Costa Minimal Surface
The Costa surface is a complete minimal embedded surface of finite topology (i.e., it has no boundary and does not intersect itself). It has genus 1 with three punctures (Schwalbe and Wagon 1999). Until this surface was discovered by Costa (1984), the only other known complete minimal embeddable surfaces in R^3 with no self-intersections were the plane (genus 0), catenoid (genus 0 with two punctures), and helicoid (genus 0 with two punctures), and it was conjectured that these were the only such surfaces.
Rather amazingly, the Costa surface belongs to the D_4 dihedral group of symmetries.
The Costa minimal surface appears on the cover of Osserman (1986; left figure) as well as on the cover of volume 2, number 2 of La Gaceta de la Real Sociedad Matemática Española (1999; right figure).
It has also been constructed as a snow sculpture (Ferguson et al. 1999, Wagon 1999).
On Feb. 20, 2008, a large stone sculpture by Helaman Ferguson was installed on the south deck of the Olin-Rice Science Center at Macalester College (photo courtesy of Stan Wagon).
As discovered by Gray (Ferguson et al. 1996, Gray 1997), the Costa surface can be represented parametrically explicitly by
![画像:1/2R{-zeta(u+iv)+piu+(pi^2)/(4e_1)+pi/(2e_1)[zeta(u+iv-1/2)-zeta(u+iv-1/2i)]}](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/CostaMinimalSurface/Inline5.svg){kind=link}
![画像:1/2R{-izeta(u+iv)+piv+(pi^2)/(4e_1)-pi/(2e_1)[izeta(u+iv-1/2)-izeta(u+iv-1/2i)]}](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/CostaMinimalSurface/Inline8.svg){kind=link}
where zeta(z) is the Weierstrass zeta function, P(g_2,g_3;z) is the Weierstrass elliptic function with (g_2,g_3)=(189.072772...,0) (OEIS A133747), the invariants correspond to the half-periods 1/2 and i/2, and first root
| e_1=P(1/2;0,g_3)=P(1/2|1/2,1/2i) approx 6.87519 |
(4)
|
(OEIS A133748), where P(z;g_2,g_3)=P(z|omega_1,omega_2) is the Weierstrass elliptic function.
See also
Complete Minimal Surface, Minimal Surface, Weierstrass Elliptic Function, Weierstrass Zeta FunctionExplore with Wolfram|Alpha
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References
Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 86-87, 2003.Costa, A. "Examples of a Complete Minimal Immersion in R^3 of Genus One and Three Embedded Ends." Bil. Soc. Bras. Mat. 15, 47-54, 1984.do Carmo, M. P. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 43, 1986.Ferguson, H.; Ferguson, C.; Nemeth, T.; Schwalbe, D.; and Wagon, S. "Invisible Handshake." Math. Intell. 21, 30-35, 1999. Ferguson, H.; Gray, A.; and Markvorsen, S. "Costa's Minimal Surface via Mathematica." Mathematica in Educ. Res. 5, 5-10, 1996. https://library.wolfram.com/infocenter/Articles/2736/.GRAPE. "Costa's Surface (Celsoe Costa)." https://archive.ins.uni-bonn.de/numod.ins.uni-bonn.de/grape/EXAMPLES/AMANDUS/costa.html.Gray, A. "Costa's Minimal Surface." §32.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 747-757, 1997.Hoffman, D. and Meeks, W. H. III. "A Complete Embedded Minimal Surfaces in R^3 with Genus One and Three Ends." J. Diff. Geom. 21, 109-127, 1985.Nordstrand, T. "Costa-Hoffman-Meeks Minimal Surface." http://jalape.no/math/costatxt.htm.Osserman, R. A Survey of Minimal Surfaces. New York: Dover, pp. 149-150, 1986.Peterson, I. "Three Bites in a Doughnut: Computer-Generated Pictures Contribute to the Discovery of a New Minimal Surface." Sci. News 127, 161-176, 1985.Peterson, I. "The Song in the Stone: Developing the Art of Telecarving a Minimal Surface." Sci. News 149, 110-111, Feb. 17, 1996.Ramos Batista, V. "The Doubly Periodic Costa Surfaces." Math. Z. 240, 549-577, 2002.Ramos Batista, V. "A Family of Triply Periodic Costa Surfaces." Pacific J. Math. 212, 347-370, 2003.Ramos Batista, V. "Singly Periodic Costa Surfaces." J. London Math. Soc. 72, 478-496, 2005.Schwalbe, D. and Wagon, S. "The Costa Surface, in Show and Mathematica." Mathematica in Educ. Res. 8, 56-63, 1999.Sloane, N. J. A. Sequences A133747 and A133748 in "The On-Line Encyclopedia of Integer Sequences."Wagon, S. "Snow Sculpting with Mathematics." Jan 25, 1999. http://stanwagon.com/snow/breck1999.Wagon, S. "Invisible Handshake." http://stanwagon.com/wagon/Misc/invisiblehandshake.html.Wolfram Research, Inc. "3-D Zoetrope at SIGGRAPH 2000." http://www.wolfram.com/news/zoetrope.html.Referenced on Wolfram|Alpha
Costa Minimal SurfaceCite this as:
Weisstein, Eric W. "Costa Minimal Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CostaMinimalSurface.html