Riemann's Minimal Example
Riemann's Minimal Example is a singly periodic minimal surface, given by a Weierstrass representation, genus 0, embedded, with a rod group.
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def-weierstrass embedded implemented minimal minimal-periodic singly-periodic tradition-classical
Properties
- Family
- minimal-periodic
- Given by
- a Weierstrass representation
- Curvature
- zero mean curvature
- Periodicity
- singly periodic
- Genus
- 0
- Ends
- 0
- Embedding
- embedded
- Orientable
- yes
- Symmetry kind
- rod group
- Fidelity
- exact
- Exactness
- numerical-integral
Definition
No (g, dh) pair is stored: the shipped immersion is a closed form in Weierstrass elliptic functions (zeta, wp) on a rectangular torus (math_art/minsurf/zoo.py, _riemann_X) -- outside the elementary expression language. The shipped implementation is authoritative; an unverified transcription would silently define a different surface.
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- B. Riemann, 'Uber die Flache vom kleinsten Inhalt bei gegebener Begrenzung', Abh. Konigl. Ges. Wiss. Gottingen 13 (1867), posthumous.
- B. Riemann, 'Ueber die Flaeche vom kleinsten Inhalt bei gegebener Begrenzung', Abh. Koenigl. Ges. Wiss. Goettingen 13 (1867) 3-52.
- W. H. Meeks III, J. Perez, A. Ros, 'Properly embedded minimal planar domains', Ann. of Math. 181 (2015) 473-546 -- any properly embedded minimal surface of genus 0 is the plane, catenoid, helicoid, or a Riemann example.