# Thread: Dimension of cantor middle-half set(fractal)?

1. ## Dimension of cantor middle-half set(fractal)?

for a cantor middle-half set K, I think it is one-dimensional because it is on a real line. But the book say that it is 1/2-dimensional.
Consider KxK, I think it is 2-dimensional because it is a square(containing many small squares) on a plane. But it is a 1-dimensional set. Why?

2. Originally Posted by ldpsong
for a cantor middle-half set K, I think it is one-dimensional because it is on a real line. But the book say that it is 1/2-dimensional.
Consider KxK, I think it is 2-dimensional because it is a square(containing many small squares) on a plane. But it is a 1-dimensional set. Why?
Well, when you think about it a line (interval) in $\displaystyle \mathbb{R}$ has measure equal to it's length while the cantor set has measure zero. It is tiny. Almost every real number is not in the Cantor set. So it sort of makes sense that it is not 1-dimensional.

A line is 1-dimensional, and a line has measure greater than zero.

The Cantor set is not a point, and so really shouldn't have dimension zero.

Does that help

3. thank you!!