The hidden pavements of Michelangelo's Laurentian Library

The Hidden Pavements of Michelangelo's Laurentian Library Jay Kappraff ver since publishing my book Connections: the Geometric Bridge between Art and Science |1], 1 have become used to playing the role of a “mathematical tourist.” I am frequently contacted by researchers keen on discussing some discovery that they have made and seeking my advice as to the mathematical aspect of their mathematical tourist attractions suck Blake Summers; Rolf Bagemihl, an archivist living in Florence; David Krell, a philosopher; Arielle Saiber, a graduate student at Yale specializing in Renaissance history; and Saori Hisano, a graduate student at Illinois Institute of Technology, the college where Nicholson teaches. From time to time we have also sought the help of other stuwork. Sometimes the work appears to dents of Nicholson's, Salvatore Camshed new light on ancient or modern geometry; other times it seems to lead nowhere. It is in this context that Ben Nicholson telephoned me three years ago. He had become privy to a set of facsimiles of fifteen 8’6” X 8/6” pavement designs— possibly created by Michelangelo— that lay hidden beneath the floorboards of the Laurentian Library in Florence [2,3]. He was trying to decipher their geometries in order to enable him and an artist, Blake Summers, to reconstruct poreale, a Florentine theologian, and Ernest McClain, a musicologist. Isaw on my first visit to the studio of Blake Summers that he was recreating the pavement designs with the aid of a giant aluminum bar which served as a compass. Summers and Nicholson were using intuitive geometry very much in the spirit of Boethius, who translated the Elements of Euclid into a language understandable to the them at full scale. This project definitely Does your hometown have any + piqued my interest. One thing led to another, and I soon found myself a part of Nicholson’s team of researchers devoted to the study of the pavements— guilds of masons during Middle Ages. In the process of analyzing the pavements we feel that we have discovered a taxonomy of ancient geometry that commingles all of the geometric systems handed down from antiquity into an integrated whole. We have identias statues, plaques, graves, the café where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to Mathematical Tourist Editor, WIET Dirk Huylebrouck, Aartshertogstraat 42, Figure 1. Entrance stairway at the Laurentian Library. (Figs. 1-7 are from Firenze Biblioteca 8400 Oostende, Belgium Medicea Laurenziana, salone de Michelangelo. By permission of the Minister of Culture. e-mail: dirk.huylebrouck@ping.be Further reproductions are strictly forbidden.) THE MATHEMATICA. INTELLIGENCER © 1999 S2RINGER-VERLAG NEW YORK Copyright ©2001. All Rights Reserved.

RAP TERITES The Hidden Pavements of Michelangelos Laurentian Library Jay Kappraff Does your hometown have any mathematical tourist attractions such Dirk Huylebrouck, Editor ver since publishing my book Blake Connections: archivist living in Florence; David Krell, the Geometric Summers; Rolf Bagemihl, an Bridge between Art and Science [1], I a philosopher; Arielle Saiber, a graduate have become used to playing the role student at Yale specializing in Renaisof a “mathematical tourist.” I am fresance history; and Saori Hisano, a gradquently contacted by researchers keen uate on discussing some discovery that they Technology, the college where Nicholstudent at Illinois Institute of have made and seeking my advice as son teaches. From time to time we to the mathematical aspect of their have also sought the help of other stuwork. Sometimes the work appears to dents of Nicholson’s, Salvatore Camshed new light on ancient or modern poreale, a Florentine theologian, and geometry; other times it seems to lead Ernest McClain, a musicologist. nowhere. It is in this context that Ben Nichol- I saw on my first visit to the studio of Blake Summers that he was recreson telephoned me three years ago. He ating the pavement designs with the had become privy to a set of facsimiles aid of a giant aluminum bar which of fifteen 8'6” x 8'6" pavement designs— served as a compass. Summers and possibly created by Michelangelo— Nicholson were using intuitive geomethat lay hidden beneath the floorboards try very much in the spirit of Boethius, of the Laurentian Library in Florence who translated the Elements of Euclid [2,3]. He was trying to decipher their into a language understandable to the geometries in order to enable him and guilds of masons during Middle Ages. an artist, Blake Summers, to reconstruct In the process of analyzing the pavethem at full scale. This project definitely ments we feel that we have discovered piqued my interest. One thing led to ana taxonomy of ancient geometry that other, and I soon found myself a part of commingles all of the geometric sys- Nicholson’s team of researchers detems handed down from antiquity into voted to the study of the pavements— an integrated whole. We have identias statues, plaques, graves, the café where the famous conjecture was made, the desk where the famous initials are scraiched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of tis mathematical significance, and either a map or directions so that others may follow in your tracks, Please send all submissions to Mathematica! Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium Medicea Laurenziana, salone de Michelangelo. By permission of the Minister of Culture. e-mail: dirk.huylebrouck@ping.be Further reproductions are strictly forbidden.)

upon usual pavements. Further details of the ries of fifteen panels, of different dewhich the pavements appear to be history and significance of the pavesigns, each about 8'6” x 8'6”. The fifbased: (1) the Vesica Pisces [1,2]; (2) ments can be found in Nicholson’s CDteen panels along one aisle mirror the the law of repetition of ratios popular- ROM, Thinking the Unthinkable House ones on the other aisle, but differ in ized by the 20th-century designer Jay [10]. subtle ways. When juxtaposed, the 15 fied six principal geometries Hambridge under the name dynamic Overall, the pavement consists of symmetry [4,5]; (3) the eight-pointed two side aisles and a figurative center pairs of panels appear to tell a story about the essentials of geometry and Brunes star discovered by Tons Brunes, aisle (Fig. 2). Desks situated on a raised number. In 1928 the pavements were the late Danish engineer [6,7,8]; (4) aset of constructions based on V2 and rewooden dais have been placed over the photographed for the first time when pavements. On the side of each desk the desks were removed temporarily ferred to by Brunes as the sacred cut are listed the books that were to be whilst structural repairs were made to [2,5,9]; (5) the ad-quadratum squarestored in it. Beneath the desks are a sethe subflooring (Fig. 3). within-a square; and (6) the golden mean [1,5]. Let me summarize what I have learned about this remarkable set of designs and briefly describe the structure of two of them. The Laurentian Library, which was designed by Michelangelo, is situated on the second floor of the San Lorenzo church complex in the heart of Florence. Work on the library was begun in 1523 by Pope Clement VII, alias Guilio Medici, the nephew of Lorenzo di Medici, as a monument to his uncle; it was opened to the public 48 years later by his distant cousin, Grand Duke Cosimo I. The Library was meant to be ahome for the books from antiquity that survived to the Renaissance. The modest Setting of the Library leaves one utterly unprepared for what one encounters upon entering. First one is confronted with a massive staircase (Fig. 1) calculated to provoke a numerological trance: There are two steps to get into the building, then series of 3 steps, 7 steps, and 5 steps, with 9 steps to the left and right. After mounting the staircase one enters the Reading Room (Fig. 2). Here the seeming regularity and normalcy hides a frenzy of paradox and ambiguity. Just look at the walls. There is no predominant structure. The wall consists of seven planes, completely disorienting the viewer. In 1774 a portentous accident occurred in the Reading Room of the Laurentian Library. The shelf of desk 74, overladen with books, gave way and broke. In the course of its ‘repair, workmen found a red and white terracotta pavement which had lain hidden for nearly 200 years beneath the floorboards. The librarian had trapdoors, still operable today, built into the floor, so future generations could view these un- Figures 2 and 3. The Laurentian Library Reading Room—with and without desks.

Figure 4. a) The Index Panel 1; and b) the Cross Panel 15. (Figs. 4-7 are details of the pavements.) The spatial conundrums, paradoxes, the books make a counterpart to those lated to an ancient musical scale based and errancies of the building fabric reto the East and follow the epistemologon the first 10 numbers [11). appear in the geometry of the paveical form devised by Aristotle. Across Let's analyze two of the pavements. ments. from the poets are the texts related to Panel 14 is referred to by Nicholson as We think that the apparent raggedness of the panels can be explithe trivium (grammar, rhetoric, and orthe Timaeus panel. It is composed of cated as accurate and premeditated inatory), then on to logic, medicine, hisfour diamonds set within circles that are terpretation of antique geometry in tory, ethics, and metaphysics. Aristotle’s cut with segments of circles, and the terms of the philosophical concerns of books were not bound in a single comwhole design is framed with a white borthe 1500s. Michelangelo was working pendium as we might find them in a der. In Fig. 5, this panel is shown juxtawith themes well understood at the bookstore today, but were found in six posed with a reproduction by Fabbrini, time, a “secret art of geometry” which different locations in the library, acfrom the circle of Michelangelo, could be read in the pavement by the cording to the part of his epistemology Michelangelo’s system of proportions. knowledgeable, that they addressed. To traverse the Michelangelo felt that the system of length of the Reading Room could be proportions developed at the time by but which is much more inscrutable today. The books in the Library were orof thought of as a journey through the full Diirer was inadequate to describe the ganized with the sacred books to the extent of the wisdom and supple human body, and that his own East and the profane to the West, a knowledge. of Monsignior Comporeale feels that joints within the body. You will notice throwback to the ancient “tree world’s system better allowed for flexible knowledge”. On the East side, tucked movement from Panel 1 (see Fig. 4a) that Michelangelo provides a scale on behind the projecting entrance door, is near the door to Panel 15 (see Fig. 4b) the right subdivided first into 2 parts, a single desk whose books include the on the other side of the Room may then 4 and 8 parts, with each unit fur- Koran, and have represented the Christian’s journey ther subdivided into 3 parts for a total books of magic. These subjects escape from baptism to enlightenment. Panel 1 of 24 equal parts. This is reminiscent the tidy categories used to order the consists of octagons and crosses symof the lambda figure Laurentian collection, and they were bolic of baptism, while Panel 15, adjaplaced out of sight. The sequence hits cent to the Pentateuch, also contains its stride with 13 desks containing the crosses Kaballah, Machiavelli, and 10 concentric sets 1 2 of 4 3 6 9 works of Italian, Latin, and Greek posquares surrounding a central square. ets, and continues on through books That the Hebrew Pentateuch is placed devoted to the quadrivium (music, ashere may be deliberate metaphor for the found in Plato’s Timaeus and referred tronomy, geometry, and arithmetic), 10-ness that pervades it: the 10 Comto there as the World Soul. This was leading to 27 desks loaded with Latin mandments, 10 generations to Abraham, one of the neo-Platonic ideas brought and Greek books of theology, ending and 10 more to the Flood. Also Ernest to the Renaissance by Ficino’s acad- 8 12 18 27 with books devoted to the Pentateuch. McClain has found that much of the nuemy. It forms the basis of the musical On the West side of the Reading Room, merology of the Hebrew Bible can be resystem studied by Pythagoras and writ-

IRRATIONAL, 3 e. ». | e 4 ‘toe k"e è e € a = 4 a wD a Y y S 4 24 | 4 73 d N A x 3 ES ES «| Ss = Figure 5. The geometric construction of the Timaeus Panel 14. ten about by Nichomachus [12,13,14], The 27 units of Michelangelo’s sysand it was used by Alberti as the basis tem of proportions can be found in is subdivided into However, itis surely deliberate that the 27 equal parts. of his system of architectural propor- Panel 14. The space from bottom to top seemingly similar space from left to tions [5,9]. So we expect the number 27 in Michelangelo’s system rather than 24. Sure enough, the head of the model projects higher, the foot projects beneath the floor plane, and an extra unit is intercalated at the hip. Now we have the 27 units of the World Soul. Figure 6. The Medici Impressa. Figure 7. The Cosimo Panel 2.

right has an extra, incommensurate inpanel. For that step, the smail differterval. The four circles of the figure 6. In the final step eight varieties of elence between the diagonal of the lipses are created to fill the diamond echo the coat of arms of Cosimo di 12 X 13 rectangle and the 13 X 13 shapes. Nicholson’s and Summer’s Medici, which is also emblazoned on square is exploited. Circles with this reconstruction of panel 2 is shown small difference as radii are drawn in Fig. 9. the pavements of the central aisle (Fig. 6). Each of the 27 units is exactly 3 solwhere the 24 circles intersect the dis in width resulting in an 81-square pitch circle to yield 48 new points grid for the entire design. The panel on the pitch circle (Fig. 8d). What appeared as a kind of errancy in the deviation of the rectangle from across from this on the other side of the . Forty-eight additional circles with a square exploded into the entire delibrary appears to be based on an 80- radius equal to the pitch circle are sign. Furthermore, in the steps leading square grid. Is it coincidental that the now drawn. These demarcate the to its creation, a series of 3, 6, 12, 24, ratio 80:81, known in musical parlance white bands of the Medici panel (Fig. 48, and 96 circles are created. This is as the syntonic comma, is exactly the 8e). the series that led to the Titius-Bode ratio by which the tones of the ancient scale, attributed to the followers of Pythagoras based onthe primes 2 and 3, differs from the Just scale, based on primes 2, 3, and 5? Such conundrums are found over and over in the structure of the pavements. Panel 2, the Medici panel (Fig. 7), is a rosette form typical of many such antique rosette forms that appeared at the time in Florence. The pavement is a rectangle of dimension 12 X 13. Nicholson feels that these numbers are significant as the number of months in the solar and lunar calendars. The small difference between a square and a rectangle is crucial to its construction. 1. In the first step in the construction, the rectangle is extended to a 13 X 13 square concentric with a 12 X 12 square, and the horizontal and vertical axes are placed in the squares. An equilateral triangle is drawn to a side of the 12 X 12 square. The distance from the center of the square to the vertex of the triangle is the radius of a standard circle of the construction called the pitch circle (Fig. 8a). Beginning where the pitch circle cuts the horizontal axis, six circles of radius equal to the pitch circle are drawn (Fig. 8b). 2. Next six additional circles are drawn beginning where the pitch circle cuts the vertical axis. 3. Twelve additional circles are drawn by repeating steps 2 and 3 for the pair of perpendicular diagonals of gle in a square establishes the pitch circle; the squares resulting in a 24-rosette b) a rosette of six circles; c} a rosette of 24 pattern (Fig. 8c). 4. Twenty-four additional circles are 28 circles; d) the mismatch of the diagonals of the 12 x 12 square and the 12 x 13 rectangle drawn half-way between the circles generates 48 additional circles; e) ninety-six of the rosette. These will be widened circles create a set of spiral bands in which into the white bands appearing in the eight classes of ellipses are placed.

Acknowledgments Dynamic Symmetry. Originally published As you can see, this project has mateby rialized for me into the ultimate of New York: Dover) mathematical tours. I wish to acknowledge the fruitful collaboration that I have undertaken with my colleagues, Ben Nicholson and Saori Dynamarhythmic Design (1932) (rpt. . Brunes, T. The Secrets of Ancient Geometry and its Use. Copenhagen: Rhodos (1967). . Kappraff, J. Geometry.” “A In Secret Geometry of at Ancient Work: A Hisano, that has made this work a great Collection of Papers in Applied Geometry pleasure. edited by K. Gorini. Math. Assoc. of Amer. Notes (In press). . Kappraff, REFERENCES 1. Kappraff, J. Connections: The Geometric Bridge between Art and Science. New J. Mathematics Beyond Measure: A Guided Tour through Nature, Myth, and Number. New York: Plenum Press (In press). York: McGraw-Hill Books. (1991). 2. Nicholson, B., Kappraff, J., and Hisano, S. . Kappraff, J. “Musical Proportions at the “A Taxonomy of Ancient Geometry Based Basis of Systems of Architectural Proporon the Hidden Pavements of Michelantion both Ancient and Modern.” In Nexus: gelo’s Architecture and Mathematics edited by Laurentian Library.” In Art and law that predicted the positions of the Science: The Proceedings of the Second K. Williams. Fuccechio: Edizioni dell’Erba planets up until Saturn. Again, is this Conference on Art and Science edited by (1996). coincidence or prescience? The pave- J. Barrallo, San Sebastian, Spain: Univ. of 10. Nicholson, B. “Architecture, ment mirroring Panel 2 is placed in an the Basque Country Press (1998), and in Geometry.” In CD-Rom: ' Thinking the 11 X 12 square perhaps symbolizing Bridges: Mathematical Connections in Art, Unthinkable House. Renaissance Society the 12 disciples and the 11 disciples Music and Science: Conference Proceedat the University of Chicago (1997). once Judas was excluded. ings edited by R. Sarhangi, Arkansas City, The question begging to be asked is Books + 11. McClain, E. “The Star of David as Jewish Harmonical Metaphor.” International Journ. KS: Gilliland Publ. (1998). of Musicology. Vol. 6, pp. 24-49. (1997). why 30 magnificent pavements would 3. Nicholson, B., Kappraff, J., and Hisano, S. be constructed and then covered up. “The Hidden Pavements of the Laurentian 12. McClain, E. Private communication. Perhaps some cryptic symbols were Library.” 13. McClain, E. The Pythagorean Plato. York concealed within the pavements in the Mathematics manner of Umberto Eco’s Name of the In Nexus Il: Architecture and edited by K. Willams. Fuccechio, Italy: Edizioni dell’Erba (1998). Rose and then hidden to prevent their 4. Hambridge, J. The Elements of Dynamic revelation. Nicholson has a more mun- Symmetry. Originally published by Brentdane hypothesis. When the library was ano’s (1929) (rot. By New York: Dover). conceived there were approximately 5. Edwards, E.B. Pattern and Design with Beach, ME: Nicolas-Hays (1978). 14. McClain, E. “Temple Tuning Systems.” International Journ. of Musicology, 3 (1994). New Jersey Institute of Technology Newark, NJ 07102 USA 1000 books in existence. When it was completed, the Library contained over 3000. The Library had to be reconfigured to accommodate the new books. In the original plan for the Library, a Revealing a Lost Body of Knowledge triangular room was supposed to have been constructed to house the rarest In the summer of 1998, Ben Nicholson, Saori Hisano, and the author preof the books. However, this room was sented their research at two major conferences: The Second International never built. Conference on Art and Mathematics directed by Vera Martha Winitzky de Unfortunately we do not know who created the pavements. The books listing the financial transactions in connection with them would most likely have listed the designer, but they have been lost. However, surely Michelangelo would at least have had a major say concerning such a crucial component of his Spinadel and Maria Emiliana Uranga Otaegui in San Sebastian, Spain; and Nexus II directed by Kim Williams in Mantua, Italy. I then made a presentation at the Bridges Conference directed by Reza Sarhangi at Southwestern College in Winfield, Kansas. We also had a meeting in Florence with the Director and Chief Librarian of the Laurentian Library. As a result of this meeting, the Library agreed to sponsor an exhibition of the pavement research, which might prompt the community to think about exposing this lost body of knowledge to the open air. library.

More or Less Mathematics Wanted in Engineering? Two images, one of the medieval plied numeracy. We will return to this cellarium at Fountains Abbey, England, question later. At school I liked and was good at and another of a tiled forecourt in mathematics. Once the prettiest girl in Vernon, France, inspired two engineers the class allowed me to kiss her in exto different conclusions. The first change for letting her copy my maths author considers that the importance of homework. That is the only time in my life, and I am a sexagenarian, that I’ve abstract mathematics is somewhat ever found a use for mathematics. More overrated, and he even makes an seriously, it has been my experience that appeal to the scientific community to very few engineers use mathematics in support his view. The second author practice. They have of course to be nucalls for another kind of intervention: merate. The insistence on mathematics for all engineers has been a damaging he asks for some mathematical deterrent to some highly capable, imagassistance by proposing an unsolved inative, and creative people. Engineerproblem to readers. Perhaps engineers ing is after all about making things work. The empirical approach, trial and error, and mathematicians will have the ability to experiment quickly is of reactions, outraged or otherwise, to much more importance. Most problems cannot be solved by mathematics anycontribute to The Intelligencer’s way, or they have to be reduced to too letters.—D.H. simple a model to be useful. I expressed this view in a rather light-hearted manner in an article pub- A. Less Mathematics and lished internally within my university. More Numeracy Wanted in To my surprise, many colleagues said Engineering that what I had said needed saying Edward Reed again and again. The professor of To give academic respectability to a trivmathematics got the vapours, refused ial piece of engineering research, it is to speak to me, and shortly afterwards standard practice to add some mathetook early retirement. I also likened matics to make it appear more signifimathematics in engineering to the cant than it is. Mathematics is the silicon wowing of ladies: as there is supposed implants of academia. This is not to be to be a lot of it about and since most confused with numeracy. Numeracy is men find little of it coming their way highly relevant for engineers and unfortunately is often lacking (see Figure 1). Most readers will have looked at the drawing, read the caption and answered the question before even reading the first paragraph. I understand this is a question in Trivial Pursuit, although I cannot confirm it. The answer universally accepted is the Great Wall of China. The real question is, is this true? I ask students, colleagues, and friends this question. Some look it up in books, like Hutchinson’s New Century Encyclopaedia (Helicon Publishing Ltd., 1995), or the Readers’ Digest Book of Facts. They choose between the supporters of the common positive answer, as proposed in the first reference, or the opponents, as in the latter. 30 Few give areasoned answer. What Iam Figure 1. What is the only man-made object really after is some common sense, apthat can be seen on Earth from the Moon?

course numerate: they needed to calculate such matters as the number of eggs required for the mortar (see Figure 2). Back to the Great Wall problem. Here is one solution. Put your thumb in front of you and hold it up to the Moon. The Moon appears as a small disc about one-sixth the size of your thumbnail. By estimation it can be said that an object as far away and as big as the Moon— which is about 3000 km across—appears about 3 mm across to the eye. If we estimate that the wall is 5m across, we can calculate by proportion what thickness it would appear to somebody on the Moon. The answer is that it is much too small for even the most powerful of telescopes, let alone the naked eye. An often expressed view that the wall can be seen because itis over 2000 Figure 2. The medieval cellarium at Fountains Abbey, England (photograph reproduced by kind permission of Yorkshire Post Newspapers). km long is patently daft. It is hoped that the wall example has demonstrated what is meant by numeracy. they believe somebody else must be skills. Bridges are built as something fit doing awfully well. for the purpose at the most economi- After learning my views on mathecal price, and they have been since time matics, many of my critics tell me that immemorial. The medieval builder had they would not like to pass over a only wood and stone, but he could conbridge that I had designed. Neither struct stone arches and knew that if a My area is mechatronics. shape, known to us as a catenary, could However, bridge-building will serve as be drawn so as to go through every would I: an example. No bridge was ever built stone, then his arch would stand up. by mathematics. They are designed and The great medieval bridges and cathebuilt by teams of engineers requiring a drals of Europe were built without vast range of skills; the more grandiose mathematicians, but the builders had to the bridge, the bigger the range of be imaginative, creative, skilled, and of >x Wa x EA ANSa _pMontvi| tiers NS rñteabones EX ‘seWandnlien N E mx N en- „= Gougnay-en Pi. >? == Vascoeuile rouyille Q i ” L NN Tyons-a-Forèt . rua o = Br zauville > Derieüiy-s-Andeile N | _FPörtreveaue RN ì Monto fort’ le Becpet 19 Brionne Lisieux Gisors gt-/ ae >” DIA Les Andelys: 3 N =>; , Nest: Clair", NN y 7 cal “Magny:-en-Vex Ordece 4 ) = ( q eVimoutiers \ = 7 > 08.7 EN N 2 Sia Neuve = Figure 3. Map of France, locating Vernon. *Conches- en- È 5 Mantesy <= uche i = = =~ Figure 4. The tiled forecourt in Vernon—is there a pattern?

As a teacher of engineering, I have B. More Mathematics Required up of five different blocks: 3 X 2, yelproblems with students and their numerical skills even if their mathematto Explain an Ingenious Tiling low; and 2 x 2 and 2 x 1 in both black Pattern in Vernon, France and white, but their arrangement apics is good. I recently did an overseas R.J. Holroyd pears to be neither regular nor irreguanother's While on holiday in France I visited the pattern might be an example of aslar. exchange with another teacher, which involved + us taking one A mathematician suggested that classes. I found his students similarly Vernon, a small town that stands on the ymptotic periodicity and referred to poor at numerical work, so maybe this Seine about midway between Paris and me to some algebra textbooks, but so is Numeracy the sea. (See figure 3.) About a mile upfar as I could see these did not shed might be boring and mathematics exstream is the small village of Giverny, any light on the problem. Out of my citing: I confess to having found this so with the house and gardens where depth at school. However it is numerical Monet spent the last half of his life. mathematics, apparently so different a universal problem. about these mathematician’s skills that are most lacking in engi- While walking around the town, I came from the usual engineering computaneering students. Perhaps mathematiacross a recently built arts centre (the tions, I would welcome any enlightencians should address themselves to Espace ment about this fascinating pattern— this problem. with an intriguingly tiled forecourt. I Culturel Philippe Auguste) or lack of pattern. was told that the workmen had a plan Leeds Metropolitan University for laying the blocks, but no one could 104 Arbury Road Leeds LS1 3HE tell me how the plan was produced. Cambridge CB4 2JF United Kingdom The pattern, shown in figure 4, is made United Kingdom EXPAND YOUR MATHEMATICAL BOUNDARIES! Mathematics Without Borders A History of the International Mathematical Union S As told by Professor Olli Lehto, the history of the Intemational Mathematical Union (IMU) is surprisingly compelling The twentieth century has been fraught with tremendous international conflict, but there has also been a great deal of Congresses throughout the world, effectively bridging cultural and political gaps by uniting scientists through their genuine love and appreciation for mathematics. For anyone strictly interested in the mathematical and organizational details of the IMU Congresses, this book will serve as an excellent resource. However, Mathematics Without Borders also takes time to focus on the individuals, many of them leading mathematicians of the twentieth century, and their stories-told against the backdrop of world events. Contents: Prologue to the History of the IMU e The Old IMU (1920-1932) e Mathematical Cooperation Without the IMU (1933-1939) e Foundation of the New IMU (1945-1951) e The IMU Takes Shape (1952-1954) e Expansion of the IMU (1955-1958) e The IMU and Intemational Congresses (1958-1962) e Consolidation of the IMU (1963-1970) e North-South and East-West Connections (1971-1978) + Politics Interferes with the IMU (1979-1986) e The IMU and Related Organizations « The IMU in a Changing World (1986-1990) 1998/368 PP., 53 FIGS./HARDCOVER/$35.00/ISBN 0-387-98358-9 Order Today! Call: 1-800-SPRINGER + Fax: (201)-348-4505 + Visit: {http://www.springer-ny.com} 5/98 Promotion #H232