Erdös and the individual property of dimensions 1 and 2.

Some time ago I read a biography of Paul Erdös titled ‘my brain is open’ written by Bruce Schechter. The book contains some popular exposition of math. As a biography the book is good, but much of the mathematics and interpretation of it could be improved, though it probably gives a more or less nice view to an outsider. For example Ramsey theory is misunderstood in the book.

Here is one problem from that book. A square can be dissected into smaller squares (chess board). Also such that some of the small squares are unequal (glue the four central squares of the chess board). Can it be dissected into smaller squares such that any two of them are unequal? Erdös conjectured that it is not possible. Why did he conjecture that? Probably because a cube (3-dimensional square) cannot be dissected into smaller cubes which are pairwise unequal. Let us see why.

Suppose that the Big Cube is dissected to smaller cubes each two of them having different size. I will show that there must be infinitely many small cubes. Hence a finite dissection is impossible. Consider the bottom side of the Big Cube. It is a square and since the Big Cube is dissected, the bottom of the Big Cube is dissected by the bottoms of the small cubes which touch the bottom of the Big Cube. If there are infitely many small cubes standing on the bottom we are done. Otherwise take the smallest of these cubes and call it [tex]a_0[/tex]. It is surrounded by bigger cubes which implies that [tex]a_0[/tex] cannot touch the walls of the Big Cube. Consider now the top side of [tex]a_0[/tex] (the roof). Clearly the roof of [tex]a_0[/tex] does not touch the roof of the Big Cube. The roof of [tex]a_0[/tex] must be filled with cubes which are smaller than [tex]a_0[/tex]. That is because [tex]a_0[/tex] is surrounded by bigger cubes which give walls around the roof of [tex]a_0[/tex]. Take the smallest cube of these and call it [tex]a_1[/tex]. Again, [tex]a_1[/tex] cannot touch the walls which surround the roof of [tex]a_0[/tex] and hence is surrounded from all sides by bigger cubes which provide walls around the roof of [tex]a_1[/tex]. Continue doing this by induction and you will obtain the cubes [tex](a_i)_{i\in\mathbb{N}}[/tex] all inside the Big Cube.

It follows that an n-dimensional cube cannot be dissected into smaller n-dimensional cubes for any n>2, because given a dissection of n+1-dimensional cube, it would give a dissetion of its sides which in turn are n-dimensional cubes.

But let us return to the Erdös’ conjecture which now seems to be plausible. It turned to be wrong. The two-dimensional square can be dissected! See the picture below.

A dissection of a square into smaller pairwise unequal squares

In order to defend the title of this post, the reader may verify that the one-dimensional unit cube can be dissected into smaller pairwise unequal 1-dimensional cubes.

Update: There are some interesting related web sites (with pictures!), suggested to me by Kerkko Luosto:

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