Objectives
Prior to researching and writing this unit, I did not understand many of the concepts that are described here. Now, I understand these concepts, and I am able to explain and relate them to myself on numerous levels. This is the objective of this unit: that any teacher can take this unit, understand it, tweak it to suit the grade level he/she teaches, and impart the knowledge to their students. My other objective is to be able to use this unit to get students to understand geometry and fractals, while having fun with the concepts.
Background
What is a Fractal?
Fractals are shapes that can be broken down into smaller shapes, where each small shape resembles the original shape. This property is also called self-similarity. Fractals originated in the 17 th century with studies by Karl Weierstrass, Georg Cantor, and Felix Hausdroff. It is said that Benoit Mandelbrot came up with the term fractal in 1975. It came from the latin word fractus meaning "uneven." Fractals as defined by Webster's Dictionary is "irregular shapes for which any suitably chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size". Fractals are useful in a lot of areas. These include, but not limited to, medicine, soil mechanics, and technical analysis.
Characteristics
True fractals often have clear systems at endlessly small scales. This is a bit difficult to explain using geometric language. Basic fractals are self similar and due to this, they are said to be immeasurably complex. They also have a simple and repetitive definition.. Fractal generating software can be also be used to generate fractals. These types of computer generated fractals are not considered as real fractals because they do not always appear to have the same attributes as real fractals. Examples of some real fractals are coastlines, snowflakes, and patterns of some flowers, clouds, mountains and fabric.
Generating Fractals
In Math, fractals can be separated according to the way they are generated. There are four ways of generating fractals. Iterated Function Systems, IFS, is the most basic one. IFS structures show self similarity, which means that their attributes at any magnification are the same as the whole. Then there are Escape-Time Fractals. These come about by a formula relation at every point in its space. An example of this type of fractal is the Mandelbrot set. Random Fractals and Strange Attractors are the remaining types of fractals. These fractals are accomplished from complex systems. They cannot be explained mathematically and are out of the scope of this curriculum unit.
Iterated Function Systems
Iterated Function Systems is a form of generating fractals that always ends up with a self similar fractal. This type of fractal was studied extensively by John E. Hutchinson in the eighties. These fractals appear in a number of dimensions but are normally initiated and etched in two dimensions. One obvious attribute of an IFS fractal is that they are normally made up of copies of themselves whereby each copy is transformed by a function system. The work of these functions is to gather all the individual points together to make smaller shapes. That is the reason why IFS fractals always seem to consist of smaller sizes of the original shape. Examples of some IFS fractals are Sierpinski's Gasket, Cantor sets and Koch's Snowflake.. In Michael Barnsley's his book "Fractals Everywhere", he said that "IFSs provide models for certain plants leaves and ferns by virtue of self-similarity which often occurs in branching structures in nature."
Escape-Time Fractals
This type of approach in developing fractals can also be known as round fractals. They are called round fractals because these types of fractals seem to be orbiting endlessly. Typical examples are the Mandelbrot set and the Julia sets. Escape time fractals are not that difficult to generate. To generate them, an algorithm is applied to every point in a complex plane. Then a unique color is assigned to the value whether or not it approaches infinity.
Random Fractals
This form of developing fractals is fairly candid. Just as the name suggest, random fractals are developed by random processes instead of determined processes. This means that even though its starting point is known, there are many possibilities the process or path might go to. Sometimes the process or path might be predictable and sometimes not so much. A perfect example of this kind of fractal can be seen when you look at coastlines on a map or atlas.
Strange Attractors
We can never talk about strange attractors without talking a little about chaos theory. In chaos theory, changing systems, with the help of some minute initial changes, will over time, lead to large unpredictable results. If anyone has seen the movie titled 'The Butterfly Effect" with Ashton Kutcher, this same principle was used as the plot of the movie. In the movie, Ashton played a 20 year old student who realizes that he has the ability to travel through time to his former self and make some changes which eventually results in changes in his present life.
With that said, in strange attractors, a series of disorderly branching of a system, say fluid flow, results in an attractor. This attractor is known as a strange attractor because they do not have unit dimensions. The directions in some strange attractors can be characterized and some not so much. A common example is the Lorenz attractor.
Classifying Fractals
Fractals can be classified using different attributes. The most efficient way of grouping fractals will be to look at the properties of the kind of self similarity they exhibit. Self similarities exist in three different ways which are exact, quasi and statistical self similarity.
Exact self similarity fractal
Just as the name suggests, the fractals appear exactly the same. They just appear in smaller sizes than the previous shape. A typical example will be in an Iterated Function System, like the Sierpinski's Triangle, after the main triangle, the same triangle appears over and over again. It just gets smaller and smaller each time.
Quasi self similar fractal
The word quasi simply means almost. That said, these kind appears quasi but not exactly the same as the original even though they do appear in smaller sizes than the previous shapes. Example, if you zoom into a Mandelbrot set, you will see that in each resulting shape, the shape is not as oval shaped as its predecessor. This type of fractal is called the almost (quasi) self similar fractal.
Statistical self similar fractal
These fractals have values which are maintained across distinct proportions. That is why random fractals are put under this kind. Also in type, the fractals do not have an equal distortion between shapes. Instead there is a value which is maintained across the scales. A typical example is the coastline of Britain. One does not expect to see little miniature versions of it coming together to create it by observing it under a microscope.
The human lung
The lung is a crucial organ in mammals for inhaling and exhaling of oxygen and carbon dioxide respectively. It can be found on either side of the heart. The lung serves as the pathway for oxygen to move from the atmosphere into the blood and carbon dioxide to move into the atmosphere from the blood. This exchange of gases is made possible by alveoli. Before the exchange of gases happens in the alveoli, the air has to first go through the nose or mouth, then through the larynx, trachea, and a system of bronchi and bronchioles. The lungs look like a waxy structure that contains adjacent hexagonal cells filled with epithelium. The lungs in mammals might look smaller when you look at its outer surface area but in actuality, it is much larger. This is due to the fact that lungs grow as branching fractals. That is, lungs develop without covering a large surface area. Human lungs are divided into two fragments, one on each side of the heart. The right lung has 3 sections while the left has 2. These sections are further divided into subsections called lobules. We will take a closer look at the lungs in detail in the strategies.
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