Are there shapes in nth dimensions (in euclidian geometry), that can be rotated 360 degrees on one axis and have more than one point not in the same place?
Alternatively, in non-euclidian geometry, are there any objects or shapes that can be rotated the full rotation of a closed circle in the relative geometry that fulfil the same properties? (For example, in a non-euclidian plane where 360 degrees is not enough for a full rotation, it would be rotated a full rotation instead.)
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Very untechnical answer here. Sorry but its late and I am tired. I am pretty sure that a 'full' rotation of any shape would be akin to the identity operator. unless the shape undergoes some form of morphing within the rotation then a full rotation would return it to its normal state. however, if my initial instinct on this is incorrect I would love to know!
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A not very serious answer here:
A USB plug! When you first stick it into the receptacle it doesn't fit. Then you turn it over, a rotation of 180 degrees, and it still doesn't fit. Then you turn it over again, another rotation of 180 degrees, and not it fits. Thus it must have changed by the total 360 degrees rotation.
The short answer is no, in euclidean space. However, i do not know about non euclidean space.
First of all, what is a rotation in higher dimensions? In 2 dimensions rotation is simple, we just move a point while keeping its distance to some other point fixed. This is simple because we can describe the rotation by a single number (the angle)
In 3 dimensions we have 2 different ways to generalise this. one way is we can hold the position fixed relative to all points on some line. If that line is an axis, then this is equivalent to holding 1 coordinate fixed, while changing the other two coordinates without changing their distance from the origin. This is also simple and can be described by a single number, again the angle.
However, there is a different way, instead we could just move a point while keeping its distance relative to another point fixed. (you may think about this as an object orbiting another object). This type of generalization of rotation cannot be described just by a single number, you need at least 2 numbers to describe it.
Because of this we consider the first way to be the usual way to generalise rotation, in 3 dimensions. However, the thing which gets generalised is not that we fixed a line, instead its that we fixed n-2 dimensions. What I mean is that in 4 dimensions for example, rotation around an axis, would be analogous to rotation around a point in 3 dimensions. To get the usual notion of rotation which can be described by an angle, we need to fix a plane, or a pair of axes. In general, in $\mathbb{R}^n$ rotation occurs around an $n-2$ dimensional shape. If you want this to be rotation about some axes, then this is equivalent to fixing $n-2$ coordinates, and then altering the remaining 2 coordinates while keeping their distance from the origin fixed.
One way to look at it is what happens to a single point of the shape. Since in general a single point of some shape will just be some point in $x\in\mathbb{R}^n,$ we can consider what happens when $x$ is rotated 360 degrees. if $x=(x_1,x_2,...,x_n)$ then we can rotate $x$ by choosing $n-2$ of the values to hold fixed, and then changing the other 2 coordinates. For example, we may choose to hold $x_3,x_4,...,x_n$ fixed, and change $x_1$ and $x_2$.
So we choose $x'=(x_1',x_2',x_3,...,x_n)$ where $x_1'^2+x_2'^2=x_1^2+x_2^2$. What is the angle between $x$ and $x'$? One way to do this might be to forget the $n-2$ fixed coordinates, and pretend like we are working in the plane. Alternatively there is a way to use cosine law to define angles in $\mathbb{R}^n$. I will leave it to you to confirm that if the angle between $x$ and $x'$ is 360, then $x=x'$.