Schwarz P Surface
Schwarz P Surface is a triply periodic minimal surface, defined as a nodal algebraic surface, genus 3 per unit cell, embedded, with Im-3m symmetry (a crystallographic space group).
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complete-infinite-total-curvature def-nodal embedded has-approximation implemented minimal minimal-periodic tradition-classical tradition-crystallographic triply-periodic
About
The oldest of the triply periodic minimal surfaces, from Schwarz's work in the 1860s. Think of it as a scaffold of tubes joining the faces of a cube to its neighbours, repeated forever in all three directions; the surface separates space into two identical interlocking halves. Schwarz built it by solving for a single patch spanning a skew quadrilateral of straight lines and then reflecting that patch across its own edges, which is the method the whole family is named for.
Formula
Properties
- Family
- minimal-periodic
- Given by
- a nodal algebraic surface
- Curvature
- zero mean curvature
- Periodicity
- triply periodic
- Genus per cell
- 3
- Ends
- 0
- Embedding
- embedded
- Orientable
- yes
- Symmetry
- Im-3m
- Symmetry kind
- crystallographic space group
- Fidelity
- approximation
Definition
Standard nodal approximation as shipped in math_art/minsurf/tpms.py. No exact definition is stored for this surface, so this approximation is all the record has; `approximates` is null rather than pointing at itself.
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- H. A. Schwarz, Gesammelte Mathematische Abhandlungen, Springer (1890) -- the P and D surfaces.
- A. Schoen, schoengeometry.com TPMS pages (mirror: schoen ch006): 'P, D, and G are each of genus three, which is the minimum possible genus for a triply-periodic minimal surface' -- independent genus-per-cell oracle.