Math Art
Plane with Catenoids (square lattice)

Plane with Catenoids (square lattice)

Plane with Catenoids (square lattice) is a doubly periodic minimal surface, given by a Weierstrass representation, immersed, with a layer group.

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def-weierstrass doubly-periodic immersed implemented minimal minimal-periodic tradition-classical

Properties

Family
minimal-periodic
Given by
a Weierstrass representation
Curvature
zero mean curvature
Periodicity
doubly periodic
Ends
0
Embedding
immersed
Symmetry kind
layer group
Fidelity
exact
Exactness
numerical-integral

Definition

The simplest doubly periodic minimal surfaces with only catenoidal ends in the quotient: ends arranged in a square lattice with limiting normals perpendicular to the periodicity plane, growth rates adjustable by the Lopez-Ros factor, and no period problem to solve. The repository's singly periodic 'translation invariant plane with catenoidal ends' (mirror ch172) is the 1-periodic sibling, also unrecorded. THE SHIPPED ROW: g = rho/sqrt(z), dh = dz/(sqrt(z-1) sqrt(z+1)) on the upper half plane; rho = 2^(k-1) is the growth knob (the page's own family parameter, so no single member is pinned). The square-cell identity -- dis = f(1)_y = -f(-1)_x with f((1,inf)) in the mirror y = +dis, f((-inf,-1)) in x = -dis, and f((0,1)), f((-1,0)) the straight half-turn axes -- is measured at rho = 1 AND 2 (worst residual ~1e-6), so the page's no-period-problem claim is checked across the family rather than at one point.

Sources

Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.