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Borderline - A Fractal Poem

© Ed Seykota, October 15, 1986

Graphics as of March 7, 2003, with software at:



Z-naught, a point is squared and when

Itís added to Z-naught again,

Z-one, the sum, in turn is squared

And with Z-naught again itís paired.

For generating where Z is,

One uses this recursive biz.

The Mandelbrot Algorithm ...

... generates a series of points.

Some points escape all gravity

And navigate infinity

While some born closer to the core

Contract and stay there evermore

And some, a very special kind

Hang out along the borderline.

Unstable - heading to infinity


Stable - heading to zero


Borderline Stable - staying in a range

The Mandelbrot Set is the set of these Z(0)'s.

This little game of vector boat

Gets its name from Mandelbrot

Whoís set upon the complex plane

Reflects a most complex terrain,

A captivating, endless scene,

So ordered, yet so serpentine.


So I begin to walk its trails

And come to know where knowing fails

And logic, coping with the joke

To no avail, finds cosmic smoke

So this my lab-book diary

Comes to be writ in poetry.


When fractal first comes into view,

Itís tidy-bounded two by two

And harmless and all globby cute

With head and arms and butt to boot

And neatly tied or so I think

Until its skin requires more ink.


For all the way around the beast

Are surface features amply creased !

And lo, within its vallied crusting

Tiny sets of mountains thrusting

Up and out are treased and nooked

With further beauties thorned and hooked.


What sort of creature can this be

Inflaming curiosity

With such a tantalizing shell?

I surge an urge to know it well.

Iíll analyze it for a thrill.

Compute it to a pulp, I will.


And so my fractal dive begins.

Computer graphics are my fins.

Memory Iíll use to store

A record of its every pore,

Then zoom in on its horny hide

To see its secrets, magnified.

I think the lump atop will do

Quite nicely for a larger view.

So aiming my PC, I hook

A blowup snapshot for my book.

But wait, whatís this? How can it be?

Some parts are still too small to see. 

Yet I have tools to match the game

So now a detail I will tame.

I choose one to be magnified

And watch it from my printer glide.

Its outlines make my spirits fall.

Its features still are way too small.

I grab a chunk, a stranded fleck

To hopefully conclude my trek,

And yes, it grows to fill my page

And oh, itís still a middle stage

And what is this? Amazing case!

A baby fractal deep in space.

And rezzing in on baby frac

Just takes me further down the track

To kids upon another slide

Whose tiny features also hide

Some smaller children of a set

Whose offspring more ilk do beget.

And on and on expands the fan

ĎTill my computer cannot scan

Another ply of forms so small

And hi-precision gobbles all

The decimals I have to spare

And still the answer is not there.

And now my head begins to spin.

I reason, reason cannot win.

I feel the zap of fractalís koan

As to the borderline Iím blown

And sucked into a galaxy

Of synchronistic ecstasy,


A place of essence without mass,

Of asymptotes and plasma gas

And all-connected universes

Tuning-in related verses

Of an odd familiar song,

While quarks in ether sing along.


And deep within this stellar soup,

Essential to the concept, ďgroup,Ē

The notion where it all begins

Is: who are outís and who are inís?

I pick the apple. I define

An embryonic borderline.


I pull this topic to the beach

Where stone meets water. Doesnít each

Small rock and pebble get a place

When tallying the interface?

And then to pocks on grains of sand,

Where lies the ocean, where the land?


With surf and turf along the beaches,

Each into the other reaches.

Sand and water melt into

A frothy fuzzy slurried stew,

With fractal sand grains swimming wild

And fractal drops on beaches piled.


And algae green, a form of life

Which further mediates the strife

Incorporates a snatch of each.

So much for life along the beach.

Now intertwining earth and air

Are ferns and bees and other fare.


And one more question if youíll hear it:

Are we flesh or are we spirit?

Does God exist and script the play

Or are we, rather, chunks of clay?

Lovers know as they entwine.

Life is just a borderline.