Triply Periodic Horgan Surface (exact fundamental piece, genus 5)
Triply Periodic Horgan Surface (exact fundamental piece, genus 5) is a triply periodic minimal surface, given by a Weierstrass representation, immersed, with a crystallographic space group.
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def-weierstrass immersed implemented minimal minimal-periodic tradition-crystallographic triply-periodic
Properties
- Family
- minimal-periodic
- Given by
- a Weierstrass representation
- Curvature
- zero mean curvature
- Periodicity
- triply periodic
- Ends
- 0
- Embedding
- immersed
- Symmetry kind
- crystallographic space group
- Fidelity
- exact
- Exactness
- numerical-integral
Definition
Genus-5 (per cell) triply periodic 1-parameter family with vertical symmetry planes over a square grid and diagonal horizontal lines, limiting in noded planes and in doubly periodic Karcher-Scherk surfaces. Its nodal-limit neck configuration is exactly that of the (non-existent) finite Horgan surface -- yet this periodic surface EXISTS, with a 1-dimensional period problem solved by an extremal-length argument Weber presents as a picture proof.
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- M. Wohlgemuth, N. Yufa, J. Hoffman, E. L. Thomas, "Triply periodic bicontinuous cubic microdomain morphologies by symmetries", Macromolecules 34 (2001) 6083-6089. doi:10.1021/ma0019499 [O,C-TO (OCTO), G', F-RD variant]
- E. Koch and W. Fischer, "Triply periodic minimal balance surfaces: a correction", Acta Crystallogr. A 49 (1993) 209-210. doi:10.1107/S0108767392007591 [C(S) = P and Y = D as minimal surfaces; the nodal geometries remain distinct]
- H. G. von Schnering and R. Nesper, "Nodal surfaces of Fourier series: fundamental invariants of structured matter",
- Z. Phys. B Condensed Matter 83 (1991) 407-412. doi:10.1007/BF01313411 [C(D), C(Y), +-Y, C(+-Y), S nodal forms]
- H. G. von Schnering, M. Oehme, G. Rudolf, "Three-dimensional periodic nodal surfaces which envelope the threefold and fourfold cubic rod packings", Acta Chem. Scand. 45 (1991) 873-876. doi:10.3891/acta.chem.scand.45-0873 [C(I2-Y**)]
- D. A. Hoffman, J. T. Hoffman, M. Weber, et al., "Table of Surfaces", The Scientific Graphics Project, MSRI (2003). [D' and further G'/OCTO forms]
- M. Weber, 'Catenoid-Scherk Limits -- aka Triply Periodic Horgan Surface', minimalsurfaces.blog (local mirror: minsurf/book/.../ch263_catenoid_scherk.md).