math picture gallery

A compact Bonnet pair. 

Bonnet asked the question wether a surface in 3-space is uniquely determined by its metric and mean curvature. In general the answer is no. There are families of isometric  surfaces of constant mean curvature as well as the so-called Bonnet surfaces. Also, there are Bonnet pairs. They can be generated by means of a spin transformation out of isothermic surfaces. But the question, wether there exist compact examples has been open for many years.

See:Alexander I. Bobenko, Tim Hoffmann, and Andrew O. Sageman-Furnas. Compact bonnet pairs: isothermic tori with the same curvatures. 2021. [ arXiv ]

This is an isothermic torus with one family of planar and one family of spherical curvature lines. This torus gives rise to a pair of Bonnet tori (not the above ones).

See:Alexander I. Bobenko, Tim Hoffmann, and Andrew O. Sageman-Furnas. Compact bonnet pairs: isothermic tori with the same curvatures. 2021. [ arXiv ]

A discrete constant mean curvature surface with four Delaunay ends.

See: T. Hoffmann. Discrete cmc surfaces and discrete holomorphic maps. In A. Bobenko and R. Seiler, editors, Discrete integrable geometry and physics, pages 97-112. Oxford University Press, 1999.

A smoke ring can be approximated by the Hashimoto flow of a curve. IN this case the complex curvature evolves with the nonlinear Schrödinger equation. Here the curve is a 8-gon (inscribed in an ellipse) and the evolution is an integrable doubly discrete Hashimoto flow.

See: Tim Hoffmann. Discrete Hashimoto surfaces and a doubly discrete smoke-ring flow. In Alexander I. Bobenko, John M. Sullivan, Peter Schröder, and Günter M. Ziegler, editors, Discrete Differential Geometry, volume 38 of Oberwolfach Seminars, pages 95-115. Birhkäuser Basel, 2008.

Three discrete Pseudospheres (particular surfaces of revolution with constant negative Gauss curvature)

See: Tim Hoffmann, Andrew O. Sageman-Furnas, and Max Wardetzky. A discrete parametrized surface theory in R3. International Mathematics Research Notices, 2017(14):4217-4258, 2017.

A discrete Smyth surface: A cmc surface with rotational symmetric metric. This one has a 6-fold symmetry.

See: T. Hoffmann. Discrete cmc surfaces and discrete holomorphic maps. In A. Bobenko and R. Seiler, editors, Discrete integrable geometry and physics, pages 97-112. Oxford University Press, 1999.