results of leading mathematicians of his period, he defi- Kretkowski was born in Wierzbinek near Torun ´ . From nitely was a good mathematician who obtained valuable 1865 to 1867 he studied at the Sorbonne in Paris, from results. which he obtained Diplo ˆme de Licencie e`s Sciences However, Kretkowski also deserves a place in the his- Mathe´matiques. He also graduated from the School for ´ tory of mathematics at the Jagiellonian University for Bridges and Roads (Ecole imperiale des ponts et chausse´es, ´ another reason. He was a very rich Polish nobleman and he now Ecole des Ponts Paris Tech). In Paris, he became devoted considerable means to supporting science, in acquainted with several open mathematical problems. particular mathematics. The contest described earlier with During his studies he published in Nouvelles Annales de the awards financed by Kretkowski was not his only con- Mathematiques solutions of geometrical problems stated by tribution: he supported scientists with many scholarships. Hieronymus Georg Zeuthen and Ludvig Oppernmann [24]. He died in 1910 and in his last will he bequeathed all his In 1879 he applied for a Ph.D. at the University in Lvov and, estate for the development of mathematics. The Academy although he passed the mathematical exam with distinc- of Arts and Sciences was asked to administer scholarships tions and presented eight papers published in Paris and dedicated for studies in leading European mathematical Krako ´ w, he did not obtain a doctorate there (for the story, centers. In his will, he had also included a donation for the see [13]). In 1882 he got a Ph.D. in mathematics from the Jagiellonian University, which made it possible to establish Jagiellonian University on the basis of a dissertation on the a chair in mathematics. And finally, he left his entire valu- applications of functional discriminants in calculus. The able collection of mathematical books—over two thousand paper was refereed by Franciszek Karlinski and Franciszek volumes, most of which were modern—to the mathemati- Mertens (the author of the famous Mertens conjecture; in cal library of the university. 1865–1884 Mertens was a mathematics professor at the Jagiellonian University). Bricard, Hilbert, and Dehn Kretkowski published more than 20 articles in mathe- In 1896, twelve years after Birkenmajer’s result, Raoul Bri- matics. In his papers, he proved several theorems on geometry, analysis, theory of polynomials, and analytical card published the paper [9] in which he proposed a much more general answer for any two polyhedra. The propo- functions. He introduced a very clever method of deter- mining the center and radius of a sphere circumscribed sition crucial for solving the problem (now called Bricard’s condition) states: around an ndimensional simplex [25]. His 57-page-long treatise about determinants published as an appendix in If two polyhedra are equidecomposable, then there exist positive integers n , m , and integer p such that [16] was complimented by Thomas Muir in the fundamental i j monograph on the development of the theory of determi- n a þ n a þ .. .þ n a ¼ m b þ m b þ ...þ m b þ pp; 1 1 2 2 q q 1 1 2 2 l l nants [28]. Kretkowski may also be regarded as one of the 60 THE MATHEMATICAL INTELLIGENCER Figure 11. Max Dehn (https://www.geni.com/people/Max- W-Dehn/6000000000128799755). Courtesy: Mary Proctor Dehn and Christopher Winter (with permission). Figure 10. The caricature of Bricard drawn by Jean-Robert (Public Domain; https://eo.wikipedia.org/wiki/Raoul_Bricard). prisms are zero. However, the Dehn invariant of a regular tetrahedron is not equal to zero. Dehn published his result in two papers. In the first one, where a are measures of dihedral angles of the first published in 1900 [14], he described two polyhedra with polyhedron and b are measures of dihedral angles different Dehn invariants. In the second one, published two of the second polyhedron. years later [15], he proved that two equidecomposable polyhedra must have the same invariants. It is worth noting Unfortunately, Bricard’s reasoning was not correct (see that Dehn’s work is also about Diophantine equations. In [2] and [7]). Soon after, in 1997, Giuseppe Sforza also 1903 Benjamin F. Kagan presented a modified version of published an example of nonequidecomposable polyhedra Dehn’s proof [22]. Kagan’s version was easier to follow by [30]. the reader. For a clear and complete description of Dehn’s Anyway, in 1900 the well-known problem of equide- method and invariants, see [1], [7], and [20]. composability of two tetrahedra was still regarded as Dehn’s method of solution was completely different open. The almost 20-year-old solution by Birkenmajer from that presented by Birkenmajer and from the idea remained unpublished. Birkenmajer was not working as a suggested by Bricard. Moreover, Dehn’s solution to Hil- mathematician and his activities took place far from bert’s Third Problem had one more advantage, from the mathematical centers. Information about his proof was point of view of Bricard’s reasoning: it proved that Bricard’s published only in Polish, in a report of a meeting of the condition was true. Note that Dehn was familiar with the Academy. No wonder that almost nobody knew about it. work of Bricard and Sforza; in [15] he mentioned their Moreover, the problem fitted into an important discussion papers and commented on the results they obtained. of the foundations of geometry that Hilbert participated This could have been the end of the story. However, in. Then Hilbert included the equidecomposability prob- things turned out differently. lem in his famous list. Hilbert’s Third Problem was solved quite soon by Max Dehn who, roughly speaking, found another invariant of Many Years Later equidecomposability which depended on all the dihedral As we know, a special place is held in mathematics by angles of the polyhedron and on the lengths of its edges conditions that characterize some objects in unique form, (for a precise definition see, for example, [1] and [23]). Now that is, necessary and sufficient conditions for some prop- it is called the Dehn invariant. Another obvious invariant erties, as in the Poincare conjecture. For many years, it was was the volume of the polyhedra. Accordingly, a necessary known that the preservation of the volume and Dehn’s condition for equidecomposability of two polyhedra was invariant were necessary for equidecomposability. It was the equality of their volumes and of their Dehn invariants. It great news when it was proved about 60 years later (in an was clear that the Dehn invariants of the cube and all almost forty-page-long paper) by Sydler [31] that these 2018 The Author(s). This article is an open access publication, Volume 40, Number 2, 2018 61 Ludwik and Aleksander Birkenmajer Institute conditions are sufficient. This result is now called Sydler’s theorem. Also, Hilbert’s Third Problem was the basis for for the History of Science further research (see very interesting expository articles [11] Polish Academy of Sciences and [12]). Nowy Swiat 72 In recent years, following the progress of different 00-330 Warszawa branches of mathematics, other equivalent conditions for Poland equidecomposability were found (see [23]). Thus it turned e-mail: smciesie@cyfronet.krakow.pl out that the Third Problem, which seemed to be much simpler than the other Hilbert’s Problems, was very stim- Mathematics Institute, Jagiellonian University ulating for the development of mathematics, even one Łojasiewicza 6 century later. 30-348 Krako ´ w Also one century later, another interesting result con- Poland cerning this topic was obtained. Until 2007 there was no e-mail: krzysztof.ciesielski@im.uj.edu.pl direct proof of Bricard’s condition. In each published proof it was a consequence of the solution of Hilbert’s Third OPEN ACCESS Problem. It was only in 2007 that a paper was published by This article is distributed under the terms of the Creative Benko (see [2]) in which he wrote: Commons Attribution 4.0 International License (http:// In this article we give a short direct proof of Bricard’s creativecommons.org/licenses/by/4.0/), which permits condition that was overlooked for a century. There- unrestricted use, distribution, and reproduction in any fore it provides a new solution to Hilbert’s problem. medium, provided you give appropriate credit to the ori Our proof is completely elementary. Since it uses no ginal author(s) and the source, provide a link to the linear algebra, it could even be presented in a high- Creative Commons license, and indicate if changes were school math club. made. It turned out that, if we suitably modify Bricard’s condition, we can get a different, necessary and sufficient condition for equidecomposability. So, after more than one hundred REFERENCES years, the gap in another method of solving Hilbert’s Third [1] M. Aigner, G. M. Ziegler, Proofs from the Book, Springer-Verlag, Problem was closed! Berlin Heidelberg 2001. It’s time for a conclusion. Speaking about Hilbert’s [2] D. Benko, A New Approach to Hilbert’s Third Problem, The Third Problem, it is definitely Dehn who should be con- American Mathematical Monthly 114(2007), 665–676. sidered the person who solved it. He was the first one to [3] L. A. Birkenmajer, Mikolaj Kopernik. Cz. 1.: Studya nad pracami publish the correct proof. Moreover, his solution was very Kopernika oraz materyaly biograficzne, Skład Gło ´ wny w Ksie ˛ garni stimulating for mathematics and immediately led to the Spo ´ łki Wydawniczej Polskiej, Krako ´ w 1900. answer to another open problem. Nevertheless, it is good [4] L. Birkenmajer, O pewnem twierdzeniu z teoryi liczb, Prace to know that almost twenty years earlier yet another solu- Matematyczno-Fizyczne 7(1896), 12–14. tion of the problem had been given by Ludwik Antoni [5] L. A. Birkenmajer, O zwia ˛ zku twierdzenia Wilsona z teorya ˛ reszt Birkenmajer. Birkenmajer not only showed an example of kwadratowych, Rozprawy Wydzialu Matematyczno-Przyrodnic- nonequidecomposable polyhedra, but also give suit- able invariants. In print, it was only announced and zego Akademii Umieje ˛ tnos ´ ci, 57A(1918), 137–149. summarized (in Polish, a language not widely known by [6] L. Birkenmajer, Zadanie konkursowe z geometryi podane przez p. the mathematical community), but the manuscript exists Dra W. Kretkowskiego, Krako ´ w 1883, Archives of PAN and PAU and shows that Birkenmajer’s reasoning was completely ´ in Krakow. Ref. ms. 6828. different from Dehn’s, Bricard’s, and Sforza’s, and, what is [7] V. G. Boltianskii, Hilbert’s Third Problem, V. H. Winston & Sons most important, correct. (Halsted Press, John Wiley & Sons), Washington DC, 1978 Let us finish with an interesting fact, in some way con- (translated from the Russian version: Tret’ja problema Gil’berta, nected with this story. It is not common knowledge that Nauka, Moscow 1977). Max Dehn was probably the first mathematician to give a [8] W. Bolyai de Bolya, Tentamen. Iuventutem studiosam in ele- correct proof of the Jordan Curve Theorem for polygons menta matheseos purae, elementaris ac sublı`mioris, methodo (see [19]). However, his manuscript containing this result, intuitiva, evidentiaque huic propria, introducendi. Cum appendice dating from 1899, remained unpublished. triplici, Maros Va ´ sa ´ rhelyini; tomus primus 1932, tomus secudus [9] R. Bricard, Sur une question de ge ´ ome ´ trie relative aux polyedres, ACKNOWLEDGMENTS ´ Nouvelles annales de mathe ´ matiques, Ser. 3, 15(1896), s. 331– We thank Krzysztof Kwasniewicz for his help. We also thank the referee for helpful comments. 62 THE MATHEMATICAL INTELLIGENCER ¨ [10] Briefwechsel zwischen Carl Friedrich Gauss und Christian [22] B. Kagan, Uber die Transformation der Polyeder, Math. Ann. Ludwig Gerling Correspondence of Gauss and Gerling), C. 57(1903), 421–424. 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Gesellschaft der Wissenschaften zu Go ¨ ttingen, Mathe- to Tarski’s circle squaring problem, Journal fu ¨ r die reine und matisch-physikalische Klasse 1900, 345–354. angewandte Mathematik 404(1990), 77–117. [15] M. Dehn, Uber den Rauminhalt, Math. Ann. 55(1902), 465–478. [27] M. Laczkovich, Paradoxical decompositions: a survey of recent [16] W. Folkierski, Zasady rachunku ro ´ z _ niczkowego i całkowego, nakł. results, Proc. First European Congress of Mathematics, Vol. II Biblioteki w Ko ´ rniku, Paris 1870. (Paris, 1992), Progress in Mathematics 120, Birkha ¨ user, Basel [17] P. Gerwien, Zerschneidung jeder beliebigen Anzahl von gleichen 1994. geradlinigen Figuren in dieselben Stu ¨ cke, Journal fu ¨ r die reine [28] T. Muir, The Theory of Determinant in the historical order of und angewandte Mathematik 10(1833), 228–234. development, vol. III, The period 1861 to 1880, Macmillan, [18] A. Goddu, Ludwik Antoni Birkenmajer and Curtis Wilson on the London 1920. Origin of Nicholas Copernicus’s Heliocentrism, Isis 107(2016), [29] Rozprawy i Sprawozdania z Posiedzen ´ Wydzialu Matematyczno- 225–253. Przyrodniczego Akademii Umieje ˛ tnos ´ ci 11(1884), 87–92. [19] H. Guggenheimer, The Jordan Curve Theorem and an unpub- [30] G. Sforza, Un’ osservazione sull’ equivalenza dei poliedri per lished manuscript by Max Dehn, Archive for History of Exact congruenza delle parti, Periodico di matematica 12(1897), 105– Sciences, 17(1977), 193–200. 109. ´ ´ [20] H. Hadwiger, Zum Problem der Zerlegungsgleichheit der Poly- [31] J.-P. Sydler, Conditions necessaires et suffisantes pour l’equiv- eder, Arch. Math, 2(1950), 441–444. alence des polyedres de l’espace euclidean a trois dimensions, [21] D. Hilbert, Mathematical Problems. Lecture delivered before the Comment. Math. Helv. 40(1965), 43–80. International Congress of Mathematicians at Paris 1900 (trans- [32] W. Wallace, J. Lowry, Question 269, New Series of the Mathe- lated by M. W. 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