Galaxy of Galaxies: section of a Mandelbrot Set


How long is the coast of Britain? asks Benoît Mandelbrot, mathematician and recently deceased* conquistador of fractal geometry. The answer, surprisingly, depends upon how closely you’re able to look.

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line, Mandelbrot writes in the introduction to an iteration on his seminal work. Look close enough and one begins to see that “coast” and “length” are merely convenient abstractions of an ever-receding horizon of unseen complexity. Our words simply refer to that which we can see without looking closely.

There is madness where Mandelbrot goes. Unspeakable, beautiful madness.

Seeing is Perceiving

Speaking of madness, there is much of it in the perception of what we see. An article in this month’s issue of Psychological Science explores the perceptions that arise from seeing what only appears on a political map:

By perceiving state borders to be physical barriers that keep disaster at bay, people underestimate the severity of a disaster spreading from a different state, but not the severity of an equally distant disaster approaching from within a state. We call this bias in risk assessment the border bias.

Marginal Revolution’s Alex Tabarrk quips, amusingly, the authors show that making the border more salient by darkening the border lines on a map can make people feel even more protected.

Of course, not all is madness, but this bit of madness is a poignant example of seeing forming the basis of perception.

Seeing is Conceiving

I looked into my wife Laurie’s eyes some years ago and saw something that led to a thought — a fascination and an obsession which grew into a romance to power a lifetime companionship. Other conceptions followed and what was seen in a moment has developed into another generation of humankind and unfathomably rich experiences.

Along with the hyper-stereoscopics of a full complement of senses, seeing feeds the mind with conceptual nutrients. Seeing provides the feedback from which the mind builds a virtual representation of our environment, upon which we can play-out our war games and love affairs in anticipation of, in preparation, for real events. Seeing truly is simultaneously an assessment of the moment and a vision of the future, for the mind was formed on the patterns depicted in a Mandelbrot set. Each moment being, to borrow Madelbrot’s words, a reduced scale image of the whole — a vision of past, present, and future in one.

Seeing puts us in touch with the concept of “our world.”

Seeing is …

… as seeing does. Our eyes gulp up an extremely narrow slice of light spectrum as it bounces off the elements around us. The data captured is only that which our eyes have been tuned to collect — tuned by aeons of exposure to our world’s evolutionary forces. The same forces work on the mind, which seeks to interpret the meaning of electrical impulses trigged with each spray of light to the eye. In fact, most of what we know as “seeing” takes place in the mind where images are formed, gaps are filled, associations are made and meaning attached. The mind does not even require feedback from the eyes to see, as each thought may form a vision independent from what the eyes have seen, and every dream is a holistic worldview, while the eyes flutter unseeing with REM syncopation. When we understand we may say, “I see.” When the mind sees, it interprets and tests, succeeds or fails, sees further then expands its range of vision, takes pleasure while learning from pain. Iterate. Iterate.

We have fashioned tools to extend our vision, devised conceptual frameworks and mathematical constructs to expand our view, and there is always a limit, a horizon, a frontier. What lies beyond we can only glimpse at its vanishing edge; where visions branch out of visions past, at once familiar and haltingly bizarre.

Seeing further, seeing deeper. Giving way to the next iteration.

Seeing is believing that, though one life may end, another will begin.

– –

*Benoît B. Mandelbrot, 20 November 1924 — 14 October 2010


  • The ImageGalaxy of Galaxies is a computer-generated graphical representation of a partial Mandelbrot set. Featured at Wikimedia Commons, it sits among a large and fascinating collection of fractal images.
  • Resources – Special thanks to Wikipeda with its wonderfully detailed and link-rich articles, and to the New York Times, and the UK’s Telegraph whose beautifully written tribute obituaries fed my mind. To the many bloggers in my feed who celebrated Mandelbrot’s life and the beauty of his achievements this past week, among them: Planet Money’s Jacob Goldstein and’s Jason Kottke. Your posts were fractal events for my own. Also to Jeremi whose Through a Glass, Textured Seasons coincides my post with fractal beauty all their own. Finally to all of my colleagues here on Doves & Serpents: your phenomenal abilities as writers and editors are an inspiration to me.
  • About - Cipher on a Wall is a weekly column and forum here on Doves & Serpents which explores the realm of mind, memories, and dreams. You can find an introductory post for Cipher on a Wall here and a full archive of posts here. My name is Matt, and I’ll be your host for the duration.
  • Updates – the approach we’ll be taking with Cipher on a Wall is to encourage lively and ongoing discussion throughout the week between each Saturday edition. To help with this I’ll be returning to each post and adding updates in the form of additional thoughts, observations, related news, elevation of comments, links, additional resources. etc. Just know there will be updates so it’ll be worth checking back occasionally throughout the week.

Update 1

MoMA curator Paola Antonelli interviews Benoît Mandelbrot. Posting mostly as an opportunity to see that man behind the notion. (via SEED Magazine) …