Circles of More than 360 Degrees
Date: 11/10/2001 at 10:46:30 From: Benn Grant Subject: Circles with less than and more than 360 degrees... If one has a circle cut out of paper, and one "removes" some degrees from it by cutting out a sector and joining the remaining edges, one gets a cone. What happens if one cuts along the radius and *adds* degrees by adding a sector (say from another similar paper circle? It would deform the two-dimensional circle into something, but not a cone, I think. The opposite of a cone? What would that be? This question started when I was trying to imagine a world where the degree measured the same amount of angle, but where a circle was not 360 degrees (although I know that 360 degrees is a circle by definition of a degree, that is not the question I am asking.) I guess what I am asking is that if circles that have fewer degrees transform into cones, what do circles that have more degrees transform into? Also, do these inquiries relate to physics and the ideas of curved and flat spacetime? And/or open and closed universes? Necessarily? Thanks. Benn Grant
Date: 11/11/2001 at 16:58:25 From: Doctor Mitteldorf Subject: Re: Circles with less than and more than 360 degrees... Dear Benn, Here's a point of view that may help you get a handle on this question. I'm going to ask you to think in terms of triangles rather than circles for a bit. Draw triangles on a flat surface, and no matter what size they are, the sum of the angles is 180 degrees. But if you draw triangles on the surface of a sphere (say the Earth, starting at the North Pole), then the "straight" sides of the triangles are really great circles. For small triangles close to the pole, the sum of the angles is very close to 180, but as the triangles get larger, the sum of the angles gets larger as well. What does a surface look like that has the opposite of this property? Think of a saddle. Draw a small triangle on the seat of a saddle, and the sum of the angles is close to 180 degrees. However, if you draw larger triangles, the sum of the angles can be somewhat less. Let's come back to circles now. For a flat surface, the ratio of the circumference to the radius of the circle is always 2pi. But on the surface of a sphere, you can draw larger and larger circles (expanding from a pole) and if you think of the radius as the great-circle distance measured along a meridian, then the ratio circumference/ radius gets smaller and smaller as the circles expand. Similarly, drawing larger and larger circles on a saddle, you'll find that the ratio of circumference to ratio starts at 2pi and gets larger and larger. If you were an ant exploring your world on a sphere, you might feel that the world was closing on itself, and there was more of it than you expected the farther you get from your starting place. An ant exploring a saddle might feel that his world was bewilderingly large - that there was more space to be explored than he thought there should be within a given distance from his starting point. Now, what does this have to do with your question? Well, a cone has some of the attributes of a positively-curved surface like a sphere, even though it's flat. (Obviously, if you draw a triangle on a piece of paper and bend the paper into a cone, the sum of the angles is still 180. But if the triangle encloses the vertex of the cone, that's a different story. It's an easy and interesting experiment to try!) Think of a cone as bending "in the same direction" as a sphere, but all the curvature is concentrated at a single point. What you want to do with "extra degrees" corresponds to deforming your paper into a saddle. You can't do this with a flat piece of paper, because there's no way to concentrate a negative curvature all at one point, as with a cone. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.