Scherk's Fourth Surface (1835)
Scherk's Fourth Surface (1835) is a singly periodic minimal surface, given by a Weierstrass representation, immersed with singularities, with a rod group.
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def-weierstrass implemented minimal minimal-periodic singly-periodic singular tradition-classical
Properties
- Family
- minimal-periodic
- Given by
- a Weierstrass representation
- Curvature
- zero mean curvature
- Periodicity
- singly periodic
- Ends
- 0
- Embedding
- immersed with singularities
- Symmetry kind
- rod group
- Fidelity
- exact
- Exactness
- numerical-integral
Definition
The least-cited of Scherk's 1835 surfaces, given implicitly as his equation 20: singly periodic with two annular and two helicoidal ends, and singular at the two points where the horizontal symmetry curve meets the straight line shared by the helicoidal ends. Weber recovers its Enneper-Weierstrass representation via the Schwarz-Bjorling formula on the x = pi level symmetry curve. THE SHIPPED ROW is that recovery in fully closed form (G = i(1+z)/sqrt(1-z^2), dh = -2 sqrt(1-z^2)/z, explicit f -- no integration), and the gate is Scherk's own equation 20: every built point satisfies it POINTWISE to ~1e-15 (through arccosh, both radial regions), plus the closed form's consistency with (G, dh) and the helicoidal-end winding advancing the axis by exactly the 4 pi assembly period.
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- H. Karcher, "Embedded minimal surfaces derived from Scherk's examples", Manuscripta Math. 62 (1988);
- H. F. Scherk (1835); A. Enneper (1864); M. Weber, https://minimalsurfaces.blog/ (6-Ended Scherk g0; Alternating Fence of Half-Catenoids, 2024; Fence of Catenoids; Helicoidal Karcher-Scherk; Periodic Enneper; Enneper-Scherk; Translation-Invariant Torus with 1 Enneper and 3 Annular Ends, notebook by Ramazan Yol, 2024).
- M. Weber, 'Scherk's Fourth Surface', minimalsurfaces.blog (local mirror: minsurf/book/.../ch341_scherks_fourth_surface.md).
- H. F. Scherk, 'Bemerkungen ueber die kleinste Flaeche innerhalb gegebener Grenzen', J. Reine Angew. Math. 13 (1835) 185-208 (equation 20).