**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*

- HW 4.13 – Unit 4 Test Review 1
- Video Archive
- Adding/subtracting radicals
- Multiplying/distributing radicals
- Rationalizing denominators
- Evaluating nth roots
- Simplifying radicals with variables
- Simplifying radicals with negative radicands
- Multiplying complex numbers
- Powers of
*i* - Rationalizing imaginary denominators
- Solving quadratics with non-real roots

- H.4 Multiply complex numbers
- H.5 Divide complex numbers
- H.8 Powers of i
- J.6 Solve a quadratic equation by factoring
- J.9Solve a quadratic equation using the quadratic formula
- L.4 Simplify radical expressions with variables I
- L.6 Nth roots
- L.7 Multiply radical expressions
- L.8 Divide radical expressions
- L.9 Add and subtract radical expressions
- L.10 Simplify radical expressions using the distributive property

From the exercises on pages 430-432, do 7, 10, 12, 17

]]>- “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 degree 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 degree rotation, 2/3 a full rotation, points straight at the 3-cycle ball on the bottom. A 72-degree rotation, 1/5 of a full, points straight at the first 5-ball. Rotations of 144, 216, or 288 degrees point straight at the second, third, and fourth 5-balls.

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

]]>- Notes
- HW 4.12 – Solving Quadratics with Non-real Roots
- Lesson Video (solving quadratics with non-real roots)
- Video Archive

- H.4 Multiply complex numbers
- H.5 Divide complex numbers
- H.8 Powers of i
**J.6 Solve a quadratic equation by factoring****J.9 Solve a quadratic equation using the quadratic formula**- L.4 Simplify radical expressions with variables I
- L.6 Nth roots
- L.7 Multiply radical expressions
- L.8 Divide radical expressions
- L.9 Add and subtract radical expressions
- L.10 Simplify radical expressions using the distributive property

First, we observed that orbits along the horizontal *x*-axis of the Mandelbrot Set bifurcate in the same way that they do in the Feigenbaum plot, and for good reason. Furthermore, we noticed that elsewhere along the edge of the central cardioid of the Mandelbrot Set, we can find balls that yield cycles of any magnitude, and in general **the size of the ball is inversely related to the magnitude of its cycle** (bigger ball = smaller cycle).

When we look at the Julia Sets we get in each of these balls, we realize that each of them have a central area, centered at the origin, and each with an infinity of “pinch points” found along its edge. The number of areas pinched together at this point **exactly matches** the size of the orbit contained! Given a mystery Julia Set, then, we can determine the size of its contained orbit just by its shape!

Finally, we observed that every ball connected to the central cardioid has an infinity of balls attached to it, also of decreasing size. The fractal nature of the Mandelbrot Set manifests itself here: the largest sub-ball corresponds to a bifurcation, the next largest to trifurcations, and so on. Thus, the 2-cycle found in the San Marco Julia Set can bifurcate into a 4-cycle, trifurcate into a 6-cycle, or any other multiple of two. The 3-cycle found in the Douady Rabbit can bifurcate to a 6-cycle, trifurcate to a 9-cycle, and so on for any multiple of 3. And this pattern extends to the Julia Sets themselves: the 3-ball off of the 2-ball creates a Julia Set that is a San Marco of Douady Rabbits:

And the 2-ball off of the 3-ball creates a Julia Set that is a Douady Rabbit of San Marcos:

With this in mind, we can play the game: Find the Julia Set. For each Julia Set in the attached file, use the shape/structure of the Julia Set to predict where in the Mandelbrot Set it can be found, and as a result the magnitude of the cycle found within it. Then verify your predictions using Fractal Zoomer.

]]>

Our test on Unit 4 will be on **Friday, January 18th**. Please do not be absent on that day! If you know for some reason that you will be absent on that day, please arrange a time **before the 18th** to take the test! Otherwise, we will have to schedule a time for you to take the test during exam week.

- Notes
- HW 4.11 – Rationalizing Imaginary Denominators
- Lesson Video (you can stop watching this video at the 8-minute mark. The examples he does beyond that point are beyond what we will do in this class)
- Video Archive

- The central “bulb” of the Mandelbrot Set is a cardioid, and each other bulb off of that central body is a perfect circle.
- Each bulb has its own cycle of orbits. Some bulbs have the same size orbit as others, but with a different “order” (1,2,3,4,5,6 vs 1,3,5,2,4,6 for example)
- If you tile the plane with Julia Sets of a sufficient density, the collection of Julia Sets make a photo-mosaic of the Mandelbrot Set. This further reinforces that there is something important between the location of C within the Mandelbrot Set and the shape of the corresponding Julia Set.
- Along the central “spike” on the left side of the Mandelbrot Set, we can find baby Mandelbrot Sets strung along that string. The main body of these baby Mandelbrots show cycles instead of fixed points. Realizing that the horizontal axis on which the Mandelbrot set is centered is the
**real number**axis, this suggests that there may be a meaningful connection between the Mandelbrot Set and Feigenbaum Plot…

Keep working with Fractal Zoomer and considering the questions posted yesterday. Please also read the section on Mandelbrot Set from pages 74-81 in your copy of Fractals: The Patterns of Chaos (and bring that book back on Monday).

]]>