Here's probably the simplest way to view Boy's surface. Only the elliptical gridlines are shown. This makes it clear how the whole surface is swept out by the...
1033
C Goodman-Strauss
cgstraus
Jul 23, 2006 6:34 pm
Hi there; glad you are enjoying the files! Here are some comments and responses to some of Chris's recent posts; this is kind of a long email, in several...
1034
C Goodman-Strauss
cgstraus
Jul 23, 2006 6:34 pm
Here's a fun file; note that antarctica is wrapped one way round the torus at n=0 and the other way round at n=1. You can also replace the torus with the...
1035
Chris Young
c1572young
Jul 23, 2006 6:50 pm
Thanks for the detailed reply. I appreciate the information very much, and will refer back to it for future files in GC. ... Maybe once I figure out how to...
1036
Chaim Goodman-Strauss
cgstraus
Jul 23, 2006 9:10 pm
Hi there, here are a couple more files along those lines. In the first, we see a sphere, with latitudes and longitudes shown, stereographically projected into...
1037
Chris Young
c1572young
Jul 23, 2006 10:41 pm
Here's a more symmetrical version, since there are 24 gridlines, divisible into 3 groups to match the 3-fold symmetry....
1038
Chris Young
c1572young
Jul 23, 2006 10:51 pm
Trying to re-write the parametric equations in terms of a spinning, expanding-and-contracting ellipse, with vertex at the origin. Here, at least, you can see...
1039
Chris Young
c1572young
Jul 23, 2006 10:54 pm
http://mathworld.wolfram.com/BoySurface.html has a very good article on all this. (Wasn't able to reach it earlier.)...
1040
Chris Young
c1572young
Jul 23, 2006 11:20 pm
Here's an illustration of the equations from the Mathworld article: http://mathworld.wolfram.com/BoySurface.html I have to use the other set of gridlines to...
1041
Chris Young
c1572young
Jul 25, 2006 5:06 am
By putting these files side-by-side and then crossing your eyes slightly and focusing on the "middle" of the three apparent images, you can get a pretty good...
1042
Chris Young
c1572young
Jul 25, 2006 5:22 am
Not a major gripe or anything, but I've been meaning to mention this for a while: It looks like dots are transparent if the previous expression used ...
1043
Chris Young
c1572young
Jul 25, 2006 11:50 am
I thought that in an earlier file the 3D lines were getting dimmed when in the background, which was a helpful depth cue. Does this only happen in certain...
1044
Chris Young
c1572young
Jul 25, 2006 12:05 pm
Here's a trick to keep the checkering stable and line up with the grid even when doing progressive "stretching of the skin" over the framework. The key is the ...
1045
Chris Young
c1572young
Jul 25, 2006 12:11 pm
Maybe somebody can suggest some useful demo in Projective Geometry involving these surfaces, which are models of the real projective plane, somehow...
1046
Chris Young
c1572young
Jul 25, 2006 1:05 pm
Last post on the subject for a while, I promise. Just have to correct my statement re the ellipses: Obviously, the vertex isn't always at the origin, as I...
1047
Chris Young
c1572young
Jul 25, 2006 1:06 pm
Here's a trick to make the yellow grid lines progressively drawn, as well. This makes it a lot simpler to follow the way a complex surface is constructed. Just...
1048
Chris Young
c1572young
Jul 25, 2006 1:16 pm
I had the checkering lagging behind the trace dots. Using "²" instead of "<" fixes this....
1049
Chris Young
c1572young
Jul 25, 2006 1:59 pm
Can't tear mysefl away. Here are a few last attempts to grasp what's going on with these surfaces. Here, I'm just drawing a third of the surface (one of three...
1050
Chris Young
c1572young
Jul 27, 2006 5:07 am
It would be great if there were some way to use the [h s v] and [r g b] color mappings when drawing 2D curves. Then the successive curves could be...
1051
Chris Young
c1572young
Jul 27, 2006 9:41 am
You can use the "d" slider to "assemble"; the three modules of the Roman surface. The "n" slider controls the progressive tracing of the surfaces. The "a"...
1052
Chris Young
c1572young
Jul 27, 2006 9:59 am
Another view. Fixed it so that the modules on the right remain the same size....
1053
Chris Young
c1572young
Jul 27, 2006 10:23 am
Use the animation slider ("d" here) to move the lock piece back and forth: To actually do some math with this, you could figure out how the Boy's surface model...
1054
Chris Young
c1572young
Jul 27, 2006 11:03 am
Cleaned up the confusing purple grid lines. Animated fitting of interlocking third module....
1055
Chris Young
c1572young
Jul 27, 2006 11:05 am
This shows how one hole is open and the other covered on each module, although it's a little hard to make out in the right-hand pane. Making the two...
1056
Chris Young
c1572young
Jul 27, 2006 11:06 am
1057
Chris Young
c1572young
Jul 31, 2006 8:36 am
From "Geometry and the Imagination" by Hilbert and Cohn-Vossen, page 304. A surface of even connectivity, i.e., "surface whose topological behavior is midway...
1058
Chris Young
c1572young
Jul 31, 2006 8:49 am
Seems like what we have here is a dimpled tetrahedron, essentially. If it were parameterized from the center, then all the face-on views would be identical....
1059
Chris Young
c1572young
Jul 31, 2006 10:10 am
I think I may not have the longitude and latitude lines right for the inscribed sphere here. http://en.wikipedia.org/wiki/Roman_surface has an excellent...
1060
C Goodman-Strauss
cgstraus
Jul 31, 2006 7:10 pm
Hello, attached are more embeddings with cusp points: a torus: and a file with a cross cap (projective plane) and a klein bottle drawn as a pair of cross caps...
1061
C Goodman-Strauss
cgstraus
Aug 1, 2006 2:17 am
Hello, in the Advanced Applied class I'm teaching (aka A First Look at Fourier Series Stuff), we are going over solving various physically motivated boundary...
