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The Circle Squared, Beyond Refutation MAA Committee on Participation of Women Sponsors Panel On Volume 9, Number 2 THE NEWSLETTER OF THE MATHEMATICAL ASSOCIATION OF AMERICA March-April 1989 The Circle Squared, MAA Committee on Participation of Beyond Refutation Women Sponsors Panel on "How to Stan Wagon Break into Print in Mathematics" Frances A. Rosamond The circle has finally been squared! No, I do not mean that some­ one has found a flaw in the century-old proof that 7r is transcen­ This panel at the Phoenix meeting was intended to provide en­ dental and that a straightedge and compass construction of 7r ex­ couragement for women mathematicians, but its "how to" lessons ists. I am instead referring to a famous problem that Alfred Tarski will be useful to all mathematicians. The essentials of writing a posed in 1925: Is it possible to partition a circle (with interior) into research paper, expository article, or book from first conception finitely many sets that can be rearranged (using isometries) so that to final acceptance were discussed by authors and editors. Panel they form a partition of a square? In short, Tarski's modern circle­ members were: Doris W. Schattschneider, Moravian College; Joan squaring asks whether a circle is equidecomposable to a square. P. Hutchinson, Smith College; Donald J. Albers, Menlo College; A proof that the circle can indeed be squared in Tarski's sense has and Linda W. Brinn. Marjorie L. Stein of the US Postal Service was just been announced by Mikl6s Laczkovich (Eotvos t.orand Univer­ moderator. The organizer was Patricia C. Kenschaft, Chair of the sity, Budapest). MAA Committee on the Participation of Women. This article summarizes the panelists' suggestions. Complete pa­ As occasionally happens in mathematics, this particular problem is pers by panelists will appear in the NEWSLETTER of the Associa­ easier to solve in higher dimensions, where the isometry group is tion for Women in Mathematics. Single copies will be available richer. Because of the Banach-Tarski Paradox, a ball in three-space by sending $3 to AWM at Box 178, Wellesley College, Wellesley, is equidecomposable with a cube; in fact, and this is why it is called Massachusetts 02181. a paradox, any two balls and cubes in three-space are equide­ composable, no matter what their volume (using nonmeasurable GETTING STARTED To find a topic and start to write you must sets as pieces, obviously). But in the plane, where the isometry overcome the familiar excuses described by Doris Schattschneider: group is solvable, Lebesgue measure can be extended to a total, finitely-additive, isometry-invariant measure (Banach, 1923), which I don't have any ideas. There's no one to talk to at my institution. means that equidecomposable sets must have the same measure. I don't teach at the graduate level. It is not hard to see that any two polygons of the same area are My first paper was rejected. equidecomposable (Tarski, 1924, using the classic Bolyai-Gerwien I don't have enough time. ("Who does?" quipped Schattschneider.) Theorem that any two polygons of the same area are geometrically (continued page 9 ) equidecomposable (I.e., all pieces are triangles, and boundaries are ignored) and some easy set theory to deal with the boundaries of the pieces). This led to Tarski's circle-squaring problem. His re­ sult for polygons showed that any polygon could be squared in the Bylaw Changes for August Business Meeting page 4 modern sense. And so Tarski asked about the circle. For further Wolf Prize Awarded to Calderon and Milnor page 4 historical details and references see my book, THE BANACH-TARSKI PARADOX (Cambridge, 1985), or my article in Vol. 3, NO.4 (1981) Summer Meeting in Boulder pages 11-35 of THE MATHEMATICAL INTELLIGENCER. For a discussion of recent, but Meeting Deadlines page 11 pre-Laczkovich, work, see a forthcoming paper by Richard Gard­ MAA Program pages 12 and 13 ner in Ani DEL SEMINARIO MATEMATICO E FISICO DELL' UNIVERSITA 01 MODENA. Preregistration Form page 33 Housing Form page 34 There had been very little progress on this problem, and it had be­ come somewhat notorious because of the paucity of partial results Minicourse Registration page 35 and the difficulty of even guessing with any (continued page 2 ) 2 FOCUS March-April 1989 ( "Circle Squared, Beyond Refutation" continued from page 1 ) confidence which way it might go. Commenting on this problem from the in the introduction to THE SCOTIISH BOOK (Birkhauser, 1981), Paul Erdos has written: "This is a very beautiful problem. If it were my President's desk.. problem I would offer $1,000 for it-a very nice question, possibly very difficult. Really, one has no obvious method of attack." L1da K. Barrett, Mississippi State University Some partial results were known. Dubins, Hirsch, and Karush proved in 1963 that the circle could not be squared if "Jordan scis­ Committees,the Work of the sors" were used to form the pieces. And Richard Gardner proved in 1985 that the circle-squaring is impossible if the group of allow­ Association, and You able motions is restricted to a discrete subgroup of isometries of the As president of the MAA, I will from time to time comment on the plane. Inspired to attack the problem by a lecture Gardner gave in workings of the Association and on other topics as seen from the Italy, Laczkovich has come up with a magnificent proof. Indeed, he president's viewpoint. I would like to begin this process with a dis­ obtains the stronger result that the circle is equidecomposable to 50 cussion of the Association's committee structure. At each MAA the square using translations alone (and fewer than 10 pieces). meeting, in addition to the formal program, there is an immense The adequacy of translations is quite surprising, for it reveals a pre­ amount of activity that takes place in committee meetings. The viously unsuspected fact even in the case of triangles! Laczkovich committee schedule for the Joint Mathematics Meetings in Phoenix proves, en route to his main result, that any two polygons of the called for meetings of 23 committees. Committee meetings be­ same area are equidecomposable by translations alone. This is a gan on the Sunday before the meeting with an all-day meeting of completely new result, and definitely false for the case of geometric the Selection Committee for the new Executive Director. On Mon­ equidecomposability (i.e., the classical case where the dissection day, the Executive and Finance Committees met all day, and the is effected by straight lines). The only convex polygons that are ge­ Board of Governors met on Tuesday. Concurrent with the regu­ ometrically equidecomposable to a square using translations alone lar meeting sessions on Wednesday through Saturday, there were are the centrally symmetric ones (Hadwiger and Glur; see §10 of committee meetings, breakfast, morning, lunch, afternoon, night; V. G. Boltianskii's book, HILBERT'S THIRD PROBLEM, Washington: V. and on Sunday, there was a day-long meeting of the MAA Science H. Winston, 1978). Policy Committee. How can an average member of the Association relate to this activ­ Laczkovich's method makes heavy use of the discrepancy of a ity? How can a member with an interest in a particular committee planar set. To illustrate this notion and give a flavor of his proof, learn what the committee is doing? How can a member who would consider the main step in showing that an isosceles right triangle like to serve on a committee have an opportunity to take part? Let is translation-equidecomposable to a square. Let T be the triangle me answer these questions in order. with vertices (0,0), (1,0), and (1,1). If L is a subset of the plane then Ldenotes the reduction of L modulo the unit square. The At the meeting in Boulder, we will have available at the registration discrepancy of Lwith respect to T is the difference between the area desk a list of the committees that will be meeting, along with the of T (0.5) and the fraction of the points of Lthat lie in T. Laczkovich names of their chairs. Any member of MAA who wishes to attend a committee meeting and simply listen to the proceedings is welcome uses a sequence of sets, Ln , defined below that depend on vectors x, y, and u in the plane. His method requires points x, y in the to do so. A number of the committees (for example, the Commit­ plane such that x, y, (0,1), and (1,0) are linearly independent over tee on Computers in Mathematics Education) have over the years welcomed all with an interest to visit the committee session. In a the rationals and such that for all u in the plane the discrepancy of number of cases, new members of the committee have come from approaches zero rapidly as a function of n. Here L denotes Ln n these visitors. the set { u + kx + my 0 :S k, m < n }, and "rapidly" means that the order of magnitude of the discrepancy of l.n is The Association regularly prints a committee list showing the mem­ 7 2 bounded by (log n ) / n . If one can find points x and y that work bership of each committee, a list of all members who serve on simultaneously for T and for S, a subsquare of the unit square national committees showing which committees they serve on, and having area 0.5, then T and S can be shown to be translation­ for each of our 29 sections a list of its members who serve on na­ equidecomposable. Laczkovich shows that such points exist in tional committees.
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