# Saddle Tower

H. Karcher These examples generalize Scherk's conjugate pair of singly-periodic/doubly-periodic minimal surfaces. The singly-periodic examples stay embedded if the dihedral symmetry (and with it the number of punctures) is increased (Gauss(z)=z^k , k=ee-1). The most symmetric ones (bb=0.5/ee) can be deformed by decreasing bb. See [K1], [K2] for more details. These surfaces are parametrized by punctured spheres, but the Weierstrass integrals have periods, a vertical one in the singly periodic case, two horizontal ones for doubly periodic surfaces. The parameter lines extend polar coordinates around the punctures to the whole sphere---in these cases giving level lines on the surfaces. The degree of dihedral symmetry is, of course, a discrete property, and it is controlled by the parameter ee. Thus, ee should be set to an integer (the default is 2). For each choice of ee, changing bb gives a one-parameter family of surfaces, of which the most symmetric member is obtained by setting bb = 0.5/ee. Try setting ee to 3 and 4, and bb to 0.333 and 0.25 respectively. The wings of the singly periodic SaddleTower surfaces become *parallel in pairs* if (for ee>2 ) one sets bb = 0.0825. These stay embedded for ee=3 and ee=4. We also recommend viewing the associate family morphing. [K1] H. Karcher, Embedded minimal surfaces derived from Scherk's examples, Manuscripta Math. 62 (1988) pp. 83--114. [K2] H. Karcher, Construction of minimal surfaces, in "Surveys in Geometry", Univ. of Tokyo, 1989, and Lecture Notes No. 12, SFB 256, Bonn, 1989, pp. 1--96. For a discussion of techniques for creating minimal surfaces with various qualitative features by appropriate choices of Weierstrass data, see either [KWH], or pages 192--217 of [DHKW]. [KWH] H. Karcher, F. Wei, and D. Hoffman, The genus one helicoid, and the minimal surfaces that led to its discovery, in "Global Analysis in Modern Mathematics, A Symposium in Honor of Richard Palais' Sixtieth Birthday", K. Uhlenbeck Editor, Publish or Perish Press, 1993 [DHKW] U. Dierkes, S. Hildebrand, A. Kuster, and O. Wohlrab, Minimal Surfaces I, Grundlehren der math. Wiss. v. 295 Springer-Verlag, 1991 We also recommend viewing the associate family morphing. [K1] H. Karcher, Embedded minimal surfaces derived from Scherk's examples, Manuscripta Math. 62 (1988) pp. 83--114. [K2] H. Karcher, Construction of minimal surfaces, in "Surveys in Geometry", Univ. of Tokyo, 1989, and Lecture Notes No. 12, SFB 256, Bonn, 1989, pp. 1--96. For a discussion of techniques for creating minimal surfaces with various qualitative features by appropriate choices of Weierstrass data, see either [KWH], or pages 192--217 of [DHKW]. [KWH] H. Karcher, F. Wei, and D. Hoffman, The genus one helicoid, and the minimal surfaces that led to its discovery, in "Global Analysis in Modern Mathematics, A Symposium in Honor of Richard Palais' Sixtieth Birthday", K. Uhlenbeck Editor, Publish or Perish Press, 1993 [DHKW] U. Dierkes, S. Hildebrand, A. Kuster, and O. Wohlrab, Minimal Surfaces I, Grundlehren der math. Wiss. v. 295 Springer-Verlag, 1991