Radial Symmetry — Image Pack

Polar coordinates collapse two-dimensional geometry to two simple ideas: a distance from the center and an angle around it. From this minimal starting point you can grow elaborate symmetric forms. The rose curves r = a·cos(kθ) trace petal patterns whose count depends on whether k is even or odd. The superformula of Johan Gielis generalizes this further with six parameters, producing flowers, stars, leaves, and shells from a single closed-form expression. The result is geometry that recalls nature without imitating it — the same mathematics that describes radiolaria, snowflakes, and pollen grains. Every form in this collection is the literal locus of a polar equation, rendered as a continuous mathematical object.

r(theta) = a cos(k theta), r(phi) = (|cos(m phi / 4)|^n2 + |sin(m phi / 4)|^n3)^(-1 / n1)