By Luther Pfahler Eisenhart

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On minimal surfaces, this is true for asymptotic directions as well. ) Orthogonality of the asymptotic directions can be shown History of minimal surfaces 21 to be a requirement that is equivalent to that of zero mean curvature. Hence the orthogonality property can be used to define minimal surfaces. Gauss' paper of 1827 "Disquisiones generales circa superficies' curvas" [10] marks the birth of differential geometry. Following the advances of Gauss, it became possible to deal with surfaces by their intrinsic geometry, which includes those surface features that can be determined without reference to the external space containing the surface.

16). 15: Scherk's first and fifth minimal surfaces, d i ~ o v e r e d in 1831. The first is twoperiodic, the second one-periodic. The Weierstrass equations allow calculation of the cartesian coordinates ((x,y,z) with respect to an origin (xo,Yo,zo)) of the minimal surface at all points on the surface - except flat points - in terms of a complex analytic function R(60). 18) Integration is carried out on an arbitrary path from 60o to 601 in the complex plane, for a fixed value of 0 between 0 and n/2.

The normal vectors to the P-surface at its eight fiat points (one obscured) are indicated by the arrowed vectors. These vectors point towards the eight vertices of the cube. 21: The gyroid surface discovered by Schoen in the 1960's. ) 30 Chapter 1 The general features of the Bonnet t r a n s f o r m a t i o n can be seen in the simplest e x a m p l e , n a m e l y the i s o m e t r y b e t w e e n the catenoid a n d the helicoid (Fig. 19). U n d e r the action of the transformation, each point on the surface traces an ellipse in space, c e n t r e d at the origin.