The Costa surface is a complete minimalembedded 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 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 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
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 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.; Gray, A.;
and Markvorsen, S. "Costa's Minimal Surface via Mathematica." Mathematica
in Educ. Res.5, 5-10, 1996. http://library.wolfram.com/infocenter/Articles/2736/.Ferguson,
H.; Ferguson, C.; Nemeth, T.; Schwalbe, D.; and Wagon, S. "Invisible Handshake."
Math. Intell.21, 30-35, 1999.GRAPE. "Costa's Surface
(Celsoe Costa)." http://www-sfb256.iam.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 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.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. News127, 161-176, 1985.Peterson,
I. "The Song in the Stone: Developing the Art of Telecarving a Minimal Surface."
Sci. News149, 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.