Schwarz H
Schwarz H is a triply periodic minimal surface, given by a Weierstrass representation, genus 3 per unit cell, immersed, with a crystallographic space group.
Open Schwarz H in the interactive viewer →
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
- Genus per cell
- 3
- Ends
- 0
- Embedding
- immersed
- Symmetry kind
- crystallographic space group
- Fidelity
- exact
- Exactness
- numerical-integral
Definition
No (g, dh) pair is stored: the Gauss map is a product of powers of Jacobi theta functions on the hexagonal/square torus, continued along an unwrapped branch (math_art/minsurf/hexagonal.py); theta functions are outside the exact 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.
- 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**)]
- 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]
- 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]
- J. W. Fisher, S. W. Miller, J. Bartolai, T. W. Simpson, M. A. Yukish, "Catalog of triply periodic minimal surfaces, equation-based lattice structures, and their homogenized property data", Data in Brief 49 (2023) 109311. doi:10.1016/j.dib.2023.109311 [equation compilation; K, Q*, Triplane/F recommendations]