These objects, or similar ones (I tweak them sometimes), are available at Shapeways and/or Sculpteo.

I don't use a modeler. For each design I write a program (in Python) which writes a description of the surface of the object, as a long list of triangles. The longest such program (at the moment) is 270 lines, smaller by far than any general-purpose modeling program!

My specialty seems to be different representations of the Klein bottle. The grids of the Lawson and "bagel" designs consist of a single endless strand; that it crosses itself, rather than forming only one series of parallel stripes, shows that the surface is non-orientable.

Classic Klein bottle.
Part of my motive for making it
was to see whether I could make a graceful shape
from a *single* set of parametric equations,
rather than patching together
pieces of several different tori.

Lawson's surface, by stereographic projection from the hypersphere.

w = cos(u) cos(2v), x = cos(u) sin(2v), y = sin(u) cos(v), z = sin(u) sin(v)

A "Klein bagel": a figure-eight sweeps in a circle while making a half-turn around its midpoint. Alternatively you can think of it as what you get if you bend the edge of a Möbius strip over so that it meets itself in the middle.

Box of ellipses: x = cos(t+α), y = cos(t+β), z = cos(t+γ), where α is a multiple of 2π/3, β is a multiple of 2π/5, γ is a multiple of 2π/7.

this page created 2017 Feb 07

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