1.4. CONTRUCTING INTERESTING TILINGS 17

Figure 1.15. Projecting from two dimensions to one

successive points were projections of (m, n) and (m + 1, n). We can then

identify L with the real line, with p corresponding to the origin.

The perceptive reader will have noticed that I did not say whether S

was constructed from the open unit square or the closed unit square. If L

goes through an integer point (m, n), then both (m + 1, n) and (m, n + 1)

will be on the boundary of S. Do we include one point, the other, or both in

Λ1? The set of points p for which this problem arises has measure zero, but

is still infinite. For each irrational α, let Ωα

0

be the set of tilings constructed

this way in which L does not contain any integer points. Then let Ωα be

the completion of Ωα

0

in the tiling metric.

If m and n are integers, then the tiling obtained from p = (x0 +m, y0 +n)

is exactly the same as that obtained from p = (x0, y0). Conversely, if (x0, y0)

and (x1, y1) do not differ by an integer vector, then the tilings built from

these two choices of p are not the same. This implies that Ωα

0

is isomorphic

to the 2-torus, minus the points corresponding to lines L that hit integers.

This set of singular lines is itself a line in the 2-torus, albeit one that winds

densely.

To understand the points of Ωα that are not in Ωα,

0

we consider tilings

based on the point p = (m + , n) in the limit of → 0. As →

0+,

the

tilings converge in the tiling metric and contain the projection of the point

(m + 1, n), but not the projection of (m, n + 1). As →

0−,

the tilings

contain the projection of (m, n + 1), but not that of (m + 1, n). The two

limits are otherwise identical. The upshot is that there are two tilings in Ωα

that correspond to p = (m, n), and these points are not close in the tiling

metric! Topologically, the space Ωα consists of the 2-torus with an irrational