Posts Tagged ‘mathematics’

Fractal learning

Sunday, October 11th, 2009

One day I asked myself the question, what would learning look like if it could be visualized?

322px Mandel zoom 00 mandelbrot set Fractal learning

A fractal. Latin fractus, meaning fractured. It is recursive by definition.

What comes to my mind is the Mandelbrot set. In 1975, Benoît Mandelbrot first coined the term fractal. Mandelbrot emphasized the use of fractals as realistic and useful models of many “rough” phenomena in the real world. In The Fractal Geometry of Nature (1982) he writes:

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.

If something is rough, that’s learning. As you approach a new topic, you start from a fuzzy idea of what it could be. As it comes into focus, new details expose themselves on the fringes, enabling you to discover even more interesting perspectives you were not aware beforehand.

Fractals are seen in many parts of nature. Even fractal cosmology exists as an area of study. In a New Scientist article (2007) Labini & Pietronero asked the question, “Is the universe a fractal?“. Their study of nearly a million galaxies suggests that the matter in the universe is arranged in a fractal pattern up to a scale of about 100 million light years.

The Second Law of Thermodynamics states that the total entropy in the universe increases over time, as change happens. In layman terms that would be analogous to a room getting messed up over time as people live in it. In thermodynamics, entropy is a measure of the amount of energy in the system that is no longer available. As entropy increases in the universe, at the same time incredibly intricate and detailed order emerges from the details. Think of the human brain on planet earth for example.

250px Fibonacci spiral 34.svg Fractal learning

Fibonacci spirals also depict the fractal pattern of beauty in nature. Golden ratio is a very well known principle in mathematics and art, first originating in the Liber Abaci (Book of Calculation) in the 13th century. Good examples of forms with Fibonacci spirals include the spirals of shells, various flowerings, the branching of trees and arrangement of leaves on a stem.

The internet looks like a fractal.

So what do fractals have to do with learning?

When considering learning, we are pattern recognizers. Just like fractals, our neural networks evolve over time and extend outside of us. As our environment changes, so do we.  As we process information, in addition to entropy, new patterns emerge. By increasing the ammount of information, you increase the possibility of new patterns to be recognized by people.

In the digital world, entropy is information overload and order is the pattern that emerges from the interconnection of such information.

Knowledge is like a hologram. In holograms, even smaller pieces of it include the picture of the whole object. Knowledge is like a hologram. The experience changes as your point of view towards the object changes. The knowledge is not in a single image, but distributed on a network.

This is pattern recognition. And it’s the culmination of fractal learning. It’s a Mandelbrot set that zooms into the details indefinitely. Universe is fractal by nature. So is learning fractal by nature. It’s rough, it’s self-similar, it’s recursive and increasing the likelihood for serendipity is key for building higher structures.

Here is a recent Finnish presentation recording of my talk on the subject at a conference (Verkkoja kokemassa):

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Here are my slides from the Distance Education & Teaching conference in Madison, USA (still waiting for the presentation recording to be published):