"In 1997 the superformula was discovered, which solved the problem of the limited symmetry of superellipses and supercircles. Supershapes like pentagons and starfish, triangles and rose sepals, flowers and leaves, can now be described by a single equation, based on the generalization of Lamé's supercircles and superellipses.
The key step from Lamé’s supercircles was to convert the equations to polar coordinates and to add a single parameter for symmetry. It was extended into three dimensions as well.
This discovery was published in the book “Inventing the Circle” (2001, 2003) and as Invited Special Paper in the American Journal of Botany in April 2003. Worldwide awareness was created by websites such as Nature Science update, Science News Online, and Wolfram’s Mathematica website.
Supershapes were introduced later in the field of geometry under the general name of Gielis' curves, surfaces & -transformations, also in higher dimensions."
Quoted from GIELIS' CURVES & SURFACES
How to make shapes like this
Bram Stolk, author of the Superformula Shape Miner for iOS writes:
"Superformula was created to have fun with mathematics. Anybody can explore the colorful universe of shapes described by a powerful formula.
The Superformula, or Gielis-formula was first proposed by Johan Gielis. The formula can be used to define a wide range of shapes, including shapes that occur in nature.
When describing shapes in three dimensions, two instances of the formula are applied, each with 6 parameters. This results in a 12 dimensional parameter space in which a multiverse of interesting shapes reside.
The Superformula app is developed as a tool to mine this 12 dimensional parameter space. By swiping your finger, the space is explored, two parameters at a time. The strength of this application comes from the lightning fast calculations used to determine the shapes. The app uses an OpenGL-ES2 vertex shader to compute the shapes on the graphics hardware of the device. This means zero wait, and instant visualization of the Gielis-shape."
EXPLORATIONS OF GIELIS CURVES & SURFACES by Genicap
Using the Super Formula for building Super-Antennas by Blue Economy (Thanks for sharing.)