We wrapped up yesterday’s game of “Find the Julia Set” and reviewed some other interesting properties of the Mandelbrot Set (many of which were hinted at by Mr. Drix yesterday):
- “Area” Julia Sets come from “areas” in the Mandelbrot Set, and “string” Julia Sets come from “strings” in the Mandelbrot Set
- There are “veins” through the main cardioid of the Mandelbrot Set where we can watch the orbit of a single fixed point die towards a 2-, 3-, 4-, or any magnitude cycle. Along these veins, the fixed point forms spokes. Offset slightly from these veins and they become spirals
- Relatedly, there are balls off of the main cardioid for every possible cycle, and every variant of every cycle (1/5-cycles, 2/5-cycles, etc.). Imagine a circle centered on the cusp of the cardioid. Any angle rotation around that circle points straight at a ball on the edge of the cardioid, whose orbit-cycle magnitude exactly matches the fraction used to find that angle. A 120° rotation, representing 1/3 of a full rotation, points straight at the 3-cycle ball on the top of the Mandelbrot set, and a 240° rotation, 2/3 a full rotation, points straight at the 3-cycle ball on the bottom. A 72° rotation, 1/5 of a full, points straight at the first 5-ball. Rotations of 144°, 216°, or 288° point straight at the second, third, and fourth 5-balls.
- This note also explains why we can only find two 6 balls along the edge of the cardioid. A 1/6 (60°) rotation points to a six ball, but a 2/6 rotation points to the 3-ball (2/6 = 1/3), a 3/6 rotation points to the 2-ball (3/6 = 1/2), and a 4/6 rotation points to the other 3-ball (4/6 = 2/3). The only two 6-balls are at 1/6 and 5/6 rotations around the cusp.
Our last two topics of the course: an open problem related to the connectedness of the Mandelbrot Set, and a discussion of Newton’s Method of Approximation, and its surprising connection to the Mandelbrot Set