How is Cantor set similar to coin tosses?

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    Cantor Set
In summary, the Cantor set can be seen as a model for an infinite series of coin tosses, with the elements of the set representing sequences of heads and tails. By using a base-3 representation, the Cantor set can be easily mapped to a sequence of coin tosses. However, the ternary numbers containing 1's in this model do not have a clear interpretation.
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EnumaElish
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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?
 
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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
[tex]f:C\rightarrow \{0,2\}^\mathbb{N}: 0.a_1a_2a_3...\rightarrow (a_1,a_2,a_3,...)[/tex]
with inverse
[tex]f^{-1}:\{0,2\}^\mathbb{N}\rightarrow C:(a_n)_n\rightarrow \sum_{k=1}^{+\infty} a_k 3^{-k}[/tex]

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...##.
 
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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)?
 
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They don't really have an interpretation.
 

What is the Cantor set?

The Cantor set is a fractal set made up of points that are removed from a line segment in a specific pattern. It was first described by German mathematician Georg Cantor in 1883.

How is the Cantor set constructed?

The Cantor set is constructed by starting with a line segment and dividing it into three equal parts. The middle third is then removed, leaving two line segments. This process is repeated infinitely on each remaining line segment.

What are the properties of the Cantor set?

The Cantor set is self-similar, meaning that it is made up of smaller copies of itself. It is also uncountable, meaning that it contains an infinite number of points. Additionally, the Cantor set has a fractal dimension of ln(2)/ln(3), which is approximately 0.63.

How is the Cantor set similar to coin tosses?

The Cantor set and coin tosses both involve a process of repeatedly removing a portion of the original set. In the case of the Cantor set, the middle third is removed, while in coin tosses, one option is removed (heads or tails). Both also result in a set with uncountable elements.

What are some applications of the Cantor set?

The Cantor set has applications in various fields, including computer science, physics, and economics. It is used in the construction of fractal antennas, which have applications in wireless communication. It is also used in the study of dynamical systems and chaos theory.

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