An algebraic surface with affine equation

(1)

where
is a Chebyshev polynomial of the
first kind and
is a polynomial defined by

(2)

where the matrices have dimensions . These represent surfaces in with only ordinary
double points as singularities. The first few surfaces are given by

The th
order such surface has

(6)

singular points (Chmutov 1992), giving the sequence 0, 1, 3, 14, 28, 57, 93, 154, 216, 321, 425, 576, 732, 949, 1155, ... (OEIS A057870 )
for ,
2, .... For a number of orders , Chmutov surfaces have more ordinary double points than any
other known equations of the same degree.

Based on Chmutov's equations, Banchoff (1991) defined the simpler set of surfaces

(7)

where
is even and is again a Chebyshev
polynomial of the first kind . For example, the surfaces illustrated above have
orders 2, 4, and 6 and are given by the equations

See also Goursat's Surface ,

Ordinary
Double Point ,

Superellipse ,

Tanglecube
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References Banchoff, T. F. "Computer Graphics Tools for Rendering Algebraic Surfaces and for Geometry of Order." In Geometric
Analysis and Computer Graphics: Proceedings of a Workshop Held May 23-25, 1988
(Ed. P. Concus, R. Finn, and D. A. Hoffman). New York: Springer-Verlag,
pp. 31-37, 1991. Chmutov, S. V. "Examples of Projective
Surfaces with Many Singularities." J. Algebraic Geom. 1 , 191-196,
1992. Hirzebruch, F. "Singularities of Algebraic Surfaces and Characteristic
Numbers." In The
Lefschetz Centennial Conference, Part I: Proceedings of the Conference on Algebraic
Geometry, Algebraic Topology, and Differential Equations, Held in Mexico City, December
10-14, 1984 (Ed. S. Sundararaman). Providence, RI: Amer. Math. Soc.,
pp. 141-155, 1986. Sloane, N. J. A. Sequence A057870
in "The On-Line Encyclopedia of Integer Sequences." Trott,
M. Graphica
1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael
Trott. Champaign, IL: Wolfram Media, pp. 3 and 82, 1999. Trott,
M. The
Mathematica GuideBook for Graphics. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/ .
Cite this as:
Weisstein, Eric W. "Chmutov Surface."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/ChmutovSurface.html

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