Schwarz D Surface

Schwarz D Surface is a triply periodic minimal surface, defined as a nodal algebraic surface, genus 3 per unit cell, embedded, with Pn-3m symmetry (a crystallographic space group).

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def-nodal embedded has-approximation implemented minimal minimal-periodic tradition-classical tradition-crystallographic triply-periodic

About

Schwarz's diamond surface, the P surface's conjugate: the two are the same local geometry rotated through a right angle in the associate family, so a patch of one bends into a patch of the other without stretching. Its two labyrinths follow the bonds of the diamond lattice, which is where the name comes from.

Formula

\sin\left(x\right) \sin\left(y\right) \sin\left(z\right) + \sin\left(x\right) \cos\left(y\right) \cos\left(z\right) + \cos\left(x\right) \sin\left(y\right) \cos\left(z\right) + \cos\left(x\right) \cos\left(y\right) \sin\left(z\right) = 0

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: Pn-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 (1890).
M. Weber, minimalsurfaces.blog, 'Schwarz D Surface' (mirror: minsurf ch308): 'The period lattice is the face-centered cubic lattice, and its quotient is a compact Riemann surface of genus 3' -- independent genus-per-cell oracle.