H! This is WuFeng and I would like to talk about Pi. Pi, 3.14159… is one of the oldest and most famous mathematical constants known to humanity. It is simple in definition yet has so many intriguing facts, details and applications.
Pi has many applications in computing, engineering and science. It raises many questions and facts in mathematics and computer science. This number has been a part of human scientific and engineering achievement since the early days of civilization as there are evidence of this number even during Babylonian times.
Julia sets are a fractal known for its beauty and intricate designs, however unlike other fractals, Julia Sets are not created by using self similar patterns. Rather Julia Sets images are generated by taking each pixel on on that image and calculating a color value based off of an equation.
The Julia Set video above is generated using the complex equation:
Since complex numbers are always in the form , we take the part to be our horizontal values and as our vertical values, which means is a pixel on our image. For each point, the value will be plugged back into the equation over and over again until is smaller then . The number of iterations is then counted up and used to assign to that pixel a colour value.
However due to that fact that for some and values, our function will never return a value smaller then so as a result an upper limit or max iteration must be placed to stop calculations and automatically assign a default color.
To generate different Julia sets, one only needs to vary the value. The video above are all values of such that .
Fractals are everywhere. Their jagged form and self similarity allow for intriguing and beautiful representations of nature. One example are in plants. Take for example the following rule.
At the end of each branch add the shape of branches shown in the picture and then keep doing this over and over again for several generation. However unlike a fractal, stop after a small number of iterations. True fractals are created through an infinite number of iterations where as this tree is what is called a Pre-Fractal.
With a few variations in this rule by varying the size and angle of the new branches added will allow for a great combination of unique trees.
A more formal way of defining these rules is called the Lindemayer system after Hungarian biologist Aristid Lindenmayer. This system uses techniques from a field of linguistics called formal language, which deals with understanding of the possible structures of sentences.
More intricate rules can be used to generate different plants such as ferns and broccoli.
In mid 20th century, mathematician Lewis Fry Richards discovers a strange paradox. For any coastline, the length of the coastline depends greatly on the ruler being used. The finer the ruler with more likely one will be able to measure bays can jagged edges that would be ignored by the larger one, thus the smaller the ruler the longer the measurement of the coastline. This brings up two paradoxical questions. With a fine enough ruler, is the coastline infinite and if so how can there exist only a finite amount of land?
It turns out, according to an article by Benoit Mandelbrot, the coastline of landmasses such as England may actually be infinite however the area is finite. It also turns out that there are shapes such as some fractals that also exhibit the same property. The Koch’s snowflake, is a fractal known an infinite perimeter but only a finite area. This is because on each iteration of the snowflake, the sides gets smaller but the number of sides increases at a faster rate. This means that once the sides are sufficiently small the number of sides is infinite.
The self similar nature and infinite nature of fractals can also be used to generate realistic coastlines with bays and jagged rocky outcrops. Fractals can also be used to generate entire islands and planets, as well as plants and trees. This technique is employed in many films and video games to produce realistic and awesome scenery.
It was once believed that the world comprised of four elements, fire, wind, earth and water, held together by the mysterious ether. These elements were comprised of geometric shapes known was the platonic solids. Kepler, before working with Tycho Brahe, hypothesized that the obits of each planet was dictated by a platonic solid. Yet, even though these ideas were disproved and forgotten, the platonic solids can be seen throughout nature.
A platonic solid is a three dimensional shape that is formed from a regular polygon. For example the Cube is formed from six squares and the tetrahedron from four regular triangles. Also, there is only five platonic solids but these shapes can be found in crystal and chemical structures as well as some microorganisms and viruses.
Cubes, tetrahedrons and octahedrons occur naturally in crystal structures. Salts and various metals naturally take a cubic structure such as table salt and iron. Tetrahedrons also occur naturally in chemical structures for compounds such as methane, Phosphate and sulfate. Icosahedrons can be found naturally in Protozoa and viruses.
Platonic solids can also be created with carbon and hydrogen atoms. These are called Platonic Hydrocarbons, however of the five Platonic Solids only three can currently be synthesized.