Supershape Pagoda
Shape · (x, y, z) = f(u, v, t)
(cos(u) · superformula(v − π/2, 4 · k, n₁, 5, 5) · sin(v), −superformula(v − π/2, 4 · k, n₁, 5, 5) · cos(v), sin(u) · superformula(v − π/2, 4 · k, n₁, 5, 5) · sin(v))
Open in the app The dials and keys named below are the app's.
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. A solid takes two of them multiplied: one around the vertical axis for the section and one from bottom to top, with φ = v − π/2, for the profile.
Here the section is a plain circle. With m = 0 the formula is (1 + 0)^(−1/n₁) = 1 whatever the exponents are, so the section factor is just cos u and sin u, and the solid is a surface of revolution. Everything it shows is in the profile.
Where the rims come from
The profile is superformula(v − π/2, 4k, n₁, 5, 5), with n₁ = 1.6 on a slider. Its m = 4k = 14, with k on a slider at 3.5. Its radius swings between 1 and 1.92. It is largest where the cosine and sine terms are equal, at 3.5φ = π/4 + jπ/2 for each whole number j.
Over the half turn from bottom to top that happens six times, at φ ≈ ±0.22, ±0.67 and ±1.12. Turned about the axis, each maximum becomes a flat rim and each minimum a waist between two.
Near the top the profile still swings, but cos φ shrinks it toward the axis. What is left of the last lobe opens as a cup, and the final stretch to the pole rises as the spike in its middle.
Try
- Press Front in the deck. The outline is the profile itself, the six rims seen edge on.
- Drag k to 1, which is m = 4. Only two rims are left, at φ ≈ ±0.79.
- Drag n₁ to 4. The rims round off into gentle bulges, the radius now reaching only 1.30.
- Turn Mesh down from its top stop to see the rings and meridians of the wireframe underneath.