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. Pick any two points that are members of the set, and we can’t draw a line that separates those two points without crossing the set somewhere.**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 (defined by the equation*x**² + y*² = 25, for example) 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, for example defined by*x**² + y*² < 25 —*is*simply connected, as anything not a part of 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*.

Our last meeting will be on **Wednesday, January 22nd in room G116**. Our last topic will be a a brief history lesson of the discovery of the Julia Set that starts in Newton’s Method for Approximation.

By next Wednesday, 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*