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Schwarz surface

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A polyhedral surface inscribed in a finite circular cylinder, such that the areas of a sequence of such surfaces under a suitable choice of the parameters can tend to any limit (including infinity).

Figure: s083570a

The construction of the Schwarz surface (see Fig.) is such, that when the maximal diameters of its faces tend to zero, they are not at all close, with respect to their location in space, to the tangent plane to the surface of the cylinder. Thus, a face of the Schwarz surface cannot approximate the element of the surface of the cylinder with increasing precision.

This surface was introduced by H.A. Schwarz in 1880.

References

[1] G.M. Fichtenholz, "Differential und Integralrechnung" , 3 , Deutsch. Verlag Wissenschaft. (1964)


Comments

References

[a1] M. Berger, "Geometry" , I , Springer (1987) pp. 263
[a2] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) pp. 208 (Translated from French)
How to Cite This Entry:
Schwarz surface. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Schwarz_surface&oldid=12887
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098