The foundation of mathematics taught to high-school students today is based on the geometry derivable using compass-and-straight-edge tools, first developed by Plato’s student Euclid and set forth in his master work The Elements written around 300BC. Because the ubiquity of compass-and-straight edge tools hasn’t changed since then, neither has the teaching of mathematics, for thousands of years. Now at long last a change may be afoot. Current students have a new math tool not available to the generations before them: the electronic calculator. Ultimately the use of this tool will far exceed faster and more accurate addition. The heart of a calculator is a microprocessor, and these tiny computer chips can perform repetitive operations in seconds that would take a lifetime for a student to work out on paper. Such repetitive operations are the key to a new kind of geometry based not on lines but fractals. Euclid’s idealized lines and circles are rarely seen in nature, which seems based much more on the previously indescribable fractal shapes of clouds and snowflakes, zebra stripes and seashells, the folds of land seen on a mountain range or the outlines of tree trunks and leaves found there. Even the wiring and organization of the human brain is believed to be fractal, and in some way this fact is now widely suspected to give rise to consciousness itself. In recent years new types of formulas for fractal geometrical shapes as fundamental as Euclid’s have been discovered, but it takes a computer chip instead of a compass or straight edge to figure out how to draw them. Two seminal works in these areas, perhaps not as well known as Euclid’s Elements (yet), are Fractals Everywhere and A New Kind of Science. Unfortunately, these are works for professional mathematicians and scientists (although there are some in-depth reviews that try to explain what they are about). An excellent starting point on fractals and the like for younger math students (which is where we all start at some time) is the award-winning Complexity & Artificial Life Research Concept
(CALResCo) site, which conveniently has a tour bus of their pages for your educational convenience…