# How is Cantor set similar to coin tosses?

• I
Science Advisor
Homework Helper
The wikipedia page on the Cantor set states that it is a model for an infinite series of coin tosses. In what sense are recorded coin outcomes similar to a set of points on the real line?

## Answers and Replies

The Cantor sets can be defined as ##\{H,T\}^\mathbb{N}## equipped with the product topology and with ##\{H,T\}## the discrete space. Instead of using ##H## and ##T##, I might as well write ##\{0,2\}^\mathbb{N}##. An element of the Cantor set is then a sequences ##(x_n)_n## which takes on elements ##0## and ##2##. It is then very easy to find a homeomorphism between this set of sequences and the Cantor set ##C##. Indeed, we set
$$f:C\rightarrow \{0,2\}^\mathbb{N}: 0.a_1a_2a_3...\rightarrow (a_1,a_2,a_3,...)$$
with inverse
$$f^{-1}:\{0,2\}^\mathbb{N}\rightarrow C:(a_n)_n\rightarrow \sum_{k=1}^{+\infty} a_k 3^{-k}$$

So more practically, given an element of the Cantor set ##C##, we can see this as a sequence of coin tosses by looking at the base-3 representation. The number ##0.0202020202...## is an element of the Cantor set and corresponds to the sequence ##HTHTHTHTHTHT....##.

• EnumaElish
Science Advisor
Homework Helper
To push the analogy a little, what would the ternary numbers containing 1's correspond to in the coin toss experiment? Could those be interpreted as "experiments we did not measure" (they are superposed)?

They don't really have an interpretation.