4-Noid, Two Symmetry Planes
4-Noid, Two Symmetry Planes is a minimal surface, given by a Weierstrass representation, immersed.
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aperiodic def-weierstrass immersed implemented minimal tradition-classical
Properties
- Family
- minimal
- Given by
- a Weierstrass representation
- Curvature
- zero mean curvature
- Periodicity
- not periodic
- Ends
- 0
- Embedding
- immersed
- Fidelity
- exact
- Exactness
- numerical-integral
Definition
Karcher's 2-parameter family of 4-noids (4-punctured spheres, four catenoidal ends) with two orthogonal symmetry planes; the parameters control the position of the ends and their relative growth rates, and the Jorge-Meeks 4-noid is the intersection of the two 1-parameter sections that 3DXM exhibits separately as 'Symmetric 4-noid' (opposite end pairs of different size) and 'Skew 4-noid' (varying angle between end pairs, morphing toward two catenoids joined by a handle -- the family that convinced Hoffman that designed handles were promising).
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).
- The 3DXM Consortium, 'Symmetric 4-noid', Virtual Math Museum, virtualmathmuseum.org (local mirror: vmm/book/surface/ch013_symmetric_4noid.md).
- The 3DXM Consortium, 'Skew 4-noid', Virtual Math Museum, virtualmathmuseum.org (local mirror: vmm/book/surface/ch014_skew_4noid.md).
- M. Weber, '4-Noids with Two Symmetry Planes', minimalsurfaces.blog (local mirror: minsurf/book/.../ch173_4_noids_with_two_symmetry_planes.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). (pages 30ff).