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My question is probably rather simple, but I cannot find appropriate name or am too stupid to find a definition of such property.

I'd like to give an example:

$f({\bf r}_1, ..., {\bf r}_N, ) = \sum^{N}_{i=1}{\left| {\bf r}_{i} \right|}$

where ${\bf r}_i$ - is a position of $i$-th point in a $N$ dimensional space

It basically is a total distance of all points from the origin. So this function has such property that I can transform arguments, and the function does not change it's value. E.g: I can rotate whole system around origin.

My question is what is called such a property? I have two candidates but I don't think they are right:

  1. function is symmetrical (it has rotational symmetry round the origin)
  2. function is degenerated in respect to transformation (rotation round the origin)

There is also another transformation, that does not change the value - swapping two positions.

What is the proper mathematical term to describe it.

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    $\begingroup$ The function is invariant to rotation. $\endgroup$
    – user856
    Commented Feb 6, 2014 at 7:03
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    $\begingroup$ invariant under rotation. $\endgroup$
    – Robert Israel
    Commented Feb 6, 2014 at 7:15
  • $\begingroup$ I think symmetric is fine. To me, a symmetric function would be a function that is invariant under the action of the symmetric group on its arguments. Is anything wrong with that? $\endgroup$
    – k.stm
    Commented Feb 6, 2014 at 7:17
  • $\begingroup$ Also, I think rotational symmetry (or invariance under rotation) is a bad choice of words because that would suggest that the function was only invariant under cyclic permutations of all its arguments. $\endgroup$
    – k.stm
    Commented Feb 6, 2014 at 7:20
  • $\begingroup$ Thanks for answers, but Y U POST ANSWERS AS COMMENTS PEOPLE. =) Please let me up-vote and accept. $\endgroup$
    – luk32
    Commented Feb 6, 2014 at 7:37

2 Answers 2

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A function whose value does not change when its arguments undergo a (set of) transformation(s) is said to be invariant under the (set of) transformation(s). [As per the comments]

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Another term used in physics is to say that the function is isotropic.

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  • $\begingroup$ Thanks for pointing out this special case. This function is indeed isotropic (in the physical sense at least). But this is kind of dangerous. There can be axial anisotropy, and then the function still would be invariant under rotation, just along the axis, not origin anymore. Though my problem was a general name, I need to include permutations of arguments too. $\endgroup$
    – luk32
    Commented Feb 6, 2014 at 8:31

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