What makes something fractal

Fractal geometry can also provide a way to understand complexity in "systems" as well as just in shapes. The timing and sizes of earthquakes and the variation in a person's heartbeat and the prevalence of diseases are just three cases in which fractal geometry can describe the unpredictable. Another is in the financial markets, where Mandelbrot first gained insight into the mathematics of complexity while working as a researcher for IBM during the s.

Mandelbrot tried using fractal mathematics to describe the market - in terms of profits and losses traders made over time, and found it worked well. In , Mandelbrot turned again to the mathematics of the financial market, warning in his book The Mis Behaviour of Markets against the huge risks being taken by traders - who, he claimed, tend to act as if the market is inherently predictable, and immune to large swings.

Fractal mathematics cannot be used to predict the big events in chaotic systems - but it can tell us that such events will happen. As such, it reminds us that the world is complex - and delightfully unpredictable.

More of Jack Challoner's writings can be found at Explaining Science. Georg Cantor experimented with properties of recursive and self-similar sets in the s, and in Helge von Koch published the concept of an infinite curve, using approximately the same technique but with a continuous line. And of course, we've already mentioned Lewis Richardson exploring Koch's idea while trying to measure English coastlines.

These explorations into such complex mathematics were mostly theoretical, however. Lacking at the time was a machine capable of performing the grunt work of so many mathematical calculations in a reasonable amount of time to find out where these ideas really led. As the power of computers evolved, so too did the ability of mathematicians to test these theories. We think of mountains and other objects in the real world as having three dimensions.

In Euclidean geometry we assign values to an object's length, height and width, and we calculate attributes like area, volume and circumference based on those values. But most objects are not uniform; mountains, for example, have jagged edges. Fractal geometry enables us to more accurately define and measure the complexity of a shape by quantifying how rough its surface is. A relatively simple way for measuring this is called the box-counting or Minkowski-Bouligand Dimension method.

To try it, place a fractal on a piece of grid paper. The larger the fractal and more detailed the grid paper, the more accurate the dimension calculation will be. In this formula, D is the dimension, N is the number of grid boxes that contain some part of the fractal inside, and h is the number of grid blocks the fractals spans on the graph paper.

However, while this method is simple and approachable, it's not always the most accurate. It looks simple, but depending on the fractal, this can get complicated pretty quickly.

You can produce an infinite variety of fractals just by changing a few of the initial conditions of an equation; this is where chaos theory comes in. On the surface, chaos theory sounds like something completely unpredictable, but fractal geometry is about finding the order in what initially appears to be chaotic. Start counting the multitude of ways you can change those initial equation conditions and you'll quickly understand why there are an infinite number of fractals.

Some fractals start with a basic line segment or structure and add to it. A dragon curve is made this way. Others are reductive, beginning as a solid shape and repeatedly subtracting from it. The Sierpinski Triangle and Menger Sponge are both in that group. More chaotic fractals form a third group, created using relatively simple formulas and graphing them millions of times on a Cartesian Grid or complex plane.

The Mandelbrot set is the rock star in this group, but Strange Attractors are pretty cool too. These images are all expressions of mathematical formulas. After Mandelbrot published his seminal work in on fractals, one of the first practical uses came about in when Loren Carpenter wanted to make some computer-generated mountains.

Using fractals that began with triangles, he created an amazingly realistic mountain range [source: NOVA ]. In the s Nathan Cohen became inspired by the Koch Snowflake to create a more compact radio antenna using nothing more than wire and a pair of pliers. Today, antennae in cell phones use such fractals as the Menger Sponge, the box fractal and space-filling fractals as a way to maximize receptive power in a minimum amount of space [source: Cohen ]. While we don't have time to go into all the uses fractals have for us today, a few other examples include biology, medicine, modeling watersheds, geophysics, and meterology with cloud formation and air flows [source: NOVA ].

This article is intended to get you started in the mind-blowing world of fractal geometry. If you have a mathematical bent you might want to explore this world a lot more using the sources listed on the next page. Less mathematically inclined readers might want to explore the infinite potential of the art and beauty of this incredible and complex source of inspiration.

Take a blank sheet of paper, and draw a straight line from the center to the bottom. Now draw two lines, half as long as the first, coming out at 45 degree angles up from the top of the first line, forming a Y. Do that again for each fork in the Y. The process by which shapes are made in fractal geometry is amazingly simple yet completely different to classical geometry. While classical geometry uses formulas to define a shape, fractal geometry uses iteration.

It therefore breaks away from giants such as Pythagoras, Plato and Euclid and heads in another direction. Classical geometry has enjoyed over years of scrutinisation, Fractal geometry has enjoyed only The shapes that come out of fractal geometry look like nature. This is an amazing fact that is hard to ignore. As we all know, there are no perfect circles in nature and no perfect squares.

However with simple formulas iterated multiple times, fractal geometry can model these natural phenomena with alarming accuracy. Fractal geometry does this with ease. This blog post shall give a quick overview of how to make fractal shapes and show how these shapes can resemble nature. It shall then go on to talk about dimensionality, which is a cool way to measure fractals.

It ends by discussing how fractal geometry is also beneficial because randomness can be introduced into the structure of a fractal shape. The post requires almost no maths and includes lot of pretty pictures. In normal geometry shapes are defined by a set of rules and definitions. For instance a triangle consists of three straight lines that are connected. The rules are that if you have the length of all three sides of the triangle it is completely defined, also if you have the length of one side and two corresponding angles the triangle is also defined.

Fractal geometry also defines shapes by rules, however these rules are different to the ones in classical geometry. In fractal geometry a shape is made in two steps: first by making a rule about how to change a certain usually classically geometric shape.

This rule is then applied to the shape again and again, until infinity. In maths when you change something it is usually called a function, so what happens is that a function is applied to a shape recursively, like the diagram below.

After it has repeated an infinite amount of times, the fractal shape is produced. What are these functions then? What do you mean by repeating infinitely? As always, this is best explained by an example…. A good fractal shape is called the von Koch curve. The rules, or function, are extremely simple. First you start with a straight line. Replace the middle segment with an equilateral triangle, and remove the side of the triangle corresponding to the initial straight line.

This is what happens to the straight line, our initial shape, when it goes through the function the first time, the first iteration.

Now, the shape it has produced is fed back into the function again for a second iteration:. Remember the rule was that any straight line would be split into thirds, so now 4 lines are split up and made into triangles. It scores approximately 2. Similarly, your lungs are about 2. Packing such a huge surface area into your body provides you with the ability to extract enough oxygen to keep you alive.

Fractals can be found everywhere in the world around you, from a humble fern to the structure of the universe on the largest of scales. Even certain parts of your anatomy are fractal, including your brain. If you are mindful of fractals, you will be struck by the sheer variety of places you can find them as you go about your daily routine - from clouds, plants and the landscape to church windows and laboratories ….

Fractal mathematics not only allows us to begin modelling the shapes of nature, it can also reawaken our childlike wonder at the world around us. With thanks to Jon Borwein. Festival of Social Science — Aberdeen, Aberdeenshire.

Edition: Available editions United Kingdom. Become an author Sign up as a reader Sign in. Fractals can be found everywhere in the world around you. Michael Rose , University of Newcastle. Fractals are exquisite structures produced by nature, hiding in plain sight all around us.