Squaring The Plane
In 1975, Solomon Golomb raised the question whether the whole plane can be tiled by squares whose sizes are all natural numbers without repetitions, which he called the heterogeneous tiling conjecture. This problem was later publicized by Martin Gardner in his Scientific American column and appeared in several books, but it defied solution for over 30 years. In Tilings and Patterns, published in 1987, Branko Grünbaum and G. C. Shephard stated that in all perfect integral tilings of the plane known at that time, the sizes of the squares grew exponentially.
Recently, James Henle and Frederick Henle proved that this, in fact, can be done. Their proof is constructive and proceeds by "puffing up" an L-shaped region formed by two side-by-side and horizontally flush squares of different sizes to a perfect tiling of a larger rectangular region, then adjoining the square of the smallest size not yet used to get another, larger L-shaped region. The squares added during the puffing up procedure have sizes that have not yet appeared in the construction and the procedure is set up so that the resulting rectangular regions are expanding in all four directions, which leads to a tiling of the whole plane.
Read more about this topic: Squaring The Square
Famous quotes containing the word plane:
“Even though I had let them choose their own socks since babyhood, I was only beginning to learn to trust their adult judgment.. . . I had a sensation very much like the moment in an airplane when you realize that even if you stop holding the plane up by gripping the arms of your seat until your knuckles show white, the plane will stay up by itself. . . . To detach myself from my children . . . I had to achieve a condition which might be called loving objectivity.”
—Anonymous Parent of Adult Children. Ourselves and Our Children, by Boston Women’s Health Book Collective, ch. 5 (1978)