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Every month around 4,000 articles, written by researchers around the world, are published in countless scientific journals. In January 2002 the American Mathematical Association (AMA) chose to highlight a paper written by the American mathematician Thomas Hales for its significance.

For once a strictly theoretical mathematical piece could possibly also appeal to tradesmen. Tile layers who cover bathroom, kitchen, and porch floors with tiles of all shapes and forms may find the paper of interest. Maybe one or the other among them has pondered the question as to which shape has the smallest perimeter while covering the same region as other tiles. This is the very issue addressed in the article selected as outstanding by the AMA for its mathematical, if not its practical, relevancy.

A tile layer can take tiles of triangular, square, pentagonal, hexagonal, heptagonal, and octagonal shape—all having the same surface area—measure their perimeters and verify which has the smallest circumference. So far, so good. But it would be too soon to start mixing the grout. If the tile layer tried to cover the kitchen floor with pentagonal tiles, he would soon notice that gaps appear between the tiles. Pentagons are unusable as floor coverings because, when aligned next to each other, they do not fit together seamlessly. It is the same story for heptagons, octagons, and most other regular polygons—in none of these cases can the kitchen floor be completely covered without leaving any spaces in between the tiles.

The ancient Pythagoreans were quite familiar with this fact of geometry. They knew that among all regular polygons, only triangles, squares, and hexagons could cover

an area without leaving any spaces between them. Any other regular polygon invariably produces gaps.

So the tile layer’s choices are quite limited. All he can do is examine which of the three admissible shapes has the smallest perimeter. Take an area of 100 square centimeters: The triangular-shaped tiles have a circumference of 45 centimeters, the squares a circumference of 40 centimeters, and the hexagons—with a mere 37 centimeters—have the smallest circumference. Pappus of Alexandria (approximately AD 290–350) was already aware that hexagons were the most efficient regular polygons. So were honeybees. They want to store as much honey as possible in containers using the least possible amount of wax. So they build honeycombs in hexagonal shapes.

The reason the hexagon has the smallest perimeter is that, of the three possible tile shapes, it is most similar to the circle. And among all geometric shapes the circle has the smallest perimeter. To encircle an area of 100 square centimeters, the circle requires a perimeter of only about 35 centimeters.

Can we now claim that the problem is solved? By no means. Who says that the floor covering must consist of a single tile shape? And why should the shape have to be regular or straight edged? Indeed, the tiles do not even have to be convex; imagine tiles that bulge inward or outward. So floors could very well be tiled with varying shapes that would, by the way, only enhance their aesthetic appeal, as M. C. Escher has so masterfully shown us in his prints.

The general question that mathematicians have asked themselves is, which tile taken from the multitude of imaginable forms and shapes has the smallest perimeter? For 1,700 years it has been conjectured that the solution is the hexagonal honeycomb. All that was missing was a proof.

Hugo Steinhaus (1887–1972), a Polish mathematician from Galicia, was the first to make significant inroads. He proved that as far as tiles that consisted of only a single shape were concerned, hexagons represented the least-perimeter way to cover a floor. This was progress

since, in contrast to Pappus, Steinhaus also allowed irregular-shaped tiles. In 1943 the Hungarian mathematician László Fejes Toth (1915–2005) took the next step. He proved that among all convex polygons it was the hexagon that had the smallest perimeter. Contrary to Steinhaus, Fejes Toth did not require the floor to be covered with just one kind of tile but permitted the use of collections of differently shaped tiles. And yet his theorem ignored tiles that were not straightedged.

A completely general proof was provided only in 1998 by Thomas Hales. Just a few weeks earlier he had solved the oldest open problem in discrete geometry, the 400-year-old Kepler’s conjecture. The question was how identical spheres could be packed as tightly as possible. Hales proved that the densest way to pack spheres is to stack them in the same manner as grocers stack oranges: Arrange them in layers, with each sphere resting in a small hollow between three spheres beneath it. Hale’s proof made headlines throughout the world. But the young professor did not waste his time bathing in glory.

On August 10, 1998, the Irish physicist Denis Weaire of Trinity College in Dublin read the news in the newspaper. Without wasting any time, he sent Hales an e-mail in which he drew his attention to the honeycomb problem and added the challenge: “It seems worth a try.”

Fascinated, Hales set to work. To prove Kepler’s conjecture, he had spent five years and literally burned the fuses of computers. In comparison, the new problem was a picnic. He merely required pencil and paper and half a year’s work.

Hales started by dividing the infinitely large floor space into configurations of finite size. Then he developed a formula that brought a tile’s area into relation to its perimeter. Next, he turned his attention to convex shapes. For every convex tile—a tile that bulged outward—there had to be corresponding tiles that bulged inward. With the help of the area-to-perimeter formula, Hales was able to prove that tiles that bulged inward require more perimeter than was saved by outward-bulging tiles. Overall this meant that round-edged polyhedra provide only dis-

advantages. They were thus ruled out as contenders for the title of smallest-perimeter tiles.

Since only straightedged tiles remained as candidates, the rest was clear. After all, Fejes Toth had already proven that regular hexagons represent the best combination of tiles from among all straightedged polygons. Thus Hales had provided conclusive proof that bees do exactly the right thing when constructing hexagonally shaped honeycombs.