- #1
Dragonfall
- 1,030
- 4
How can I write down a closed form expression for the endpoints used in the construction of the Cantor set? i.e., 0, 1, 1/3, 2/3, 1/9, 2/9, 7/9, 8/9, etc.
No, it isn't. when n= 1, the first "cut", the enpoints are 1/3 and 2/3 so that the intervals left are [0, 1/3], [2/3, 1] but when you remove the middle third of those, the endpoint are 0, 1/9, 2/9, 1/3= 3/9, 2/3= 6/9, 7/9, 8/9, 1. In your notation, k/3n, with n= 2, k is not 4 or 5. It gets worse with higher values of n.AKG said:The closed form is k/3n for all n, and for all 0 < k < 3n.
HallsofIvy said:Dragonfall, think in terms of base 3. Writing a number between 0 and 1 in base 3, the first "digit" (trigit?) may be 0, 1, or 2. If 0, the number is between 0 and 1/3; if 1, between 1/3 and 2/3; if 2, between 2/3 and 1. When you remove the middle third you remove all numbers have a "1" as first digit. Now, all numbers between 0 and 1/3 must have .00, .01, or .02 as first two digits, all numbers between 2/3 and 1 must have .20, .21, or .22 as first two digits. When you remove the middle third of each you remove all numbers that have 1 as the second digit. Do you see what happens in the limit?
A Cantor Set is a fractal set constructed by repeatedly removing the middle third of a line segment. It is named after Georg Cantor, a mathematician who first described it in the late 19th century.
A Cantor Set has an infinite number of endpoints, as the process of removing the middle third of a line segment continues infinitely.
The endpoints of a Cantor Set are uncountably infinite, meaning that they cannot be put into a one-to-one correspondence with the counting numbers. This is because the endpoints are not discrete points, but rather form a continuous line.
The endpoints of a Cantor Set do not follow a specific pattern or structure, as they are created through a recursive process of removing the middle third of a line segment. However, they do exhibit self-similarity, meaning that each section of the set is similar to the whole set.
No, the endpoints of a Cantor Set cannot be represented as a decimal or binary number. This is because the endpoints are not discrete points, but rather form a continuous line, making it impossible to assign a specific numerical value to each endpoint.