We discussed an open problem related to the connectedness of the Mandelbrot Set today. It seems silly, but “connected” is actually a tough idea to define mathematically. We discussed the following ideas:
- Not disconnected: That is, the set is not divided into pieces. We can’t draw a line “between” pieces of the set without crossing the set. The Mandelbrot set is not disconnected
- “Simply” connected: Both the set and its complement (everything not a member of the set) are not disconnected. A circle is not simply connected, because it divides the plane into regions inside and outside of the circle, which are disconnected from each other by the circle. A disk — a circle that includes its interior — is simply connected, as anything not in the disk is outside the disk. The Mandelbrot Set is simply connected
- “Path” connected: Any two points of the set can be joined by a path that stay in the set. We used the Topologist’s Sine Curve to define an example that is not “path” connected. The Mandelbrot Set, though, is path connected.
- “Locally” connected: Around any point of the set, we can draw a circle small enough such that the set is not connected inside of it. We used the idea of a comb defined by the line segments y = 1/2^x from x: [0,1] as an example of a space not locally connected. Mathematicians are still unsure whether the Mandelbrot is Locally Connected or not, an open problem known as the MLC.
The last topic, which we will get into tomorrow, is a brief history lesson of the Julia Set that starts in Newton’s Method for Approximation.
For tomorrow, please read this article from the November, 1991 issue of Science News, published just after an important discovery was made about the border of the Mandelbrot Set, as well as this follow-up of sorts from 2017 in Scientific American