Abstract
A minimal surface is one for which, like a soap film with the same pressure on each side, the mean curvature is zero and, thus, is one where the two principal curvatures are equal and opposite at every point. For every closed circuit in the surface, the area is a minimum. Schwarz1 and Neovius2 showed that elements of such surfaces could be put together to give surfaces periodic in three dimensions. These periodic minimal surfaces are geometrical invariants, as are the regular polyhedra, but the former are curved. Minimal surfaces are appropriate for the description of various structures where internal surfaces are prominent and seek to adopt a minimum area or a zero mean curvature subject to their topology; thus they merit more complete numerical characterization. There seem to be at least 18 such surfaces3, with various symmetries and topologies, related to the crystallographic space groups. Recently, glyceryl mono-oleate (GMO) was shown by Longley and McIntosh4 to take the shape of the F-surface. The structure postulated is shown here to be in good agreement with an analysis of the fundamental geometry of periodic minimal surfaces.
This is a preview of subscription content, access via your institution
Access options
Subscribe to this journal
Receive 51 print issues and online access
$199.00 per year
only $3.90 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
References
Schwarz, H. A. Gesammelte Mathematische Abhandlungen Vols 1, 2 (Springer, Berlin, 1890).
Hilbert, D. & Cohn-Vossen, S. Geometry and the Imagination, 271 (Chelsea, New York, 1952).
Schoen, A. H. NASA Tech. Note D-5541 (1970); Not. Am. Math. Soc. 16, 519 (1969); 15, 929 (1968); 15, 727 (1968); 14, 661 (1967).
Longley, W. & McIntosh, T. J. Nature 303, 612–614 (1983).
Luzzati, V. & Spegt, P. A. Nature 215, 701–704 (1967).
Fontell, K. Molec. Crystal Liq. Crystal 63, 59–82 (1981).
Mackay, A. L. Silicate Membranes as Minimal Surfaces (Poster/Abstract 73-P1-5a Abstr. IUC Reg. Conf. Copenhagen, 1979).
Andersson, S. Angew. Chemie 22, 69–81 (1983).
Andersson, S., Hyde, S. T. & Schnering, H. G. von, Zeit. f. Krist. 168, 1–17 (1984).
Donnay, G. & Pawson, D. L. Science 166, 1147–1150 (1969).
Nissen, H-L. Science 166, 1150–1152 (1969).
Weatherburn, C. E. Differential Geometry Vols 1, 2 (Cambridge University Press, 1939).
Coxeter, H. S. M. Introduction to Geometry, 285 (Fig. 15.8c) (Wiley, New York, 1961).
White, S. H. Biophys. J. 23, 337–347 (1978).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mackay, A. Periodic minimal surfaces. Nature 314, 604–606 (1985). https://doi.org/10.1038/314604a0
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1038/314604a0
This article is cited by
-
A Novel Metastable Structure in Polycrystalline Metals With Extremely Fine Grains: Schwarz Crystal
Metallurgical and Materials Transactions A (2024)
-
Multiscale modeling of functionally graded shell lattice metamaterials for additive manufacturing
Engineering with Computers (2023)
-
Bicontinuous cubic phases in biological and artificial self-assembled systems
Science China Materials (2020)
-
Topological nodal line semimetals predicted from first-principles calculations
Frontiers of Physics (2017)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.
