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When talking about rotations in 2D and 3D, why is it important to consider whether they are commutative or not? Are there other properties of rotation that we should be considering?


What makes the difference between the different behaviour with 2D and 3D? Is it related to some mathematical logic with rotation(sth like rotation is a 3D change)?


What makes the difference that 2D is commutativity while 3D is not? Is this changed by anything related to some mathematical fact with the motion of rotation?


If the rotation matrix can be divided into three rotations around orthogonal x,y,z axes respectively, which order of x,y,z do 3D software and CG library use to calculate the rotation?


Water would not spill out because there is a cap on the bottle


I don't understand this example, it seems like the result is the same if we keep the axis fixed relative to the bottle. 90 degree rotations about Y do nothing, and then the rotations about Z and X still occur one after the other both times. How is there a different result here?


What will happen if it is extended to 4d?


For 4D, if you consider Time as one dimension, things would be fun.


Are there any examples in graphics where due to rotations not being commutative, something bad happened?


What's the mathematical reason that 3D rotation is not communitive?