Mathematical Asthetics
...ich are straight-sided figures with equal sides and angles. There are only, and can only be five platonic solids. These are the Tetrahedron, Cube, Octahedron, Dodecahedron and the Icosahedron. Tetrahedron Cube Octahedron Dodecahedron Icosahedron The reason that it is only possible for five to exist is that each vertex must have at least three faces come together, if it had only two it would collapse in on it’s self. Also the sum of the interior angles at each vertex must be less than 360 degrees or it would not fit together. For these reasons there are only five shapes that can be made. This was discovered by Plato in ancient Greece. He believed that each of these shapes were the fundamental building blocks of nature. He believed that each one was a different element. Tetrahedron was fire, the cube was earth, the octahedron was air, the icosahedron was water and the dodecahedron represented the cosmos as it was so different to the other four shapes. Aside from being considered amazing for their mathematical properties among mathematical circles, the platonic solids a very pleasing to look at. Many people all over the world consider the shapes of the platonic solids to be very beautiful. The cube is one of the most recognisable shapes in the world today. The Platonic Solids have intrigued and amazed people for centuries and will continue to do so. The Golden Ratio is another piece of mathematics that has astounded people for thousands of years. This diagram illustrates the golden ratio. Point C on the line divides the line so that the ratio of AC to CB is equal to the ratio of AB to AC. Some basic algebra shows that the ratio of AC to CB is 1.618 or half the sum of one and the square root of five. Although this might sound very useless the golden ratio appears in some very strange places. It is said that if you draw a rectangle around the face of Leonardo da Vinci's “Mono Lisa” the ratio of the height to the width is equal to the golden ratio. The dodecahedron is related to the golden ratio; both the surface area and the volume of unit edge length involve the golden ratio. If you have a golden rectangle and you cut a square off of it, you will still have a golden rectangle. This procedure can be continued each time resulting in the same way continually getting smaller golden rectangles. The Fibonacci sequence is very intriguing. It is a sequence where any number is the sum of its two predecessors. For example 5 is a Fibonacci number and the two numbers that came before it in the sequence would be 2 and 3. 2 + 3 = 5. Here are the first 16 numbers in the sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 and 610. The Fibonacci sequence has many different influences on our culture and in nature. But firstly it is very closely related to the golden ratio. If you get the ratio of two successive Fibonacci numbers and divide each by the number before it. Then plot these results o a graph to find the mean ratio. The ratio is approximately 1.618034, which is the value of the golden ratio. 1/1=1 2/1=2 3/2=1.5 5/3=1.666 8/5=1.6 13/8=1.625 21/13=1.61538 Also with the Fibonacci sequence we can do something similar to the golden rectangle, but it is called the Fibonacci squares. First you start with a square then add a square of the same size to form a rectangle. Then add a square to the longest side of the rectangle to form a larger rectangle. The longest side will always be a Fibonacci number. Also the larger it gets the closer it will be to a golden rectangle. There is another thing that is very interesting about the Fibonacci squares. If you draw a quarter of a circle in each square you will draw a spiral. This spiral is very similar to the spiral curve of a Nautilus sea shell. The Fibonacci sequence appears in many other place is nature. One of which is pine cones. On a pine cone the petals spiral in two different directions. Most of the time the number of petals that it takes to get around is a Fibonacci number. The Fibonacci sequence is very apparent in plants and flowers. Most leaves on plants grow in a way so that each leaf gets the same amount of sunlight and rain. If you count the amount of times you go around the stem from leave to leaf counting each and the amount of leaves until you find another that is directly under it, these are Fibonacci numbers. The sequence also relates to the arrangement of petals on flowers. For example Lily’s and Iris’s commonly have three petals. Although some Lily’s have six petals, but this is two sets of three petals. Often Buttercups have five petals, Delphiniums have 8 petals, Corn has 13 and Daisy’s have been found that had 89 petals. In the year 1202 the Fibon...