We use Mathematica to draw some notable surfaces in 3-space, making frequent use of several of Mathematica's major graphical capabilities.
Our principal reference is Alfred Gray, "Modern Differential Geometry of Curves and Surfaces," CRC Press, especially the second edition (1998) which has a rich array of graphical techniques.
An Appendix to "Elementary Differential Geometry, 2d ed." includes a summary of basic graphical commands for Mathematica (and Maple).
Contents
(1) The familiar image of the Klein bottle as a piece of laboratory glassware.
(2) A flat Möbius band in R3 (Dec '00). The usual real-world paper model is made by giving a long strip of paper a half-twist and then gluing the ends together..Such models--whether real or mathematical-- are not flat, that is, do not have Gaussian curvature zero.
We construct a flat Möbius band in R3 (and clarify this somewhat ambiguous term). Linked to this page is another that discusses the shortest such band.
(3) A multitorus (Jan 01). The usual (one-holed) torus is easy to draw since it is essentially just a product of two circles. But two or more holes are harder. This page describes a way to build a "multitorus" with any number of holes.
(4) An infinite jungle gym (Feb 01).Jungle gyms are those three-dimensional grids of pipe seen in many schoolyards. We draw a fairly large one that should suggest how an infinite one would dominate 3-space.
Remarks.
(a) As a supplement to Mathematica, it is convenient to have a modest item of software that can easily crop or resize graphics-- and make minor changes. There are many such. For the Macintosh, consider GraphicConverter (www.lemkesoft.de).
(b) Lo-tech. No special software is needed to use the Mathematica commands on these pages. Just download the relevant page, copy the desired command, and paste it into Mathematica. These commands have been thoroughly tested, but typographical error is always a hazard in writing Mathematica code in terms of HTML.
Reports of errors will be gratefully received (bon@ math.ucla.edu.