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Very frequently we index "cyclic objects" using the integers. For instance, we might say that the vertices of a polygon are $x_1, x_2, \dotsc, x_n$, where the "next vertex" after $x_n$ is $x_1$. This discontinuity gets quite annoying if I make definitions that depend on the successor of each index:

An edge exists between $x_i$ and $x_{i + 1}$ (or if $i = n$, then $x_n$ and $x_1$).

Another version which I feel is more rigorous

An edge exists between $x_i$ and $x_j$, where $j = 1$ if $i = n$, and $j = i + 1$ otherwise.

Of course, we could remedy this situation a little by defining

$$\operatorname{succ}_n(i) = \begin{cases}i + 1 & i < n\\1 & i = n\end{cases}$$

and saying

An edge exists between $x_i$ and $x_{\operatorname{succ}_n(i)}$.

I can't even drop the subscript of $\operatorname{succ}_n$ if I consider many "cyclic objects" that have different "periods" - if I'm studying many different polygons, for instance. This clumsy boilerplate obfuscates the intuition of "the next object".

Another solution abuses the notation and simply says $i + 1$ to mean $\operatorname{succ}_n(i)$ (with an earlier statement giving prior warning, of course). I would actually prefer this, but is this commonly accepted, especially in submissions to peer-reviewed journals?

Or is there an even better solution than those that I have enumerated above?

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    $\begingroup$ You could label the vertices of your polygon as $\{x_i|i\in\mathbb{Z}/n\mathbb{Z}\}$ rather than $\{x_i|i=1,\ldots,n\}$. $\endgroup$
    – Abel
    Jun 21, 2013 at 9:41
  • $\begingroup$ How about $\operatorname{succ}_n(i) = (i+1)~(\bmod~n)$? $\endgroup$
    – Adriano
    Jun 21, 2013 at 9:41
  • $\begingroup$ @Abel that would require group-theoretic knowledge which would be a barrier to the clear, intuitive message of "the next vertex". But yes I think that is an interesting alternative to consider. $\endgroup$
    – Herng Yi
    Jun 21, 2013 at 10:13
  • $\begingroup$ @Adriano I would like to avoid using a lengthy and possibly unfamiliar function as an index - it still obscures the intuitive meaning. $\endgroup$
    – Herng Yi
    Jun 21, 2013 at 10:14
  • $\begingroup$ Why use a notation at all when saying something like "where addition is done cyclically" is probably clear? $\endgroup$
    – lhf
    Jun 21, 2013 at 10:28

1 Answer 1

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Things get easier if you start from $x_0$. This way, you could state there is an edge between $x_i$ and $x_{(i+1)\bmod\, n}$ where $a \bmod b$ is the remainder of the division of $a$ by $b$.

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