Fractals
Introduction
Stare at this video. At the end, look at a flat surface 😉.
The objects we are going to see have something in common, they are made by a sequence of repetitions. Each one of these repetitions are called iterations.
Cantor Set
The following object was created by Georges Cantor in 1883. It starts with a line segment whose length is equal to one. In the first iteration the segment is divided in three equal parts and the middle one is removed.
Exercise 1: Draw the Cantor set.
Exercise 2: For every iteration count the number of line segments and the total length. Make two sequences from those numbers.
Exercise 3: Find out the general term of those sequences.
This object can be made also with paper and scissors.
Fractal tree
You can draw also a fractal tree following this instructions:
Draw the trunk.
At the end of the trunk, split by some angle and draw two branches whose lenght is half the first branch.
Repeat at the end of each branch until a sufficient level of branching is reached.
... or even the Pythagoras tree (fractal):
The construction of the Pythagoras tree begins with a square. Upon this square are constructed two squares, such that the corners of the squares coincide pairwise, forming a right triangle. The same procedure is then applied recursively to the two smaller squares, ad infinitum. The illustration below shows the first few iterations in the construction process (from Wikipedia).
What happens with the area of this tree? And what about the perimeter?
Sierpinski triangle
It starts with an equilateral triangle. This is one of the basic examples of self-similar sets–that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński (1882-1969), but appeared as a decorative pattern many centuries before the work of Sierpiński.
If one takes Pascal's triangle with 2n rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpinski triangle. More precisely, the limit as n approaches infinity of this parity-colored 2n-row Pascal triangle is the Sierpinski triangle.
Draw four iterations of the Sierpinski triangle and calculate the area of them, given that the area of the first triangle, T0, equal to 1.
Koch Snowflake
Draw an equilateral triangle (again). For the first iteration of this shape you will divide each side into three equal sections. Then, in the middle section of each side, you will construct another equilateral triangle. You get a star (K1).
For the second iteration you do the same to each side of the star, that is, divide into three sections and construct a new, smaller triangle from the middle one (K2).
Exercise: If the side of the original triangle is s, draw K3 and calculate its perimeter. Can you guess the perimeter of K4?
If you go on for ever with this construction, you get the Kock Snowflake.
What is the perimeter of this shape?
Menger sponge
This is a 3D object. We used to have one, made of origami, on the first floor of our building.
Fractals
All the objects above are called fractals.
A line is an object of dimension-1. A square is a 2-dimension shape. A cube is a 3D object.
Fractals, on the other hand, are weird mathematical objects whose dimension is different from 1, 2 or 3...
Fractals appear the same at different levels, as illustrated in successive magnifications of the Mandelbrot set; because of this, fractals are encountered ubiquitously in nature. Fractals exhibit similar patterns at increasingly small scales called self similarity.¹
The Mandelbrot set is made with the help of complex numbers.
Many mathematicians studied these weird objects but the first one to use the word "fractal" and give a complete theory about them was Brenoit Mandelbrot (1924-2010), in 1975.
Its mathematical study requires advanced knowledge y has to do with factional dimensions, Caos Theory, systems of iterated functions...
But, the construction of these drawings is easy for a computer. After all, they are made only of equations. That is the reason they are used to generate backgrounds in video-games, for example.
Many fractal structures are found in real life, like romanesco brocoli, any coastal line or the clouds.
Fractal art
Some artists have found in fractals a way to express their art. That is called "Fractal Art" and its based in auto-similarity.
What does fractals have to do with complex numbers?
Try this exercises:
Addition
Multiplication:
Draw six points in a cartesian plane. Join them to get a shape.
Find out their expressions as complex numbers.
Multiply all of them by 2. Use a table to set the results.
Draw the resulting numbers. What do you get?
Here, you can find more activities and a summary of the Maths behind this.
How to make the Mandelbrot set
Think of a real number. Square it. Square it again. Draw the different numbers you get in the real line. What happens? Does the dots show the same behaviour? Express the procedure with a formula.
Think of a complex number. Square it. Square it again. Draw the results using the Argand diagram and join the dots with a line. Do you get the same line always?
This item is based in this video.
Fractals with origami
The case of the missing fractals - Alex Rosenthal and George Zaidan (TEd-Lesson)
The case of the missing fractals - Alex Rosenthal and George Zaidan.
Enlaces
