### The Gielis Superformula

The Gielis superformula was discovered by the biologist Johan Gielis in 2003. The formula generates an infinite variety of shapes that seem to appear in nature. The formula has many uses - including generating backgrounds for video games. Rather than painstakingly creating shapes the way an artist would, designers can use the formula to easily produce realistic flora and fauna for the background.

I first came across the Gielis formula in an article about a video game called No Man's Sky. But the formula can reputedly be used to do a lot more than create backgrounds for video games. Genicap is a website devoted to the various uses of the formula -- the applications are wide-ranging and numerous. It's hard to believe that a fairly simple formula can cover all these areas.

Indeed, the Genicap site claims that the Gielis formula "provides a direct geometrical description and relation between circles and squares, flowers and snowflakes, molecules and space-time, sounds and vision, anything and everything." It lists applications in 3-D printing, geology, data mining, data transformation, and antenna design (a "supershaped" antenna, claimed to be much better than existing designs).

I'm certainly not qualified to evaluate any of these claims. What I did do is play around with the Gielis formula and generate some interesting shapes. Despite its power, it's relatively easy to work with the formula.

Here is the formula as presented in the Wikipedia entry devoted to the subject:

where $r$ is the radius in polar coordinates and $\phi$ is the angle that varies from 0 to $2\pi$. The generalization of the formula to 3 dimensions is also straightforward -- you can get the 3D version by multiplying two 2D-versions of the formula.

[If you can run Mathematica, you can download a notebook where you can manipulate the formula in two and three dimensions and experiment by generating various shapes.]

I'll give you a glimpse at the shapes the Gielis formula generates -- you can decide how well it generates shapes that mimic flora and fauna of distant planets in the universe.

Here is a simple two dimensional shape generated by the formula.

I simulated a bunch of shapes by generating randomized values for the variables. A collage of some of the shapes is below.

We can make curved shapes from the Gielis shapes as follows.

And finally, here is a 3-dimensional version generated by the Gielis formula.

[If you can run Mathematica, you can download a notebook where you can manipulate the formula in two and three dimensions and experiment by generating various shapes.]