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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/   
    
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Higher-Dimensional Geometry

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