echidna

OK, I'll take a dip at it & say no rotation. Possibly both flawed, but 2 reasons ...

a) Each movement as specified in the OP represents a 180 degree flip in each of the 3 geometric axes. These 3 flips will always result in coming back to the starting orientation.

Nope, paragraph b) was a dud.

FB, I'm hoping you have an authorative answer of some sort ...

echidna

No bugger it, crash & burn as they say ...

b) Topologically the 2 spheres only see each other as flat point contact, with each point on one sphere mapped to an exact point (2 actually) & orientation of the other sphere.

Make rolls 2 & 4 only 45 degree rotations (or any other angle) & try it again. More complicated but I think the orientation is still unchanged ?

SULPHUR

Let radius equal 1.Distance travelled to equator equals pi/2. Move ment along equator equals x. Movement back to north pole is - Pi/2 .What is x ?

nermal

Consider, a mark on each ball of equal size oriented in such a way that the marks are vertically aligned.
Roll on ball down 90 degrees and the marks will be symmetrically located relative to an axis drawn tangentially to both balls.
Rotate the balls together (motion is relative, right) and the marks will remain at all times symmetrically located. You can rotate just one and get the same effect, but it's not as intuitive.
Roll one ball back up, and the marks will again be vertically aligned.

Ed

SULPHUR

What I was trying to point out "badly" is that if the ball was rolled back to the original point of contact and then up wards the net movement and rotation would be zero and the answer would be yes

yguy

Correction: the movable ball ends up being rotated about the N/S axis by TWICE the number of degrees it travelled along the fixed ball's equator.

SULPHUR

Read the post above. I think it makes sense

nermal

The question is: Can the orientation of the balls be changed by the motions described. Ie. If two marks on the balls are vertically aligned (they are longitudinally oriented in the same direction) after completing the movements, when the balls are again vertically stacked, can the marks be longitudinally misaligned?
The answer is no, they cannot, so long as there is no slippage between the balls. They will always end up once again longitudinally aligned, no matter the arbitrary rotation distance about the equator.
Topologically, if you have axial symmetry (even function), you can fold the space about the y axis and any two arbitrary corresponding points in the function will meet.
This is exactly what is happening. When you roll the top ball down to the equator, you are unfolding the space.
Then rotating the moveable ball about the equator traces out an even function. Then you roll the ball back up and re-fold the space. Two points initially aligned vertically on the ball are corresponding points in the space. Therefore they will always return to alignment.
This is easily verified: Take two tennis balls. Put a magic marker dot on each, and stack them one on the other with the dots vertically aligned facing you. Do the exercise, and no matter the motion of the moveable ball about the fixed ball's equator, it will return with the dot vertically aligned in exactly its original position.

Ed

yguy

Not sure at whom this is directed, but this doesn't contradict what I said. The movable ball starts with its own N/S axis colinear to that of the fixed ball, and that's how it ends up - but it also ends up rotated about the fixed ball's N/S axis as I have described.

SULPHUR

I was playing about with a couple of balls awhile ago.I cannot see it working.Maybe I used the wrong balls !!!