I’m reading an old Martin Gardner (former Scientific American puzzle writer) puzzle book.
He introduces the topic of tarii by saying that an infinity of non-intersecting “meridians” go around the tarus, passing through the center, while an infinity of “parallels” go around the outside, in the same plane as the disk of the hole.
Then he says "There are two other less obvious infinite sets of “oblique” circles. Can you find them.
I couldn’t, but that particular question was never answered in the back. What are these obliques?
OK, you’re familiar with the flat representation of a torus? As a rectangle, with the opposite edges identified (much like the world of many video games)? The latitudinal and longitudinal circles there correspond to horizontal and vertical lines. Now draw one of the diagonals. I believe that it, and all of the lines parallel to it, would be circles if you re-wrapped the torus. The other set, of course, corresponds to the other diagonal.
I guess my video games don’t have toruses. (But then I stopped playing them when pacman replaced pong.)
But I think you’re right about them being diagonals.
At first I couldn’t see how diagonals could be called “circles”, so I tried to find an image of a torus grid on the web to draw diagonals on.
Google didn’t help with the main question, but their image finder came up with these “crop circles”.
I’m starting to see it now in terms of a bias-belted tire.
I guess that if the circles have a diameter equal to the tire diameter plus the hubcap diameter, then they would fit tightly to the tire. Seems at first that the fit would not have a round cross-section, but I suppose it does.
What those other circles are has been answered. In case you’re interested, though, there’s a lot more structure around that isn’t too difficult to see. In fact, if you’ve got a solid styrofoam torus (available at hobby and craft stores) and a bunch of string you can even play around on your own.
The circles you’re talking about turn out to be honest circles for the standard rendition of a torus, but in general what topologists care about are closed loops on the surface and they consider two to be the same (we say “homotopic”) if one can be slid onto the other without leaving the surface. As Gardner stated, two such loops are the meridians and the longitudes. Show that any two meridians that go the same way around the torus are homotopic, and likewise that any two longitudes that go the same way around the hole are homotopic.
Next, if you start with a fixed point and consider all loops from that point back to itself, we can compose two loops by going along one, then along the other. Going backwards along a loop is the “inverse” in the sense that going forward and then backwards along the same path can be deformed back to the “constant path” that just stays at the point. Show that if the meridian is M and the longitude is L, the two other circles discussed in the post are ML (first along M and then along L) and ML[sup]-1[/sup].
If you’re feeling really ambitious, classify all the different homotopy classes of loops on the torus.
I found out that a cheaper method is a box of Enteman’s donuts. Easy to slice and see the cross-sections are circular. Bagels might work as well, if you have more people to eat them all, like a math class of kids.
Well, first of all you don’t want to be slicing. All this goes on on the surface. Second of all, if you did slice a baked good along a disk in the solid torus bounded by even a moderately complicated loop it’d just leave you with a pile of crumbs.
Oh, and as for “cheaper”: Buying one large torus, string, and maybe some pushpins to help hold the string down is cheaper than buying and dicing up box after box from Entenmann.
That is so strange, but thank you for sharing.
The donuts (and you can get a math major to confirm this, if you like) were free, since I bought and ate them like I always do.
And the idea that “you don’t have to slice” is directly contradicted by the fact that the picture in questionwas the one that made the circle visible to me for the first time. As mentioned above.