How many degrees of rotation are there in a sphere?

Question

I know that a 2-Dimensional circle has 360 degrees of rotation, but how many are there the circle's 3-Dimensional counterpart - the sphere?

Answer 1

The clarification of your question indicates that you are after the concept of "solid angle". Compare this to the usual concept of "circular" or "planar" angle:

A planar angle (measured in radians "rad") between two straight lines originating at the center of a circle of unit radius is the length of the circular arc between those two lines (more precisely, from one line, chosen as "reference", to the other, as an angle is assigned a positive sign if you rotate counterclockwise and a negative sign otherwise).

Similarly, a solid angle (measured in steradians "sr") is assigned to the cone generated by half a straight line originating at the center of a sphere of unit radius with one point of that line moving in a closed loop at the surface of the unit sphere.

The measure of such a solid angle is simply the spherical surface area enclosed by the aforementioned loop at the surface of the unit sphere. Just like a planar angle, a solid angle can be "oriented" (i.e., assigned a sign) according to the direction in which the loop is traveled. The usual convention is to count a solid angle positively if the loop is traveled clounterclockwise seen from the outside of the sphere or, equivalently, clockwise seen from the origin at the center of the sphere. (You may memorize this by recalling that the "south" face of a loop is seen at a positive solid angle; see link below for the definition of "north" and "south").

For planar angles a complete turn is 2*pi radians because that's the length of the circumference of the unit circle. Planar angles are defined up to a multiple of 2*pi. Similarly, solid angles are defined up to a multiple of 4*pi, because that's the surface area of a unit sphere.

Indeed, consider that the solid angle A changes to -A when you reverse the direction of the defining loop but it also becomes (4*pi-A) because the loop whose south side you are now seeing is the curves that encloses the "other part of the sphere" compared to the previous situation. You may use this argument to convince yourself that multiples of 4*pi are as irrelevant to solid angles as multiples of 2*pi are irrelevant to circular angles.

Finally, note that astronomers often express solid angles in "square degrees" (also in "square minutes" or "square seconds"). The correspondance with steradians is easy to derive if you consider only tiny angles (so tiny that the relevant patch of the "celestial sphere" is essentially flat). A square patch of sky one angular degree on a side cuts a surface on a unit sphere of area roughly equal to (pi/180)^2 and this number is DEFINED as the exact value of a square degree in steradians (a square minute is 60^2=3600 times smaller, a square minute is 3600^2 = 12960000 times smaller than a square degree). However, the area of a patch of sky (the solid angle it "subtends") can onlly be calculated in the implied simple-minded way (angular width times angular height) when the patch is small enough for its curvature to be negligible.

That curvature is very important for the whole celestial sphere (above and below the horizon) which corresponds to a solid angle of 4*pi steradians. To obtain this solid angle in square degrees, just divide 4*pi by the above value of a square degree expressed in steradians (there no magic here: the usual way of converting units does apply).

4*pi / (pi/180)^2 = 129600 / pi = 41252.96... square degrees

Note that this is pi times less than the product of 360 by 360. The solid angle corresponding to the whole sphere used to be called a "spat" (that's the 3D equivalent of a "turn" for 2D angles) but I have not seen the term used recently, at least not properly (see "spat" in the list pointed to by the 3rd link below).

Answer 2

In a circle, the radius line can rotate through 360° before it overlaps. The circle, however can rotate only 180° in a sphere before it overlaps. This, of course, neglects the center and the diameter of the circle about which it is rotated. These are repeated an infinite number of times.

Answer 3

Solid Angle Of Sphere

Answer 4

I would say 360 deg
Since if you rotated a sphere 360 deg(in same dir'n ) you would back where you started.

But you can rotate in an infinite no. of directions.

Answer 5

Infinity. It can turn 360 degrees in any number of directions.