# M-Set AS METAPHOR

In 1803 William Blake wrote the well-known lines:

"To see a World in a Grain of Sand
And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour."

There is something magical about this idea of looking at a tiny grain of quartz and seeing a whole world – a microcosm that might even contain its own numberless grains of sand.  Blake’s implied relativity of spatial and temporal scales is intriguing and, given the durability of this worlds-within-worlds concept in art, literature and science, the blurring of distinctions between the very large and the very small must strike some kind of harmonious chord in the human mind.

Could this concept apply to the physical world? To be honest, we must answer that we cannot be absolutely sure on this point. A few modern cosmological theories, such as the Inflation addendum the Big Bang model, do permit an infinite hierarchy of “universes”, but most cosmological thinking still retains the usual notions of a finite universe and an absolute size scale extending from smallest to largest objects. In the boundless realm of mathematics, however, the story is quite different.  An intensely beautiful mathematical “world” called the Mandelbrot Set provides an awe-inspiring metaphor for the poetic vision that large and small may be relative, rather than absolute, concepts.

The M-Set was discovered by the French mathematician Benoit Mandelbrot in 1980, and has been widely publicized in books such as The Fractal Geometry of Nature by Mandelbrot and Chaos by James Gleick, and in scientific magazines (for example see the beautiful pictures and excellent summary in the July 1985 issue of Scientific American).  For those who are mathematically inclined, here is a brief outline of how the M-Set is created.  Start with the expression z ® z2+c.  Choose two complex numbers z and c, and solve the expression z2+c to get a new value of z.Put the new z into the z2+c term and compute another z value.  Continue this process on a computer for many iterations.Color coding the rate at which different values of c cause z to shoot off to infinity, stabilize in the realm of finite numbers, or go to zero, creates the visual embodiment of the “ M-world”.One of the many wonders of this infinitely complex “world” is that it can be created by just a few simple lines of computer code that are repeated recursively.  From these little algorithmic loops came the most baroque/rococo “universe” that anyone had ever seen.

As in Blake’s poem, this “world” has no bottom.  There are layers within layers within layers, literally without end, of whorls, mandellas, and spider webs; there are wheels-within-wheels, and indeed whole worlds-within-worlds.  The M-Set is a truly psychedelic wonder of the first order.  Here we have an almost palpable archetype for the concept of infinity.  I would use the phrase tangible archetype, except for the fact that one of the defining features of the M-Set is that nowhere in the labyrinth can one find a surface smooth enough for a tangent.  Upon magnification even surfaces that appeared to be smooth explode with quills and scrolls and lighting bolts and spiral staircases.  Smoothness does not exist in the M-world, not on any of its infinite size scales.

And there is something more, something truly sublime.  Start with a small patch of the M-Set.  Then imagine looking at it through an ideal microscope with unlimited magnifying power.  As you observe the M-Set on ever-smaller scales, down through literally endless layers of ornate structure, you occasionally come upon a rapidly expanding vortex of dazzling color with a small black structure at its center.  As more detail is revealed, you witness something that is almost a miracle.  The black spot appears to be the M-Set itself!  Deep within the outermost M-world are tiny, but equally infinite copies of the M-world, and if you keep enlarging one of these microscopic M-worlds you will eventually find even tinier M-worlds.  There is no end to this hierarchy, no bottom-most level, just endless recursive worlds within worlds within worlds... .  Scale (i.e., size) is no longer fixed and absolute, but rather is purely relative.  The M-Set’s beautiful symmetries of space, time and scale convey an immediate aesthetic pleasure, and also compel one to think about these strange concepts of self-similarity (copies within copies), infinity and relativity of scale.

In addition to the books and the article mentioned above, there are two even more dramatic resources for basking in the mysteries of the M-Set.  The Public Broadcasting System has commissioned and shown a remarkable hour-long program on this subject.  The title of the program is “The Colors of Infinity”.  It is hosted by Arthur C. Clarke, and the exploration of the M-Set via advanced computer graphics is a tour-de-force.  Video copies of “The Colors of Infinity” can be purchased for about \$30 from Films for the Humanities, Inc. of Princeton, New Jersey.  Those who would prefer to explore the M-Set interactively on a computer are in luck.  There are quite a few websites that allow one to do first hand exploration, such as:

or

Our present science tends to favor reductionism.  We surmise that the physics of our world has a bottom-most or most fundamental level and all phenomena are built up from these quarks, or strings, or whatever is currently in fashion.  Mathematics, on the other hand, need not be so limited.  Here the mind is set free to dream of universes with the most exquisite symmetries and infinities.  I urge you to explore the M-Set.  The epiphanies you experience will be worth the effort.

Robert L. Oldershaw
Amherst College
Amherst, MA 01002

e-mail: rloldershaw@amherst.edu