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Gyroid

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Gyroid

The gyroid, illustrated above, is an infinitely connected periodic minimal surface containing no straight lines (Osserman 1986) that was discovered by Schoen (1970). Große-Brauckmann and Wohlgemuth (1996) proved that the gyroid is embedded.

The gyroid is the only known embedded triply periodic minimal surface with triple junctions. In addition, unlike the five triply periodic minimal surfaces studied by Anderson et al. (1990), the gyroid does not have any reflectional symmetries (Große-Brauckmann 1997).

Gyroid metal sculpture (Bathsheba Grossman)

The image above shows a metal print of the gyroid created by digital sculptor Bathsheba Grossman (https://www.bathsheba.com/).

An exact Enneper-Weierstrass parameterization uses

g(z)=z
(1)
f(z)=(e^(itheta))/(sqrt(z^8-14z^4+1)),
(2)

where theta=arctan[K(1/2)/K(sqrt(3)/2)] and an analytic branch of the square root is chosen on each parameter patch. Here K(k) is the complete elliptic integral of the first kind with elliptic modulus k, giving theta approx 38.0147 degrees (Gandy and Klinowski 2000).

See also

Minimal Surface, Schwarz D Surface, Schwarz P Surface, Triply Periodic Minimal Surface

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References

Anderson, D. M.; Davis, H. T.; Nitsche, J. C. C.; and Scriven, L. E. Adv. Chem. Phys. 77, 337-396, 1990.Gandy, P. J. F. and Klinowski, J. "Exact Computation of the Triply Periodic G ('Gyroid') Minimal Surface." Chem. Phys. Lett. 321, 363-371, 2000. https://doi.org/10.1016/S0009-2614(00)00373-0.Garstecki, P. and Hołyst, R. "Scattering Patterns of Self-Assembled Gyroid Cubic Phases in Amphiphilic Systems." J. Chem. Phys. 115, 1095-1099, 2001.Gòzdz, W. and Hołyst, R. "High Genus Periodic Gyroid Surfaces of Nonpositive Gaussian Curvature." Phys. Rev. Lett. 76, 2726-2729, 1996.Große-Brauckmann, K. "Gyroids of Constant Mean Curvature." Experiment. Math. 6, 33-50, 1997.Große-Brauckmann, K. and Wohlgemuth, M. "The Gyroid Is Embedded and Has Constant Mean Curvature Companions." Calc. Var. Partial Differential Equations 4, 499-523, 1996.Grossman, B. "The Gyroid." https://www.bathsheba.com/math/gyroid/.Hyde, S. T.; Andersson, S.; Blum, Z.; Lidin, S.; Larsson, K.; Landh, T.; and Ninham, B. W. The Language of Shape. Amsterdam, Netherlands: Elsevier, 1997.Hyde, S. T.; Andersson, S.; Ericsson, B.; and Larsson, K. "A Cubic Structure Consisting of a Lipid Bilayer Forming an Infinite Periodic Minimal Surface of the Gyroid Type in the Glycerolmonooleate Water System." Z. Kristallogr. 168, 213-219, 1984.Mackay, A. L. "Periodic Minimal Surfaces." Nature 314, 604-606, 1985.Numberphile. "Butterflies and Gyroids." Dec. 17, 2020. https://www.youtube.com/watch?v=VFRrzzb1dvU.Osserman, R. Frontispiece to A Survey of Minimal Surfaces. New York: Dover, 1986.Schoen, A. H. "Infinite Periodic Minimal Surfaces without Selfintersections." NASA Tech. Note No. D-5541. Washington, DC, 1970.Terrones, H. "Computation of Minimal Surfaces." J. de Physique 51, No. 23, Suppl. Colloq. C7, 345-362, 1990.Weber, M. "Gyroid." https://minimalsurfaces.blog/home/repository/triply-periodic/gyroid/.

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Gyroid

Cite this as:

Weisstein, Eric W. "Gyroid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Gyroid.html

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