Johan Gielis
The superformula, American Journal of Botany, 2003
Belgian botanist and mathematician who published the superformula in 2003, one equation that draws many of the shapes of plants, shells and cells.
Superformula
One formula with four numbers that draws starfish, petals, shields and squares. Nested as a family, each ring a step further along, it blooms from spiky to round.
About this pattern
Johan Gielis, 2003 · Gabriel Lamé, 1818
Gabriel Lamé's superellipse, |x|ⁿ + |y|ⁿ = 1, rounds a square into a circle and on into a star as n goes down; Piet Hein made a table of it. In 2003 the Belgian botanist Johan Gielis published a generalisation in the American Journal of Botany, written as the distance from the centre in each direction.
One number, m, says how many lobes there are. Two more, n₂ and n₃, shape the lobes, puffed up or pinched; and the last, n₁, sets how deep the cuts between them go. With a handful of values it draws a starfish, a flower, a cross-section of a stem, a cell, a shield, even a triangle with soft sides, which was Gielis's point: many of the shapes of living things are one shape seen with different numbers.
Here a whole family is drawn at once. From the centre out, each ring takes n₁ a step further from the first value to the last, so the family runs from pinched and spiky to full and round, and a twist turns each ring a little on the one inside, until the lobes spiral.
In motion the rings grow outward, a new one opening at the centre as the outermost fades. The formula also has a three-dimensional form, often called a supershape: one superformula runs round the equator and a second from pole to pole, multiplied together the way a sphere's circles are, so that the same few numbers make seed pods, lanterns, soft cubes and spiky stars.
Choose Solid to draw one as lines of latitude and longitude, the nearer lines heavier, and drag to turn it; in motion it turns once round. Dragging the flat rings tilts them like a card.
Behind the mathematics
Gabriel Lamé1795–1870
The superellipse it generalises, 1818
French mathematician whose generalised ellipse equation became the foundation of the superellipse and many smoothly squared forms.