Supershape Flower

What it draws

Gielis's superformula r(ϑ) = (|cos(mϑ/4)|^n₂ + |sin(mϑ/4)|^n₃)^(−1/n₁) gives a radius for each angle, as the preset Gielis Supershape shows in the plane. Wavelace has it as a function, superformula(θ, m, n₁, n₂, n₃), which computes exactly this. The solid multiplies two of them. One runs around the vertical axis, with u, for the section. The other runs with φ = v − π/2 for the profile.

The section is superformula(u, 5, n₁, 4, 4), with n₁ on a slider at 0.6. The profile is superformula(v − π/2, 3, 0.1, 1.2, 1.2), so its power is −1/0.1 = −10.

The preset opens with Span of v (×π) at 2. That runs the profile a whole turn, where a solid usually takes half a turn from bottom to top, and the extra half turn is what makes the flower.

Five arms, three layers

Around the axis the section's radius swings between 1 and 3.17, five times a turn, so it is a five-armed star. Its steep exponent −1/n₁ = −1/0.6 draws the arms out long and thin.

The profile's power of −10 is steeper still. Its radius is 1 in three narrow lobes, at φ = 0°, 120° and 240°, and falls to 0.06 between them. Each lobe is a thin sheet of the star.

The lobe at 0° lies flat, its arms reaching 3.17. The other two point up and down at 60° from the horizontal. There cos φ is negative, which sends the section through the axis to the far side: those arms fall between the flat ones, 36° over, and reach half as far, 1.59, at heights ±0.87.

Try

• Set Span of v (×π) to 1. Only the flat lobe is left, a five-armed star as thin as a leaf, 0.13 as tall as it is wide.
• Press Top in the deck. Ten arms, the flat five and the tilted five between them.
• Drag n₁ to 2. The section now swings only between 1 and 1.41, and the arms widen into rounded petals.
• Press Side in the deck to see the three layers edge on.

Read more

• Superformula
• Superquadrics
• Spherical coordinate system