From: israel@math.ubc.ca (Robert Israel)
Newsgroups: sci.math.research
Subject: Re: Image of Boy's Surface
Date: 4 Jul 1997 00:44:13 GMT
In article <33BC21EC.249D@worldnet.att.net>, Marjorie Piette & Steve Ellis writes:
|> Can someone direct me to a nice image of Boy's surface? (As I
|> understand it, Boy's surface is an immersion of the projective plane in
|> 3-space.) An electronic version I could paste into a document would be
|> nice. A Mathematica program for constructing such an image would be
|> fine. Thanks.
Joe Fields has a picture of it at
http://zariski.math.uic.edu/~fields/gifs/BoysSurface.gif
with a description of how to make a paper model at
http://zariski.math.uic.edu/~fields/topology/Boys_surface.html
Pictures of a skeleton of the surface are at
http://http.cs.berkeley.edu/~sequin/SCULPTS/BoysSky.gif
and http://http.cs.berkeley.edu/~sequin/SCULPTS/BoysCar.gif
(from a sculpture by Carlo H. Séquin, see http://http.cs.berkeley.edu/~sequin/SCULPTS/sequin.html)
A parametric representation suitable for Maple or Mathematica is:
x = cos(s)(sqrt(2) cos(s) cos(2 t) + 2 sin(s) cos(t))/D(s,t)
y = cos(s)(sqrt(2) cos(s) sin(2 t) - 2 sin(s) sin(t))/D(s,t)
z = 3 cos(s)^2/D(s,t)
where D(s,t) = 1 - sqrt(2) sin(s) cos(s) sin(3 t)
0 <= t <= Pi, -Pi/2 <= s <= Pi/2
See Apéry, Adv. Math. 61 (1986) 185-266.
Robert Israel israel@math.ubc.ca
Department of Mathematics (604) 822-3629
University of British Columbia fax 822-6074
Vancouver, BC, Canada V6T 1Y4