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 *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 ^{2}+c. Choose two complex numbers z and c, and
solve the expression z^{2}+c to get a new value of z.Put the new z into the z^{2}+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 “

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 *tangible*
archetype, except for the fact that one of the defining features of the

And there is
something more, something truly sublime.
Start with a small patch of the

In addition to
the books and the article mentioned above, there are two even more dramatic
resources for basking in the mysteries of the

http://www.softlab.ntua.gr/mandel/mandel.html

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

Robert L. Oldershaw

Amherst College

Amherst, MA 01002

e-mail: rloldershaw@amherst.edu

File Uploaded: 31st October 2000

Paper Updated: 1st June 2001