Fractals are great for teaching. They engage many learning styles and are open ended vistas for exploration and further understanding.
Fractals are so useful and ubiquitous that some have criticized the concept as vague or too general. They cite the fact that early attempts to define the concept failed. See the end of this post for a discussion on the precise definition of fractal sets.
Since fractals call on our artistic sensibilities, our abilities to craft and design, our mathematical precision as well as estimation, they are an ideal addition to any classroom, and I teach about them every chance I can. Fortunately there are many prepared curricula that incorporate fractal geometry into traditional school mathematics.
At the college level, I would suggest Yale university’s free online introductory course in fractal geometry . This course gives a gentle introduction to fractals and fractal geometry, then builds up to fractal models for finance and other advanced topics. Don’t know where to start? Check the complete lesson plans included in the course!
For the K-12 level, fractals obviously make great activities and research projects. Teachers working with state standards take note: fractals align with the CCSSM standards:
Understand similarity in terms of similarity transformations.
Drexel University has great resources for lesson plans and introductory materials. Also, think-maths.co.uk has foldable activities you can use in the classroom right away!
Fractals have a precise definition, involving the concept of complex dimension. Fractals are famous for generalizing the concept of dimension from integer values to real number values. The work of Professor Michel Lapidus establishes a further extension of the concept of dimension to the complex plane, and gives a precise definition for the notion of ‘fractality’, which I will describe below. This interpretation is due to myself only, please see Professor Lapidus’ Fractal Geometry and Complex Dimension, 2 ed. for a precise explanation.
Benoit Mandelbrot, a great popularizer of fractals, described dimension as a rate of change of the number of elements of a predetermined size required to fill a given shape, as the scale of those elements is varied. Allowing for the complex solutions to this rate equation (heuristically) yields the complex dimensions of a given object. This set of complex numbers give information about the precise scaling behavior of fractal sets, describing, for example, how a Cantor set is invariant under scaling by multiples of 3, but as the scale is varied continuously, this invariance has a multiplicative periodicity of 3.
A precise discussion of the notion of complex dimension requires the notions of fractal strings and geometric zeta functions. The informal heuristics given above can be found in the works of the scientist Didier Sornette.
PhDtutors 