By K. Böhmer, G. Meinardus, W. Schempp

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**Sample text**

Naturally, what we are really asking about is the proportion of diagonal to side. For convenience, let’s take the side of the square to have length 1, and write d for the length of the diagonal. Now look at this design. We have four unit squares coming together to make a 2 by 2 square. Notice that their diagonals also form a square. This square has sides of length d, so we can think of it as a unit square scaled by a factor of d. In particular, its diagonal must be d times as long as that of a unit square, so it must have length d2 .

Since the inside and outside angles combine to make a half turn, the inside angles must be 1 1 -1 2 --. 2 3 6 In particular, six of these triangles will fit together at a corner. 28 M E A S U R E M E N T Hey, this makes a regular hexagon! So as a bonus, we get that the angles of a regular hexagon must be twice those of the 1 triangle, in other words -3 . This means that three hexagons fit together perfectly. So it is possible to have knowledge about these shapes after all. In particular, now we can see why the original mosaic design works.

What is the point of making up these imaginary shapes and then trying to measure them? It’s certainly not for any practical purpose. In fact, these imaginary shapes are actually harder to measure than real ones. Measuring the diagonal of a rectangle requires insight and ingenuity; measuring the diagonal of a piece of paper is easy—just get out a ruler. There are no truths, no surprises, no philosophical problems at all. No, the issues we’re going to be dealing with have nothing to do with the real world in any way.