Sierpinski Gasket coordinate description

Question

I was reading Gerald Edgar's "Measure, Topology, and Fractal Geometry" when I came across this exercise

Let coordinates $(u,v)$ be defined in the plane with origin at one corner of the triangle $S_0$, and axes along two of the sides of $S_0$. Then $(u,v)$, with $0\leq u\leq 1, 0\leq v\leq 1$, represent a point of the Sierpinski gasket iff the base $2$ expansion of $u$ and $v$ never have $1$ in the same place.

($S_0$ is the starting triangle with side one) How can I prove it? I don't see the exercice being true. I'm confused about this point []in the picture, which i think has coordinates $(0.1,0.1)$ in base $2$, contradicting the thing I want to prove.

Answer 1

Technically, the points meeting the stated criteria do belong to the Sierpinski gasket. However, they comprise a proper subset of the gasket--that is, the "only if" assertion does not hold, as you correctly observed that $(u,v) = (1/2, 1/2)$ is a point belonging to the gasket yet does not satisfy the binary expansion criterion.

In fact, as a necessary property of strict self-similarity, there is an infinite family of such points. Consider the set of affine transformations that map the entire gasket to one of its three sub-triangles. The image of $(1/2, 1/2)$ under the composition of any finite combination of these mappings will also be a point in the gasket that does not satisfy the binary expansion criterion. So for instance, $(1/4, 1/4)$, $(1/4, 3/4)$, $(1/8, 1/8)$, $(1/8, 3/8)$, $(1/8, 5/8)$, $(1/8, 7/8)$, $(1/16, 1/16)$, are also such exceptional points.

Answer 2

The statement of the exercise might be a little bit imprecise, but it can be made correct. Indeed, Edgar seems to hint that this might be needed: in the paragraph following the exercise, he states:

Another description of the condition on the base 2 expansions of $u$ and $v$ is to say that the sum $u+v$ can be computed (base 2) without carrying. You will also need to take into account the numbers with two different expansions in base 2.

I believe a plausible interpretation of the exercise is

Let coordinates $(u,v)$ be defined in the plane with origin at one corner of the triangle $S_0$, and axes along two of the sides of $S_0$. Then coordinates $(u,v)$ with $0 \le u,v \le 1$ represent a point of the Sierpinski gasket if and only if there is a choice of base 2 expansions for $u$ and $v$ such that a $1$ never appears in the same place of both coordinates.

For example, if a $u$-coordinate has an expansion which ends in $0\overline{1}$, it can instead be written with an expansion ending in $1\overline{0}$; and if a $v$-coordinate has an expansion which ends in $1\overline{0}$, it can instead be written with an expansion ending in $0\overline{1}$. If this choice is always made, a point $(u,v)$ is in the Sierpinski gasket if and only if $u$ and $v$ do not have any $1$s in the same place.

For the point suggested in the original question: $$ (0.1_2, 0.1_2) = (0.0\overline{1}_2, 0.1_2).$$ Note that these two expansions do not, in fact, have any $1$s in the same place.

Addendum: As a general bit of advice for reading Edgar's excellent text (as well as many other texts in mathematics), it is good to think about the exercises as tools which are meant to get you thinking about the material. I suspect that Dr Edgar is aware that the statement of the exercise is not exactly complete, and that he wants you to think about these kinds of edge cases. The fact that you figured out that something was missing indicates that you were doing your due diligence. Молодец.