Toroidal Karcher-Scherk Tower (genus 1)
Toroidal Karcher-Scherk Tower (genus 1) is a singly periodic minimal surface, given by a Weierstrass representation, immersed, with a rod group.
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def-weierstrass immersed implemented minimal minimal-periodic singly-periodic tradition-classical
Properties
- Family
- minimal-periodic
- Given by
- a Weierstrass representation
- Curvature
- zero mean curvature
- Periodicity
- singly periodic
- Ends
- 0
- Embedding
- immersed
- Symmetry kind
- rod group
- Fidelity
- exact
- Exactness
- numerical-integral
Definition
Vertical handles added to Karcher-Scherk saddle towers with at least 6 ends, first mentioned in Karcher's Tokyo notes; the mirrored page shows 5-ended members with triangular-prism symmetry in a 1-parameter family of end angles. Distinct from the genus-1 Costa-Scherk tower record, which needs a half-period phase shift. THE SHIPPED ROW BUILDS ONE FindRoot-SOLVED (tau1, a1) MEMBER PER WING ORDER k, straight from the notebook's tables: k=3 (1.0, 0.38900635790684035), k=4 (0.4, 0.13619041259273051; vertical period T = 1.077748, matching the recorded T_z oracle), k=5 (0.5, 0.21484200805849266), k=7 (0.2, 0.12152998199565702), k=8 (0.15, 0.056545236758766944) -- each a member of the 1-parameter end-angle family, not a canonical surface; the wing ends are trimmed by a mask disk rather than the notebook's incomplete-elliptic-F chart.
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, 'Toroidal Karcher-Scherk Surfaces', minimalsurfaces.blog (local mirror: minsurf/book/.../ch157_toroidal_karcher_scherk_surfaces.md).
- H. Karcher, 'Construction of minimal surfaces', Surveys in Geometry, Univ. of Tokyo, 1989, and Lecture Notes No. 12, SFB 256, Bonn (1989) 1-96 (the 'Tokyo notes' the mirrored pages cite for their formulas).