Math Art
P-Gyroid-D Associate Family

P-Gyroid-D Associate Family

P-Gyroid-D Associate Family is a triply periodic minimal surface, given by a Weierstrass representation, genus 3 per unit cell, embedded or immersed depending on parameters, with varies symmetry (a crystallographic space group).

Open P-Gyroid-D Associate Family in the interactive viewer →

def-weierstrass implemented minimal minimal-periodic tradition-classical tradition-crystallographic triply-periodic

Formula

Gauss map g
−i⁢3+z⁢−1+z2⁢i⁢z−1+z⁢−3+z\frac{-i \sqrt{3 + z} \sqrt{-1 + z}}{2 i \sqrt{z} - \sqrt{1 + z} \sqrt{-3 + z}}
Height differential dh
−iz⁢1+z⁢−3+z\frac{-i}{\sqrt{z} \sqrt{1 + z} \sqrt{-3 + z}}

Properties

Family
minimal-periodic
Given by
a Weierstrass representation
Curvature
zero mean curvature
Periodicity
triply periodic
Genus per cell
3
Ends
0
Embedding
embedded or immersed depending on parameters
Orientable
yes
Symmetry
varies
Symmetry kind
crystallographic space group
Fidelity
exact
Exactness
numerical-integral

Definition

Extracted from the shipped row's source by AST (complete multi-line lambdas, the shipped rPD coordinate 1-forms (_pgd_om in math_art/minsurf/weierstrass.py): dh = phi3, g = phi3/(phi1 - i*phi2)) and reproduced numerically against the shipped callables over the complex plane, exactly -- no scalar slack. theta = 0 is Schwarz P; the Bonnet angle e^{i theta} sweeps the associate family through the gyroid (38.0148 deg) to Schwarz D (90 deg). Square roots are per-factor principal branches, the shipped code's own convention on its fundamental domain.

Sources

Weierstrass data reproduced numerically against the shipped implementation (math_art/minsurf/zoo.py, row PGD) by sampling g and dh over rings in the complex plane at the row's default parameters: g: matches over 144 complex samples (worst 8.52e-14); dh: matches over 144 complex samples (worst 1.11e-16).