Very good work, very nice pictures.
Just a thought....R and r can be illustrated more clearly,initially I mistook them for the external and difference between external and internal radii (we are always more comfortable with 2 dimensions ) .
New forumlae to remember ......
Surface Area = 4 × π² × R × r
Volume = 2 × π² × R × r²
The best part was the metamorphosis of a torus into a sphere
Last edited by ganesh (2005-09-12 21:19:59)
Character is who you are when no one is looking.
pov-ray is good but I prefer max or maya
But the picture is still a great render! : )
Last edited by kylekatarn (2005-09-13 02:28:04)
Great pics, but I think the torus-into-sphere is a bit too slick. That's not good mathematics, there's a topological discontinuity when you lose that central hole. Don't give people the idea that a sphere is a kind of torus, because it ain't :-)
Good point phayes, I will look at changing that.
I have removed the "no hole" images from the torus page, but it looks kinda sad now
But I now have some really interesting questions:
Can a torus become a sphere?
Is it still a torus when the hole disappears?
Apparently a torus can have "n" holes, including zero. There is a detailed article at mathworld ( http://mathworld.wolfram.com/Torus.html ), but it does not say about the extreme case becoming a sphere
Another reference, the National Science Digital Library ( https://ask.nsdl.org/default.aspx?id=11434&cat=1166 ) has a reply to a question that says that a torus can degenerate into a sphere!
Perhaps I could explain on the torus page that the torus-with-a-hole-in-the-middle is the "Standard Torus", and that there were other special kinds of tori?
After reading around these various references, I think that there are two slightly different views of the essence of torushood.
If you think of it geometrically, as being the envelope you get when you wipe a circle around a circular orbit, then a sphere does seem to be a degenerate torus, got by shrinking the orbit to a point, as you originally suggested. But if you think of a torus as a topologist does, as a kind of surface, the essence of torushood is the hole (or, if you like, the 'handle'). This is what makes the torus have a different homotopy group, why you can cut along a circle which doesn't separate the surface, etc...
Seen from the topology perspective, that degenerate case isn't a sphere at all, it is a very odd entity: two spheres, A and B, in the 'same place' but with distinct surfaces, which pass through one another at the poles: so if you go to the north or south pole on A and keep going, you move into B, and vice versa. So if you draw a great circle through the poles and cut the surface in half along it, it won't fall apart into two hemispheres, but will contract back 'through' itself into a cylinder: just like, well, a torus. (To see why, consider that to get a torus you need to sweep the circle through 360 around the 'orbit', but you only need to sweep it 180 to get a complete spherical surface when you rotate about its center. So the full toroidal 360 sweep gets you two spheres. Another way is to visualize the spindle torus, see http://mathworld.wolfram.com/Torus.html, and imagine the extreme case where the 'inner' surfaces come to coincide with the 'outer' ones in the limit. Now imagine what happens to a circle on the original 'normal' torus which went 'through' the hole. In the spindle torus it goes from the outside, 'into' the torus, then back out. In the limit, it starts 'outside' on sphere A, say, then goes 'inside' at the pole onto sphere B, then back 'out' at the opposite pole. The 'same' circle starting at the other side of the sphere only intersects this circle at the actual poles: everywhere else it is on different surfaces.)
But, Im sure that this is all rather too, er, hairy for this website, and it might be better to just stick to the geometrical view of things. In which case, my objection to the original graphic is misplaced, and I hereby withdraw it.
Hey, this has been quite interesting. Thanks for prodding me to check out other points of view here ! :-)