The Costa minimal surface is a complete minimal surface of finite topology and genus
one with three ends. Costa (1984) constructed the surface as an immersion with three
embedded ends, and Hoffman and Meeks (1985) proved that it is an embedded
surface . Before Costa's construction, the only known complete
minimal surfaces in embedded
surfaces were the plane , catenoid ,
and helicoid .
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
where Weierstrass zeta function , Weierstrass
elliptic function with A133747 ),
the invariants correspond to the half-periods 1/2 and
(4)
(OEIS A133748 ), where Weierstrass
elliptic function .
See also Complete Minimal Surface ,
Embedded Surface ,
Minimal
Surface ,
Weierstrass Elliptic Function ,
Weierstrass Zeta Function
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References Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Costa, C. J. "Example of a
Complete Minimal Immersion in Bol. Soc.
Bras. Mat. 15 , 47-54, 1984. https://doi.org/10.1007/BF02584707 . do
Carmo, M. P. Mathematical
Models from the Collections of Universities and Museums Ferguson, H.; Ferguson,
C.; Nemeth, T.; Schwalbe, D.; and Wagon, S. "Invisible Handshake." Math.
Intell. 21 , 30-35, 1999. 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. Hoffman, D. and Meeks,
W. H. III. "A Complete Embedded Minimal Surface in J. Diff. Geom. 21 ,
109-127, 1985. https://doi.org/10.4310/jdg/1214439467 . Nordstrand,
T. "Costa-Hoffman-Meeks Minimal Surface." http://jalape.no/math/costatxt.htm . Osserman,
R. A
Survey of Minimal Surfaces. 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 . Referenced
on Wolfram|Alpha Costa Minimal Surface
Cite this as:
Weisstein, Eric W. "Costa Minimal Surface."
From MathWorld https://mathworld.wolfram.com/CostaMinimalSurface.html
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that were also embedded
surfaces were the plane, catenoid,
and helicoid.
dihedral group of symmetries.
![1/2R{-zeta(u+iv)+piu+(pi^2)/(4e_1)+pi/(2e_1)[zeta(u+iv-1/2)-zeta(u+iv-1/2i)]}](/images/equations/CostaMinimalSurface/Inline5.svg)
![1/2R{-izeta(u+iv)+piv+(pi^2)/(4e_1)-pi/(2e_1)[izeta(u+iv-1/2)-izeta(u+iv-1/2i)]}](/images/equations/CostaMinimalSurface/Inline8.svg)
is the Weierstrass zeta function, 
is the Weierstrass
elliptic function with 
(OEIS A133747),
the invariants correspond to the half-periods 1/2 and 
, and first root
is the Weierstrass
elliptic function.
of Genus One and Three Embedded Ends." Bol. Soc.
Bras. Mat. 15, 47-54, 1984. 
Ferguson, H.; Gray, A.;
and Markvorsen, S. "Costa's Minimal Surface via Mathematica." Mathematica
in Educ. Res. 5, 5-10, 1996. 
with Genus One and Three Ends." J. Diff. Geom. 21,
109-127, 1985.