Editing Squaring the square (section)

== Squaring the plane ==

[[File:squaring_the_plane.svg|thumb|Tiling the plane with different integral squares using the Fibonacci series
<div style="margin-left:1em;text-indent:-1em">1. Tiling with squares with Fibonacci-number sides is almost perfect except for 2 squares of side 1.</div>
<div style="margin-left:1em;text-indent:-1em">2. Duijvestijn found a 110-square tiled with 22 different integer squares.</div>
<div style="margin-left:1em;text-indent:-1em">3. Scaling the Fibonacci tiling by 110 times and replacing one of the 110-squares with Duijvestijn's perfects the tiling.</div>]]
In 1975, [[Solomon Golomb]] raised the question whether the whole plane can be tiled by squares, one of each integer edge-length, 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 describe a way of tiling of the plane by integral squares by recursively taking any perfect squared square and enlarging it so that the formerly smallest tile has the size of the original squared square, then replacing this tile with a copy of the original squared square. The recursive scaling process increases the sizes of the squares [[exponential growth|exponentially]] – skipping most integers – a feature which they note was true of all perfect integral tilings of the plane known at that time.

In 2008 James Henle and Frederick Henle proved Golomb's heterogeneous tiling conjecture: there exists a tiling of the plane by squares, one of each integer size. 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.{{r|henle}}