== Numbering tiles ==

If we want a straightforward way to linearly index tiles without necessarily knowing the image size, we can use the following enumeration:

|| 0 || 1 || 3 || 6 || 10 || 15 || 21 ||
|| 2 || 4 || 7 || 11 || 16 || 22 || 29 ||
|| 5 || 8 || 12 || 17 || 23 || 30 || 38 ||
|| 9 || 13 || 18 || 24 || 31 || 39 || … ||
|| 14 || 19 || 25 || 32 || 40 || … || ||
|| 20 || 26 || 33 || 41 || … || || ||
|| 27 || 34 || 42 || … || || || ||

One way to generate these values is using the '''Cantor polynomial''':
{{{
#!latex
$n = \dfrac{(x + y) (x + y + 1)}{2} + y$
}}}

Efficiently inverting that polynomial is not trivial. Here is one way to do it:
{{{
#!latex
$k = E(\dfrac{\sqrt{8n+1}-1}{2})$

$y = n - \dfrac{k (k + 1)}{2}$

$x = k - y$
}}}