The straight side of sliced circles.
Cut a circle out of a sheet of paper. Then cut the circle into pieces so that the pieces, when fitted back together, form a square having the same area as the original circle. Such a task seems impossible. How do you get rid of the curves? But a Hungarian mathematician has now proved that it is theoretically possible to cut a circle into a finite number of pieces and rearrange them into a square. Miklos Laczkovich of Eotvos Lorand University in Budapest accomplishes this mind-bending feat in a 39-page manuscript now under study by mathematicians throughout the world. So far, no one has detected any flaws in his reasoning.
His work bears on the fundamental questions of what mathematicians mean by the notion of curvature and how they decide when two objects have the same area. The proof shows that present ideas about area are correct but curves and straight lines are so different they can be converted into each other only by using strange manipulations.
The problem solved by Laczkovich relates to an ancient riddle known to Archmides and other Greeks scholars. At issue is whether one can use just a ruler and compass to draw a square with an area equal to that of a given circle. The problem remained unsolved for centuries despite the efforts of numerous mathematicians, both amateur and professional. In the end, the solution hinged on the properties of the number pi, the ratio of a circle's circumference to its diameter. A circle and a square have equal areas only if the ratio between a square's side and a circle's radius equals the square root of pi. In 1882, mathematicians proved that pi is what they call a transcendental number, effectively ruling out the possibility of constructing a square out of a circle using only ruler and compass.
Laczkovich tackled a version of the problem originally devised in 1925 by mathematician and philosopher Alfred Tarski. Tarski removed the ruler-and-compass restriction and asked whether there is any way to cut up a circle into pieces that could be rearranged into a square of the same area. In the previous year, Tarski and Stefan Banach had proved a remarkable analog of the same conjecture in three dimensions, showing paradoxically that a sphere could be cut up and rearranged not only into a cube of the same volume but also into a cube of twice the volume. In fact, a sphere sliced up in just the right way could be rearranged into virtually and shape of any size.
Mathematicians who studied Tarski's circle problem strongly suspected no way existed to cut up a cirlce to make a square without losing even a single point out of the circle. In 1963, Lester E. Dubins, Morris W. Hirsch and Jack Karush of the University of California, Berkeley, proved the problem couldn't be solved by cutting a circle into "ordinary" pieces -- which have well-behaved, relatively smooth boundaries -- no matter how many such pieces are used.
Laczkovich has now proved that "squaring the circle" is possible, provided that the pieces have the right form. His pieces encompass an array of strange, practically unimaginable shapes. Although some resemble those in an ordinary jigsaw puzzle, others are collections of single, isolated points, curved segments or twisted bits riddled with holes. Remarkably, assembling a square from these pieces of a circle is possible simply by sliding the pieces together. No piece has to be rotated to fit into place. The resulting square has no gaps and no overlapping pieces. Laczkovich estimates this effort requires about 10 pices -- almost as many pieces as there are water molecules in the Mediterranean Sea.
Laczkovich's proof applies not only to circles but to almost any plane figure with a mathematically well-behaved boundary. Any such figure can be cut and rearranged into a square of the same area with no gaps or overlaps.
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|Date:||Jul 8, 1989|
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