Algebraic geometry / 01

Quintic Slice Explorer

Geometry

A real two-dimensional slice of the quintic threefold — the solution set of p⁵ + q⁵ = 1 — projected from four real dimensions into three. Each coloured patch belongs to a pair of fifth-root sheets (k, ℓ).

p⁵ + q⁵ = 1 X = Re(p), Y = Re(q) Z = cos(α)·Im(p) + sin(α)·Im(q)

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Projection

Projection angle α Imaginary extent |v|

Surface

Mesh resolution Sheet selection Wireframe Auto-rotate

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Mathematics & controls

With t = u + iv, p = exp(2πik/5) cos(t)^(2/5) and q = exp(2πiℓ/5) sin(t)^(2/5). We use 0 ≤ u ≤ π/2 and −V ≤ v ≤ V, with k, ℓ = 0,…,4. The five root choices generate the 25 sheets. This is a finite visualization window, not the entire compact manifold.

The displayed metric is not the Ricci-flat Calabi–Yau metric. This is a projected algebraic curve (real dimension 2), not the full threefold (real dimension 6).