# Supershape Flower

Shape · (x, y, z) = f(u, v, t)

`(superformula(u, 5, n₁, 4, 4) · cos(u) · superformula(v − π/2, 3, 0.1, 1.2, 1.2) · sin(v),  −superformula(v − π/2, 3, 0.1, 1.2, 1.2) · cos(v),  superformula(u, 5, n₁, 4, 4) · sin(u) · superformula(v − π/2, 3, 0.1, 1.2, 1.2) · sin(v))`

[Open in the app](https://www.wavelace.com/app#p=165) · [This page](https://www.wavelace.com/presets/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](https://en.wikipedia.org/wiki/Superformula)
- [Superquadrics](https://en.wikipedia.org/wiki/Superquadrics)
- [Spherical coordinate system](https://en.wikipedia.org/wiki/Spherical_coordinate_system)