The Mandelbrot Set and Fractal Geometry

(Notes to the CRP: There is a fractal coloring in page at the end of this lesson which can be used as a body engager.

You may also want to have an extra sheet of paper or a whiteboard to write out all the symbols and the equations as you go for your speller to see, as the equations can get a little complex

There are also two different VAKTIVITIES at the end of this lesson which require some extra materials. All the materials are listed in each activity)

Typically, when we think of GEOMETRY, we think of straight lines and angles; this is what is known as EUCLIDEAN geometry, named after the ALEXANDRIAN Greek mathematician EUCLID. This type of geometry is perfect for a world created by humans, but what about the geometry of the natural world? That’s where BENOIT MANDELBROT (interestingly, Mandelbrot, directly translated from German, means Almond Bread) changed the idea of geometry and gave us what is known as FRACTAL geometry.

Spell: STRAIGHT GREEK NATURAL
Today we are talking about the Mandelbrot Set and ________ geometry.
FRACTAL
What are we talking about today? THE MANDELBROT SET AND FRACTAL GEOMETRY
What do we call the geometry that considers straight lines and angles?
EUCLIDEAN
Name one of the types of geometry mentioned in the paragraph. EUCLIDEAN / FRACTAL
Name another branch of maths other than geometry. ALGEBRA / TRIGONOMETRY / CALCULUS / PROBABILITY / STATISTICS
Which branch of maths do you find the most interesting and why?
Mandelbrot translates to Almond Bread in __________. GERMAN
What does Mandelbrot translate from German to English? ALMOND BREAD
Fractal geometry is useful for the _____ world. NATURAL

Simply put, fractals are IRREGULAR geometric figures that are characterized by being SELF-SIMILAR repeating versions of themselves. Whether you look at the figure close up or from a distance, the figure looks the same and is composed of the same figure that repeats itself. If we look at the TRIANGLE below, which is called the SIERPINSKI triangle, we can see that it is self-similar and it repeats. The big triangle is made up of 3 smaller triangles, and when you look at each smaller triangle, you will see that that triangle is also made up of 3 smaller triangles.

Spell: VERSION COMPOSED SMALLER
Fractals are _______ geometric figures. IRREGULAR
Fractals are characterized by being ______ repeating versions of themselves. SELF-SIMILAR
What is the name of the self-repeating triangle? SIERPINSKI
What are fractals characterized by? SELF-SIMILAR REPEATING VERSIONS OF THEMSELVES
Give an antonym for the word similar. DISSIMILAR / UNLIKE / DIFFERENT
Look at the Sierpinski triangle; what type of triangle is used? EQUILATERAL TRIANGLE
Discuss some things in nature that you find similar to one another.

(Note to the CRP, write out all the symbols on a separate page or a whiteboard in front of your speller as well as all the equations).
To get an understanding of fractal geometry, we need to start with COMPLEX numbers. Complex numbers are made up of real numbers and what are called IMAGINARY numbers – i. While it might be imaginary, it is often known as the √-1 (square root). But, if we MULTIPLY a negative number with another negative number, it makes it a positive, so does the √-1 actually exist? This is why it is called imaginary. A real number is a number we use in EVERYDAY life, and this could really be any number. So, we might have 2i or -i.

Spell: UNDERSTANDING KNOWN REALLY
What type of number do we need to start with? COMPLEX NUMBER
What are complex numbers made up of? REAL NUMBERS AND IMAGINARY
NUMBERS
Name one of the components of a complex number. REAL NUMBER / IMAGINARY NUMBER
What happens when we multiply a negative number by another negative number? NUMBER BECOMES A POSITIVE
What is an imaginary number known as? THE SQUARE ROOT OF -1(√-1)
How is a real number described in the paragraph? A NUMBER WE USE IN EVERYDAY LIFE
Give a real number between 15 and 30.
Give an imaginary number between 15 and 30. (The answer must have i next to it, e.g., 16i.)

How, then, do we get a complex number? Well, we need our normal x-axis, and then we also need a second axis that runs PERPENDICULAR (the upright or vertical
line) to the x-axis. This will be our imaginary number line. These two AXES together are known as a COMPLEX NUMBER PLANE. Complex numbers always have the FORMULA, a+bi. This means that if we plot a point on our graph at 2 and 2i, our formula will read 2+2i. Any number on the complex plane will be what we call a complex number.

Spell: NORMAL AXIS ALWAYS
Our second axis runs _________ to the x-axis. PERPENDICULAR
The perpendicular axis is for our _______ numbers. IMAGINARY
What are these two axes together known as? COMPLEX NUMBER PLANE
What is the formula for a complex number? a+bi
Look at the complex number planes below and give the complex number for each one.

Answers:
1.1+1i
2.-2+2i
3.4+-1i
4.-4+-4i
5.2+3i

For spellers still at the acquisition stage you can use the below questions:
1.___ + 1i
2.-2 + ____i
3.___ + ____i
4.-4 + – ___i
5.2 + ___i

VAKT – point to the dot on each graph above

Mandelbrot sets are a type of fractal that we will start with. Mandelbrot sets have to fall within the BOUNDS of -2 and 2, if the point falls out of that bound, it is no longer considered part of the set. If we had to use all the numbers that fall within the bounds and plot them on the complex plane, we will end up with the Mandelbrot set. To work this out, we use the following FUNCTION (a special relationship where each input has a single output), f(z)=z²+c, where c is a complex number. We are always going to start with 0 and then repeat or ITERATE the function with each answer. If the number keeps growing, then it is not part of the Mandelbrot set. However, if the number stays within the limit, then it is probably part of the Mandelbrot set.

Spell: POINT CONSIDERED CERTAIN
What is the limit for the bounds for the Mandelbrot set? -2 and 2
The function used to work out if a number is part of the Mandelbrot set is f(__) = z² + c. Z
What is the function that we use to work out if a number is part of the Mandelbrot set? f(z)=z²+c
What does c stand for in the function? COMPLEX NUMBER
What number must we always start with? 0
How do we know that a number is part of the Mandelbrot set? STAYS WITHIN THE BOUND
How do we know that a number is not part of the Mandelbrot set? NUMBER KEEPS GROWING

Let’s have a look and do some examples using the function f(z) = z²+c.
We always start with z=0, and let’s say for our first ATTEMPT, we are going to use 1+i as our complex number. Our function will read as:
f (0) = 0²+1 = 1
Now we need to iterate this. Our answer above is now our value for z.
f (1) = 1²+1 = 2 Let’s keep going. f (2) = 2²+1 = 5
f (5) = 5²+1 = 26
Our number keeps growing, and it will continue to grow to INFINITY; it will grow without bounds. So, our complex number of 1 is not part of the Mandelbrot set.
Let’s have a look at the complex number -1+i and see if that gets us into the Mandelbrot set. Remember, we always start with 0. f (-1) = 0 + (-1²) = -1
So now our function (f) is -1.
f (-1) = -1 + (-1²) = 0
We are back to 0, and if we use our 0 again with our current complex number, we will end up with -1. This means we are going to OSCILLATE (swing between) between 1 and 0. Therefore, our complex number of -1+i does fall on the Mandelbrot set. Have a look at the image below, and you will see that -1 falls in what is known as the main disc of the Mandelbrot set.

Spell: CURRENT THEREFORE IMAGE
Which word was used to mean alternate or swing between numbers? OSCILLATE
Which number did we use above? In the example, is in the Mandelbrot set; remember that the bounds are between -2 and 2? -1
What is our complex number made up of -1? -1+i
In your own words, how would you describe the Mandelbrot shape?
VAKT: Point to the -1, which is the center of the main disc in the image above.

Using the following complex numbers, do the iterations, and state whether they fall within the bounds for the Mandelbrot set or not; we are going to stick to our complex number as a+i. (There are fill-in-the-blank questions below the answers for spellers still in the acquisition phase.)

1.-0.25 + i
2.3 + i
3.0.5 + i
4.-0.15 + i
5.-1.5 + I

Answers:
1.f (0) = 0 + (-0.25²) = -0.0625 f (-0.0625) = -0.0625 + (-0.25²) = -0.125 f (-0.125) = -0.125 + (-0.25²) = -0.1875 f (-0.1875) = -0.1875 + (-0.25²) = -0.25 f (-0.25) = -0.25 + (-0.25²) = -0.3125 f (-0.3125) = -0.3125 + (-0.25²) = -0.375 f (-0.375) = -0.375 + (-0.25²) = -0.4375 f (-0.4375) = -0.4375 + (-0.25²) = -0.5 f (-0.5) = -0.5 + (-0.25²) = -0.5 = 0.4375 f (-0.4375) = -0.4375 + (-0.25²) = -0.5 (therefore) -0.25 will fall within the Mandelbrot Set bound (the last two numbers will carry on repeating)

2. f (0) = 0 + 3²= 9 f (9) = 9 + 3²= 18 f (18) = 18 + 3²= 27
3 will not fall within the Mandelbrot Set bound

3. f (0) = 0 + 0.5² = 0.25 f (0.25) = 0.25 + 0.5² = 0.5 f (0.5) = 0.5 + 0.5² = 0.75 f (0.75) = 0.75 + 0.5² = 1 f (1) = 1 + 0.5² = 1.25 f (1.25) = 1.25 + 0.5² = 1.5 f (1.5) = 1.5 + 0.5² = 1.75 f (1.75) = 1.75 + 0.5² = 2
0.5 will not fall within the Mandelbrot Set bound

4.f (0) = 0 + (-0.15²) = -0.0225 f (-0.0225) = -0.0225 + (-0.15²) = 0 f (0.045) = 0 + (-0.15²) = 0.0225
-0.15 will fall within the Mandelbrot Set bound

5.f (0) = 0 + (-1.5²) = -2.25
-1.5 will not fall within the Mandelbrot Set bound

1. f (0) = ____ + (-0.25²) = -0.0625 0
f (_____) = -0.0625 + (-0.25²) = -0.125 0.0625
f (-0.125) = -0.125 + (-0.25²) = -_______ -0.1875
f (-0.1875) = -0.1875 + (____²) = -0.25 -0.25
f (-0.25) = _____ + (-0.25²) = -0.3125 -0.25
f (-0.3125) = -0.3125 + (-0.25²) = ______ -0.375
f (-0.375) = -0.375 + (-0.25²) = ______ -0.4375
__ (-0.4375) = -0.4375 + (-0.25²) = -0.5 f
f (-0.5) = ____ + (-0.25²) = 0.4375 -0.5
f (______) = -0.4375 + (-0.25²) = -0.5 0.4375
-_______ will fall within the Mandelbrot Set bound (the last two numbers will carry on repeating) -0.25

2. f (0) = 0 + __²= 9 3 f (9) = ___ + 3²= 18
9 f (18) = 18 + 3²= ___ 27
3 will _____ fall within the Mandelbrot Set bound. NOT

3.f (0) = 0 + ___ ² = 0.25 0.5 f
(____) = 0.25 + 0.5² = 0.5 0.25 f
(0.5) = 0.5 + 0.5² = ____ 0.75 f (0.75)
= 0.75 + 0.5² = ____ 1 f (1) = 1 + 0.5² = 1. ___
25 f (1.25) = 1.25 + 0.5² = __.5 1 f (1.5) = 1.5 + 0.5² = _____
1.75 f (1.75) = 1.75 + 0.5² = ___ 2
0.5 will not fall within the Mandelbrot Set ________. BOUND

4.f (0) = 0 + (___²) = -0.0225 -0. 15 f (_____) = -0.0225 + (-0.15²) = 0 -0.0225 f (0) = ___ + (-0.15²) = -0.0225 0
_____ will fall within the Mandelbrot Set bound. -0.15

5.f (0) = 0 + (___²) = -2.25 -1.5
-1.5 will not fall within the Mandelbrot ___ bound. SET

If we look at the image above, we can see that the entire Mandelbrot set is
ENCOMPASSED by a circle. Mathematicians know that the RADIUS of this circle is 2. What mathematicians are still trying to figure out, however, is the area – there is an estimate which is roughly 1.506484 square units.

Spell: IMAGE FIGURE ROUGHLY
What is the Mandelbrot set encompassed by? CIRCLE
What is the radius of the circle? 2
What is the estimated area of the Mandelbrot? 1.506484 SQUARE UNITS
What is the formula for the area? AREA = LENGTH X WIDTH
Why do you think the exact area of the Mandelbrot set is still unknown?
Why do you think fractals are important in math?

Now that we have a basic understanding of how the Mandelbrot set works let’s think about why they are IMPORTANT and what they are used for. Well, the Mandelbrot set is just one type of fractal. There are many others, and we see many of these in everyday life. Have a look around you, and chances are you will notice something that might have fractal PROPERTIES to them. In nature, we see fractals all the time, LIGHTNING, plants, snowflakes, and even rivers or veins in our bodies.

Spell: UNDERSTANDING AROUND NOTICE

We see many fractals in _________ life. EVERYDAY

In _______, we see fractals all the time. NATURE

Name one of the places mentioned where we might see fractals. LIGHTNING / PLANTS / SNOWFLAKES / RIVERS / VEINS

Where else have you noticed fractals?

An interesting fractal to look at is the KOCH Snowflake. The idea behind the Koch snowflake is that the shape has an INFINITE perimeter but a finite area.
Essentially, you would take a straight line, divide it into three equal parts, take out the middle line, and place it at a 60° angle to the first line. You would then do this again to the last piece of your line; if you carry on doing this, you will eventually end up with what is called the Koch curve. You can create the snowflake using a similar process with triangles.

Spell: INTERESTING ESSENTIALLY DIVIDE
We are now talking about the ________ curve or snowflake. KOCH
The Koch snowflake has an infinite __________. PERIMETER
The Koch snowflake has a _______ area. FINITE
At what angle do we place the middle line segment to the first line segment? 60 DEGREES
Give a synonym for the word infinite. ENDLESS / IMMEASURABLE / LIMITLESS / VAST / UNLIMITED
What shape is used to create the snowflake? TRIANGLE

So, while fractals might be fun to play with and really pretty to look at, they also have some especially useful real-world properties. The ANTENNAE in a cell phone uses fractals, but scientists are even looking at fractals in cancer research to work out how the cancer cells actually grow. Fractal geometry is also used in computer science to COMPRESS images – interestingly, when images are compressed using fractal geometry, we don’t get PIXELIZATION. Fractals are all around us, so keep an eye open for all those beautiful patterns we see in nature.

Spell: PRETTY PROPERTIES INTERESTINGLY
Fractals have some especially useful _________ properties. REAL WORLD
What part of a cell phone uses fractals? ANTENNAE
How are scientists exploring cancer cells with fractals? LOOKING AT HOW CANCER CELLS GROW
In computer science, fractals are used to ________ images. COMPRESS
How are fractals used in computer science? USED TO COMPRESS IMAGES
Give an antonym for the word compress. EXPAND / ENLARGE / INCREASE / SWELL / INFLATE
Name two other terms associated with computer science. DATA / BANDWIDTH / BYTE / BUG / CLOUD / STORAGE / CODE
When using fractal geometry to compress images, what does not happen? PIXELIZATION
Discuss something in your life that you would like to compress.

VAKT: Watch this video to learn about fractal use in cell phones

Creative Writing:

1. Fractals and the Mandelbrot set are self-repeating and self-similar versions of themselves. However, the shape can become more complex, and we see different variations of the shape as we zoom in closer to the picture – almost as though the Mandelbrot has a variety of different alter egos. Create an alter ego for yourself and discuss the traits, characteristics, and mission in life.

2. The Mandelbrot set is sometimes thought of as an island with an infinite sea or ocean around it. Imagine the Mandelbrot Island in your head, and discuss what you think the terrain of the island would be like, who the people are, what their culture is, and what type of wildlife would be on the island.

VAKTIVITY 1: Times Tables Patterns

We can also see fractals in our timetables. Times tables are quite simple to learn, some of them have patterns, and you can rote learn them quite easily; they are also super useful to know. Watch this video to see how timetables can create patterns.

Now that you have seen how timetables create patterns let’s create our own.
There are a few different ways to do this.

You will need:
A piece of wood (roughly a square) or an embroidery hoop, or a polystyrene ring, or a piece of paper
A compass – if you are using wood or paper
Protractor
Some string or some coloring pencils if you are using paper

How to do it:
Decide on the timetable you would like to use. Draw a circle using your compass onto the paper or the wood.
Decide how many points you would like on your circle – use your protractor to ensure that each point is at an equal distance.
Let’s say you have chosen the 2 times table, and you have a circle with 12 points.
Label each dot, starting with 0.

Now, we are going to draw a line to show the connections – like you saw in the video.
2 x 0 = 0
2 x 1 = 2
2 x 2 = 4
2 x 3 = 6
2 x 4 = 8
2 x 5 = 10
2 x 6 = 12
2 x 7 = 14
2 x 8 = 16
2 x 9 = 18
2 x 10 = 20
2 x 11 = 22
2 x 12 = 24
If you are using wood, you can draw the circle onto the wood, and measure each point; at each point, hammer in a nail, then tie a piece of string to the nail, and take it across to the correct nail – tie it in place there. Carry on until you have created a pattern. You can do something similar with polystyrene and pins. If you are using paper, you can simply draw your pattern.

ACTIVITY 2: Create a Sierpinski Triangle (This is a good activity for a group)
Instructions for this activity can be found here: https://www.whatdowedoallday.com/sierpinski-fractal-triangle/