Coordinates equipped with equivalences in other conventional (or unconventional) ways

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I know coordinates with equivalences of the type :

(a,b,c, ...) ~ (a+k, b+k, c+k, ...)

(a,b,c, ...) ~ (ak, bk, ck, ...)

Does exist some other type of coordinates (and its geometry associated) that make use of equivalences on some other way.

Once, I see a commentary by John Conway where it is mentioned something like

(a,b,c, ...) ~ (ak+k, bk+k, ck+k, ...)

Where n components are naturally put in n-2 dimensions, but, it happens that I am unable to find that reference in the internet (in the context of simplicial coordinates).

If the above is correct, exist some other ?