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 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: 7,963 Thanks: 1148 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: 6,823 Thanks: 723 A sphere has a solid angle of 4π. Thanks from nietzscheswoman
October 16th, 2017, 03:02 PM   #5
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 Originally Posted by nietzscheswoman If a 2D circle has 360 degrees (of freedom?), how many degrees (of freedom) does a sphere have? 360^2 ? or more? Got it. Thanks. I'll just stick with "degrees" then.
Degrees and degrees of freedom are measures of quite different things.

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
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Quote:
 Originally Posted by nietzscheswoman 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?
Degrees are irrelevant to spheres and also every other shape. They are a completely arbitrary choice off measurement. For example, for some applications people use mils of which they are 6400 per circle. The point is, a circle can be arbitrarily divided into how ever many pieces you like and the resulting unit can be defined. It doesn't mean it is instrinsically important whatsoever.

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: 915 Thanks: 271 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
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 Originally Posted by SDK Degrees are irrelevant to spheres and also every other shape. They are a completely arbitrary choice off measurement. For example, for some applications people use mils of which they are 6400 per circle. The point is, a circle can be arbitrarily divided into how ever many pieces you like and the resulting unit can be defined. It doesn't mean it is instrinsically important whatsoever. 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.
Wow. What an incredible answer. I get it. Thank you.

 October 22nd, 2017, 11:10 PM #10 Senior Member   Joined: Apr 2014 From: UK Posts: 965 Thanks: 342 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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