A Panorama of Fractals and Their Uses

Michael Frame and Benoit B. Mandelbrot
Mathematics Department, Yale University

When fractals were a brand new idea, and even when The Fractal Geometry of Nature, was a new book being reviewed, futurists disagreed, as is their wont. A few envisioned a whole new field, but many did not. They found it more likely (as we did) that fractal ideas would spread far and wide among existing fields, influencing them to varying degrees but replacing or absorbing none. This second scenario is what is presently witnessed.
In the absence of a new field, no one is expected to produce a comprehensive treatise on fractals, but guidance among far divergent fields is indispensible. Some publications respond by including lists of diverse uses that tend to read like incomplete directories of a university's departments. They serve no positive purpose, antognize some readers and make us cringe. The panorama on which we settled instead could be viewed as a detailed wordbook or lexicon that orders alphabetically a wide mixture of entities of all sorts. The connectivity and mutability of the web make possible dynamical cross-references.
The pages that follow are drawn to serve three separate purposes.
They began modestly as a glorified index for the topics treated in the book Fractals, Graphics, and Mathematics Education.
They developed by addressing themselves specifically to the traditionally trained mathematicians who endeavor to help humanities and social science students to connect with mathematics as the science of patterns and changes. In that spirit, they expanded to incorporate examples that are classics in the fractalist community but known to few adults trained in mathematics.
Last, but not least, we view the pages that follow as an early draft or work-in-progress. Initially we include mainly, but not exclusively, examples from the arts, humanities, and social sciences. These are generally less familiar to mathematicians, yet are very important to non-science students. The desire to post something while we are still living dictated some restraint in the initial offering of the Panorama.
Some entries are fairly long, to give a flavor of the ways in which this document can be a resource for teachers. Other entries are mere hints, included with the hope of inducing experts to submit contributions.
We emphasize the current panorama is not intended to be an encyclopedia of all uses of fractals. Rather, it represents some we have found effective in the classroom. Many readers will note their favorite applications are missing. We encourage them to contact us frame@math.yale.edu about contributing to this website and to the eventual printed version of the panorama.
Simple ideas that fundamentally alter our perception of the world are very rare. Overenthusiastic zealots of fractal geometry believe already there is sufficient evidence to guarantee that role; rabid critics assert fractal geometry is a passing fad, soon to disappear from sight. (How long must we wait before a passing fad passes? Thirty years seems a bit long.) We believe fractal geometry will make fundamental changes in science and far beyond, but are waiting for the evidence to accumulate. One of the roles of this Panorama is to share some evidence we have already seen. Another role, more important in our eyes, is to invite you to share your examples with us. In this way, we hope this Panorama indeed will grow into A Panoramic Encyclopedia of Fractals and Their Uses.
We're waiting to hear from you.