Recently I was reading the moebius strip section of cut-the-knot, and it reminded me of something one of my students and I did a couple of months ago. I thought it would be fun to see what a Moebius strip would look like if the area wrapped twice around the edge instead of the edge wrapping twice around the area. In other words, maintain the edge of the strip as a perfect circle and create a one sided surface that threads the hole.
Here are the parametric equations for it:
w = a*pi*cos(u)
x = (sin(v*w)+(cos(v*w)-cos(w))*sin(u))*cos(u)/sin(w)
y = ((cos(v*w)-cos(w))*cos(u)^2-sin(v*w)*sin(u))/sin(w)
z = (cos(v*w)-cos(w))*sin(u)/sin(w)
u and v are the parameters, w is just an intermediate variable for convenience, and x, y, z are the coordinates. The constant a sets the size of the bulge where the surface wraps around the outside of its circular edge. A value of a=3/4 looks nice.
My student Brandon Hoff plotted this using Mathematica and put it on his website. The figures are rotatable by dragging the mouse, but the lighting Mathematica used for them leaves much to be desired. They look burnt on the underside, but you can see the shape. It is interesting to mentally follow the surface as it winds through the center while you rotate the figure.