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*

### Like this:

Like Loading...