Math Art
Catenoid Field (square-torus member)

Catenoid Field (square-torus member)

Catenoid Field (square-torus member) 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

Doubly periodic field of half-catenoids growing alternately up and down: parametrized by a twice-punctured rectangular torus with Gauss map g(z) = c*Je(z), Je a Jacobi-type elliptic function with two simple poles and zeros. Weber's repository page 'Doubly Periodic Catenoids' (mirror ch025) describes a closely related doubly periodic catenoid arrangement; whether it is the same family has NOT been established here, so that page is deliberately not cross-referenced. THE SHIPPED ROW IS THE tau = i (square-torus) MEMBER with growth factor bb as the knob; J_F is realised as the balanced theta-11 quotient with zeros {0, 1/2} and poles {tau/2, 1/2 - tau/2}, normalised per member by the unit-circle branch-value symmetry (|b1 b2| = 1). Measured, and gated in the zoo self-test: the loop around each half-catenoid puncture translates by (0, 0, 0) -- the record's 'no period problem' made quantitative -- and both deck translations are purely horizontal ((-0.4926, 0, 0) and (0, -0.6642, 0)), so the field lies between two parallel planes.

Sources

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