https://www.youtube.com/watch?v=aJZHzEM_khE 2:37 --rwg On 2014-10-31 14:28, Dan Asimov wrote:
Sorry, was trying to do too many things at once. Got so wrapped up in finding a good picture of the (2,3,5) spherical tiling that I forgot about the stereographic projection.
Such a picture is about 1/15 of the way down this page, for instance:
< http://westy31.home.xs4all.nl/Geometry/Geometry.html >
-6-^Dan
On Oct 31, 2014, at 11:48 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Stereographic projection (from a north-pole-less n-sphere to n-space) is really quite interesting.
Besides the surprising fact that angles are preserved is the even more surprising fact that (n-1)-spheres are carried to (n-1)-spheres.
Stereographic projection helps immensely with visualizing things on the 3-sphere, since (except for its N pole) it gets projected to 3-space. So, all the great circles of the Hopf fibration of S^3 gets projected to circles in R^3 -- though in R^3 they have variously every radius >= 1.
(Except for the Hopf circle through the N pole (0,0,0,1), which is projected to the z-axis of R^3.)
I admit to puzzlement over this passage in the article:
----- Henry and Saul then turned to a well-known spherical tiling, called a (2,3,5) tiling, made up of triangles, but arranged to make diamonds, larger triangles and pentagons. They discovered that in the nineteenth century, the German mathematician August Möbius - yes, he of the eponymous strip - had drawn a sketch of how the tiling would look under a stereographic projection. Henry and Saul decided to make it.
“Maybe the only reason it hadn’t already been done before 3D printing is that any errors in the geometry get magnified, so you need a very precise way to make the model,” Henry says. -----
I'm really not sure what it is that "hadn't already been done before", since the (2,3,5) tiling has been depicted by computer graphics innumerable times. One such picture appears on the page
< https://en.wikipedia.org/wiki/Triangle_group >.
--Dan
On Oct 31, 2014, at 1:49 AM, Alex Bellos <alexanderbellos@gmail.com> wrote:
One for Halloween:
Stereographic projections as light sculpture...amazing 3D-printed jack-o-lanterns by Henry Segerman and Saul Schleimer
http://www.theguardian.com/science/alexs-adventures-in-numberland/2014/oct/3...