**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*

To review, take a look at the following exercises from pages 408-411: 19, 20, 21, 36, 38, 41, 42, 52, and 55. You can find answers here. Leave a comment or send me an email with questions!

Your next Personal Progress Check has been assigned. Log into AP Classroom and complete the Unit 4 MCQ Part B PPC. **Your response is due by the start of class time (9:00 for 1st period, 2:00 for 7th) on Monday, January 20**

**Update (11/22): I have reopened the PPC so that those of you who did not complete it within the expected timeframe can refer to the questions as review for your quiz on Monday. Completing it now will not be for credit.**

- “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

]]>- Complete
**one side**of the sheet (your choice which one) for a free HW credit on your record. - Complete
**both sides**of the sheet for a free HW credit*and*extra credit on the quiz.

You must show your work to count!

]]>We will have a quiz on chapter 15 when we return from exam week: Monday, January 27

Your next Personal Progress Check has been assigned. Log into AP Classroom and complete the Unit 4 MCQ Part B PPC. **Your response is due by the start of class time (9:00 for 1st period, 2:00 for 7th) on Monday, January 20**

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.

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From page 409, do questions 29, 31, and 35

Your next Personal Progress Check has been assigned. Log into AP Classroom and complete the Unit 4 MCQ Part B PPC. **Your response is due by the start of class time (9:00 for 1st period, 2:00 for 7th) on Monday, January 20**

Today, you should review. There are several links below to work on, but in class you’ll be given instructions to **complete at least one IXL module** to a score of 80+.

- Go to http://www.clever.com and click “Log in as a student”
- Search for Ithaca High School (if necessary) and select “Log in With Google”
- The system will connect to your ICSD Google account and you’ll be taken to a “homeroom” page. From there, you can click the IXL icon to log in.

- Simplify radical expressions with variables I
- Simplify radical expressions using the distributive property
- Evaluate rational exponents

There are others linked below, but you are expected to complete one of these three.

- HW 4.7 – Quiz Review
- 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)
- 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… - 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.

This last point suggests there is a significant, meaningful connection between the **location** and **shape** of a Julia Set in the Mandelbrot Set. This will be what we’ll explore tomorrow.

For now, 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 Thursday!).

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