ROSE

the petal unfolds from one equation

r = cos(kθ)

three parameters, infinite symmetry

LISSAJOUS

frequency ratio is the fingerprint

two sine waves

perpendicular in time

HYPOTROCHOID

the inner gear traces eternity

R, r, d

three numbers contain all spirographs

SUPERFORMULA

one equation, all of nature’s forms

Gielis, 2003

from starfish to snowflake