# Does Cantor's Comb Have Other Applications in Math?

• MathematicalPhysicist
In summary, it was mentioned that the Cantor set has two known applications in math - it can be used to visualize the Cantor set and it falls under the topic of fractals. Its use in fractals is due to its self-similarity property and possible formation by iteration, although its exact fractal dimension is uncertain. It was also suggested that it must have a fractal dimension, even if it is 0.
MathematicalPhysicist
Gold Member
does it have other applications in maths?

After doing some research in google, I found 2 results.
1) It is used to visualize the Cantor set
2) It can be put under the topic, fractals (I prefer a 2D fractal to a one-dimension one )

Besides these 2, I don't know whether it has any other application in math

I once did a homework assignment that started with the cantor set and ended with the binary number system. Along the way the equivalance of 1 and .999... or .111... binary fell out.

Originally posted by KL Kam
After doing some research in google, I found 2 results.
1) It is used to visualize the Cantor set
2) It can be put under the topic, fractals (I prefer a 2D fractal to a one-dimension one )

Besides these 2, I don't know whether it has any other application in math
how is it used in fractals?

Originally posted by loop quantum gravity
how is it used in fractals?
The 3 properties of fractals are
1. Self-similarity
2. Fractional dimension
3. Formation by iteration

For Cantor's Comb, it has property 1. I guess it can be formed by iteration but I'm not sure. Also I'm not sure whether it has fractional dimension. I'm not an expert and I think some mathematicians here can clear it up a bit.

I'm in no sense an expert or even knowledgeable on this issue, but wouldn't it have to have a fractal dimension? It could be 0, which is still a fractal dimension. I can't remember exactly, but I think Cantor's Dust has a fractal dimension between 0 and 1 , and Cantor's box has a fractal dimension of 0.

Here is a good website I found on fractals, but it dosn't seem to be working at the moment.

http://library.thinkquest.org/26242...torial/ch4.html

Last edited by a moderator:

## 1. What is Cantor's Comb and how is it used in math?

Cantor's Comb is a geometric fractal pattern discovered by mathematician Georg Cantor. It is created by connecting the midpoints of each line segment in a series of squares. In math, it is often used to demonstrate the concept of infinity and to study the properties of fractals.

## 2. What are some common applications of Cantor's Comb in mathematics?

Cantor's Comb has several applications in mathematics, including being used as a model for the construction of self-similar sets, as a tool for studying the properties of infinite series, and as a visual representation of the concept of infinity.

## 3. Can Cantor's Comb be applied in other fields besides mathematics?

Yes, while Cantor's Comb is primarily used in mathematics, it has also been applied in other fields such as physics, computer science, and art. Its self-similar and infinite properties have inspired research in various fields.

## 4. What are the limitations and challenges of using Cantor's Comb in mathematical applications?

Some of the limitations and challenges of using Cantor's Comb in mathematical applications include its complexity and the difficulty in calculating its properties, as well as the challenge of visualizing and understanding the concept of infinity.

## 5. Are there any real-world applications of Cantor's Comb?

While Cantor's Comb may not have direct real-world applications, the concepts and properties it represents have been applied in fields such as telecommunications, finance, and biology. Additionally, the study of Cantor's Comb has led to advancements in other areas of mathematics and science.

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