A338004 Decimal expansion of the angle of association yielding the gyroid relative to Schwarz's D surface.

6, 6, 3, 4, 8, 2, 9, 7, 0, 5, 1, 1, 4, 3, 4, 8, 0, 8, 0, 5, 7, 5, 6, 8, 8, 4, 7, 4, 3, 7, 2, 3, 9, 9, 5, 0, 0, 0, 5, 0, 4, 2, 8, 9, 8, 5, 1, 5, 6, 9, 6, 2, 5, 5, 4, 5, 7, 1, 8, 2, 4, 4, 9, 9, 5, 0, 5, 9, 3, 3, 1, 5, 0, 9, 3, 7, 7, 6, 8, 3, 8, 5, 0, 6, 8, 1, 0, 9, 7, 9, 1, 5, 6, 8, 7, 8, 5, 8, 9, 8, 7, 3, 3, 3, 0, 1, 0, 9, 0, 8, 3, 3, 8, 9, 1, 3, 9, 4, 5, 4

Offset: 0

Author

Jeremy Tan, Oct 06 2020

Keywords

• nonn
• cons

Comments

For every minimal surface, an associate family of minimal surfaces can be defined by adding an angle of association to the base surface's Weierstrass-Enneper parametrization.

If the base is Schwarz's D surface, an angle of association of Pi/2 yields Schwarz's P surface; this entry is the only other angle for which the resulting associate surface - the gyroid - is embedded.

Examples

			0.66348297051143480805756884743723...
In degrees: 38.0147739891080681076130861019883...
		

Links

• A. Schoen, Infinite Periodic Minimal Surfaces Without Self-Intersections, NASA Technical Note D-5541, 1970.
• A. Schoen, Triply-periodic minimal surfaces
• Wikipedia, Associate family

Crossrefs

Cf. A249282 (K(1/4)), A249283 (K(3/4)).

Programs

Mathematica

First@ RealDigits@ N[ArcTan[EllipticK[1/4] / EllipticK[3/4]], 120]

PARI

atan(ellK(1/2)/ellK(sqrt(3/4))) \\ Charles R Greathouse IV, Feb 05 2025

Formula

Equals arctan(K(1/4) / K(3/4)), where K is the complete elliptic integral of the first kind.