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. His submission consisted of a few equations and an observation that they produced interesting shapes.

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. In the de Jong system, sine and cosine keep the orbit bounded (all values remain in [−2, 2]) while the parameter product makes the dynamics sensitive enough to produce structure. The four-parameter space is manageable in an interactive sketch, but systematic sampling would require years.

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

The de Jong attractor's description and appearance leave a compelling gap. Four trigonometric operations applied iteratively to a pair of coordinates contain no growth term, branching logic, or simulation of an organic process. Across wide regions of the parameter space, however, they produce forms that read as biological: bilateral symmetry, tapering extremities, and internal foliation resembling a leaf's venation or a gill's lamellae. Their visual vocabulary is botanical and anatomical.

The resemblance suggests a relationship between smooth, bounded oscillation and geometries that natural selection tends to produce. Both solve similar constraint problems: packing a large surface area into a bounded volume, distributing material efficiently, and producing stable forms under perturbation.

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 computation could address questions of pattern and form without an engineering application. Dewdney treated strangeness as mathematical computing's best advertisement, and the column's archive still contains surprising artifacts.

We built the de Jong artifact for manipulation. The image that loads first is one configuration among uncountable others. Dragging the four sliders moves through a continuous four-dimensional manifold in which each point is a form and adjacent points are adjacent forms. No static image can represent that territory. The artifact shows these shapes as positions in a smooth space, with gradual transitions between organic-looking forms.