Living Elegance: Mathematical Patterns in Nature
Consider something as simple as a sea shell. In today’s fast paced modern life it can be easy to miss a sea shell. But if we were to pause from the distractions swirling around us we would see that the world is full of beautiful patterns. We see things like symmetry, stripes, tessellations and other patterns when we just look. Today we will focus on patterns that can be described with the help of an idea called fractals.
A fractal is self similar, meaning that it has similar patterns at large and small scales. This similarity lends it self naturally to mathematical models. Let’s consider a fractal that is built using geometric rules. We can construct a famous fractal called Sierpinski’s Triangle. The rules for constructing this triangle are simple:
- Start with an equilateral triangle
- Break the triangle into four triangles of equal size
- Color the middle triangle
- Repeat step 2 for the three other triangles in the subdivision
Here is what the first four iterations look like:
The Fibonacci sequence is constructed using a few simple rules.
- The first term is 0
- The second term is 1
- The third term is equal to the first term plus the second term (1)
- The nth term is equal to the the sum of the (n-1) and (n-2) terms
The first few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13…. There are a variety of uses for this sequence but we get something special when we use the numbers to draw boxes. We start with a 1×1 square. Then we add another 1×1 square next to it form a 2×1 rectangle. Next we add a 2×2 square to form a 3×2 rectangle. We continue in this fashion adding new squares to form a new rectangle. When we have added a few terms of the sequence we can go back and connect alternating arcs on each square we added. The result looks something like the image to the image below.
It turns out spirals with similar ratios happen all the time in nature. It forms a type of fractal because as you continue the pattern the macro view looks similar to the micro view. To help you see this spiral in more places, please flip through the slide show that follows. Whether it is galaxies, hurricanes, sea shells, broccoli,ram horns or flower seeds, this pattern repeats over and over again.
Consider a tree that grows and splits into two branches. Each branch in turn splits into two more pieces. If we keep going for a few iterations we will get something that looks like this:
We can add variation to the tree like structures we create by altering the angles and number of each branch. A key break through for exploring fractals came from a theoretical biologist named Aristid Lindenmayer. Lindenmayer created a system that used simple letters to describe the state of a plant’s leaves and branches. That system can be combined with simple rules that change the representation of the system over time. We used simple rules earlier while we explored the Fibonacci set. Our rule was that each value was based on the previous two values.
Since the letters describe the tree, we actually draw the result. Now drawing each branch would be very tedious, but luckily computers are great at doing repetitive math operations. A computer program can be constructed to draw out the equivalent “plant”. For example, they may draw a branch, and then turn 35 degrees and then begin drawing again. These three elements of a simple grammar, simple rules and a drawing program allow for a vast array of fractal creations to be explored. With these techniques it is possible to draw mountains, trees and other natural formations quickly using an algorithm or calculation instead of an elaborate 3D model that captures each facet and face.
If we look just a little we will find a wealth of patterns in every aspect of the natural world. We can use simple, repetitive rules to model systems that would otherwise be impossible to model. Fractals present opportunities for both beauty and curiosity. As we begin to understand fractals more and more with computers, we may find that the cure for cancer or a similar world changing discovery to be just a simulation away using fractals.
“Bottomless wonders spring from simple rules… without end” Benoit Mandlebrot
- Fibonacci Sequence and Fractal Spirals, Fractal Foundation, http://fractalfoundation.org/fractivities/FibonacciWorksheetAdvanced.pdf
- Fractal Triangles, Fractal Foundation, http://fractalfoundation.org/fractivities/TriangleWorksheet.pdf
- Patterns in Nature, Wikipedia, https://en.wikipedia.org/wiki/Patterns_in_nature
- Visualize It: Sierpinski Triangle, Khan Academy program by Charles Cozad, https://www.khanacademy.org/computer-programming/visualize-it-sierpinski-triangle/3208360053
- Fibonacci Numbers, The On-line Encyclopedia of Number Sequences, http://oeis.org/A000045
- Fibonacci Number, Wolfram Math World, http://mathworld.wolfram.com/FibonacciNumber.html
- Lindenmayer System, Wikipedia, https://en.wikipedia.org/wiki/L-system
- L-Systems Turtle Graphics Renderer – HTML5 Canvas – by Kevin Roast, http://www.kevs3d.co.uk/dev/lsystems/