- #1
nate808
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Lets take the cantor middle thirds set as an example (Iterate by removing the middle portion of a line segment) If this was done an ACTUAL INFINITE number of times, would you be left with line segments, points, or neither?
Nope; measure is (countably) additive. If P and Q don't overlap, then [itex]m(P) + m(Q) = m(P \cup Q)[/itex]. In particular, if Q is a set of measure zero, and P is what's left after removing Q from a line segment, then the measure of P is the same as of Q.If you take away a set with measure of zero which is not empty, from something, like a line segment, do you alter the characteristics of that line. For example, taking away a point from a line makes it unconnected. Would the length change?
Because the sum of the lengths of the intervals you removed is 1!I also don't understand how the measure would be zero.
Your usage of "it" is very ambiguous.I guess its just my perception, but it seems strange that fractals are supposed to be magnified versions of the original, so why wouldn't it still have a length no matter how many times you do it as long as the original senment had a length?
Fractal Geometry Cantor Middle Thirds is a mathematical concept that involves creating a pattern by repeatedly dividing a line segment into three equal parts and removing the middle third. This process is then repeated on the remaining line segments.
Yes, the line segments in Fractal Geometry Cantor Middle Thirds are infinite as the pattern continues infinitely with each iteration of dividing and removing the middle third.
Yes, points can exist within the line segments in Fractal Geometry Cantor Middle Thirds. The removed middle third creates gaps that can be filled with points, making the line segments infinitely detailed.
Yes, Fractal Geometry Cantor Middle Thirds is considered a self-similar pattern because the overall shape is repeated at smaller and smaller scales with each iteration.
Fractal Geometry Cantor Middle Thirds has applications in various fields such as physics, computer graphics, and finance. It can be used to model natural phenomena like coastlines and clouds, create intricate graphics, and analyze financial data.