A Letter to Scientific American

Drag the parameter sliders to explore the parameter space. Each position produces a distinct organic form from the same four equations.

In 1987, Peter de Jong wrote a letter to Scientific American. The magazine had been running a column called "Computer Recreations," a long-running series that treated computation as a site of mathematical play over engineering utility. De Jong's letter described four equations and the forms they produced when iterated. That letter, a few lines of mathematics and description, is the entire historical record. De Jong himself has left almost no other trace in the literature. What he submitted was a few equations and an observation that they produced interesting shapes, without the apparatus of a paper, a formal result, or a theorem.

The equations are:

x_{n+1} = sin(a · y_n) − cos(b · x_n)
y_{n+1} = sin(c · x_n) − cos(d · y_n)

Four parameters: a, b, c, d. Each is a real number, typically in the range [−3, 3]. From a starting point (usually (0, 0)) the equations generate a sequence of points. Plot enough of them, color by density or velocity, and organic forms emerge: leaves, bones, feathers, folded cloth. The parameter space is continuous, so the transition between forms is smooth. A small nudge to any of the four values produces a shape that is related but distinct, as if the form had been stretched or pressed from a common mold.

The Buddhabrot, Clifford attractor, and de Jong attractor belong to the same family of iterated function systems, mappings applied repeatedly to a point, with the accumulated orbit rendered as an image. What distinguishes the de Jong system is its particular texture of constraint: the sine and cosine terms ensure that the orbit stays bounded (all values remain in [−2, 2]) while the parameter product keeps the dynamics sensitive enough to produce genuine structure instead of noise. The four-parameter space is small enough to explore interactively but rich enough that systematic sampling would require years. Most of it has never been seen.

The Clifford attractor uses a similar parametric form. The family resemblance to de Jong is visible in the banded, folded structure of the orbits.

What makes the de Jong attractor interesting as an object is the gap between its description and its appearance. The equations contain no biology. They have no growth term, no branching logic, no simulation of any organic process. They are four trigonometric operations applied iteratively to a pair of coordinates. And yet what emerges consistently, across wide regions of the parameter space, are forms that read as biological: bilateral symmetry, tapering extremities, internal foliation that resembles the venation of a leaf or the lamellae of a gill. The visual vocabulary of the forms is botanical and anatomical before it is mathematical.

This is not a coincidence to be explained away. The appearance of biological structure in a purely mathematical iteration points to something real about the relationship between the mathematics of smooth, bounded oscillation and the geometry that natural selection tends to produce. Both are solutions to similar constraint problems: how to pack a large surface area into a bounded volume, how to distribute material for structural efficiency, how to produce stable forms under perturbation. The equations don't know this. But the forms remember it, or something isomorphic to it.

The Buddhabrot renders the density of orbits that escape the Mandelbrot set's boundary, producing forms that bear a similar family resemblance to biological structure.

De Jong's letter was published in the "Computer Recreations" column, which ran from 1984 to 1991 under A.K. Dewdney's editorship. The column was a minor institution of mathematical computing culture, a monthly demonstration that the tools of computation could be turned toward questions of pattern and form that had no engineering application and required no justification beyond their interest. Dewdney's instinct was that the best advertisement for mathematical computing was its strangeness, not its usefulness. He was right, and the archive of that column is a reliable source of artifacts that remain genuinely surprising.

We built the de Jong artifact for manipulation. The parameter sliders are the point. The image that loads first is one configuration among uncountable others, and the space of forms accessible by dragging the four sliders constitutes a territory that no static image can represent. What you are navigating when you move the sliders is a continuous manifold, a four-dimensional space in which each point is a form and adjacent points are adjacent forms. That continuity is the artifact's argument: that these shapes are positions in a space, that the space is smooth, and that the transition between organic-looking forms is as gradual and lawful as the transition between any other kind of mathematical object.