# Visualizing/Conceptualizing the Cantor Set on the Circle

So I understand what the cantor set is geometrically when referencing a line, but I am currently working on a problem that deals with the Cantor set on a circle, and for the life of me I can't visualize or conceptualize what this means.

The way my book define the cantor set was as follows: $$\bigcap_{n=0}^\infty C^n$$ Where: $$C^0=[0,1], C^1=[0,\frac{1}{3}]\cap[\frac{2}{3},1],....$$ And that makes sense to me. But what I don't get is this idea of the cantor set on a circle. So I guess I'm basically asking what each iteration of the cantor set on a circle looks like, and what it means to for the cantor set to be on something like a circle.

Due to lack of context, I can ony guess, but I suppose that the author of that text has in mind something like $$\varphi(C)$$, where $$\varphi\colon[0,1]\longrightarrow S^1$$ is the map defined by $$\varphi(x)=\bigl(\cos(2\pi x),\sin(2\pi x)\bigr)$$. Although $$\varphi$$ is not a homeomorphism, it turns out that $$\varphi(C)$$ and $$C$$ are homeomorphic. • @Christopher Quinn La Fond Jr.: visualizing the cantor set on the circle --- Visualize the points of the usual Cantor set in the unit interval $[0,1]$ colored yellow, and the complementary open intervals colored black. Now visualize picking up the unit interval and bending it into a circle so that the points $0$ and $1$ are merged into one point. The part that's colored yellow on the circle will be a Cantor set on the circle. Oct 17 '20 at 9:27