October 15th, 2017, 03:32 PM  #1 
Newbie Joined: Oct 2017 From: Toronto Posts: 6 Thanks: 0  Degrees of (Freedom?)
If a 2D circle has 360 degrees (of freedom?), how many degrees (of freedom) does a sphere have? 360^2 ? or more?

October 15th, 2017, 04:37 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,842 Thanks: 1068 Math Focus: Elementary mathematics and beyond 
As far as I know, "degrees of freedom" pertains to systems, usually statistical ones.

October 15th, 2017, 04:51 PM  #3 
Newbie Joined: Oct 2017 From: Toronto Posts: 6 Thanks: 0 
Got it. Thanks. I'll just stick with "degrees" then.

October 16th, 2017, 01:17 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,556 Thanks: 600 
A sphere has a solid angle of 4π.

October 16th, 2017, 03:02 PM  #5  
Senior Member Joined: Jun 2015 From: England Posts: 844 Thanks: 252  Quote:
A degree (by itself) is a measure of angle and there are indeed 360 degrees to make up a circle. But there are other forms of angular measure for instance revolutions and it takes one revolution to make a circle. Mathematicians use radians and it takes $\displaystyle 2\pi $ of these to make a circle. In some European countries they use that grad or gradien which divides the circle into 400 grads, and therefore a quarter circle into 100 degrees. This compares with degrees which divide the circle into 90 degrees. The quarter circle is important and called a quadrant. On the other hand, the degree of freedom is roughly speaking the number of independent values (called coordinates) we require to specify something. So in 2D we require 2 numbers, say an x coordinate and a y coordinate. However, a circle is only part of the whole two dimensional plane and obeys a particular relationship between the x and the y so that if we know one we can calculate the other. This is also called a constraint. So any particular circle only has one degree of freedom. Does this help? Last edited by skipjack; October 16th, 2017 at 04:17 PM.  
October 16th, 2017, 05:46 PM  #6 
Newbie Joined: Oct 2017 From: Toronto Posts: 6 Thanks: 0 
This jogs my memory and I'm thankful for the clarification. But what I have in mind is this  2D circles filling up a sphere, how many of them would fit? Visualizing it right now...it seems the answer is 180. Is that how many degrees a sphere has? Or are degrees irrelevant to spheres? 
October 16th, 2017, 06:15 PM  #7  
Senior Member Joined: Sep 2016 From: USA Posts: 415 Thanks: 228 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
The question of how many circles makeup a sphere is a much more meaningful one. It is the same as the number of points on the line segment $[0,1]$ and also the same as the number off rays inscribed in the circle. This is equal to the cardinality of the continuum.  
October 17th, 2017, 02:31 AM  #8 
Senior Member Joined: Jun 2015 From: England Posts: 844 Thanks: 252 
Good to see someone maintaining an interest resulting in a productive discussion. Your original question took us from 2D to 3D. If we divide the sphere up into patches of unit area, each patch subtends a cone to the centre with the solid angle at the vertex being measured in steradians. https://en.wikipedia.org/wiki/Solid_angle https://en.wikipedia.org/wiki/Steradian There are $\displaystyle 4\pi $ steradians to make up the complete sphere, as already noted. 
October 21st, 2017, 11:10 AM  #9  
Newbie Joined: Oct 2017 From: Toronto Posts: 6 Thanks: 0  Quote:
 
October 22nd, 2017, 11:10 PM  #10 
Senior Member Joined: Apr 2014 From: UK Posts: 878 Thanks: 319 
If you were to rotate about the diameter of the circle and lay a copy every 1 degree of rotation, it would indeed start to copy over itself after 180 1 degree rotations. There would be gaps but it would approximate a sphere or a Christmas decoration.


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