Notes:Grid iteration

See Modulo operator for a definition of $\text{Mod}(a,b)$

2D grid

Here we see $4\times 5$ shown, notice that the $21^\text{st}$ point returns to the origin (or $k\eq 20$ point returns to origin) - this is key to using this system in higher dimensions as at 20 we'd increase the 3rd dimension to place our 2d grid there, see gif below

Notice the 21st point is back at the origin and the sequence repeats. Direct link

• $$\text{Mix}_x(k,m,n):\eq \text{Floor}\left(\text{Mod}\big(\text{Mod}(k,mn),m\big)\right)$$
• $$\text{Mix}_y(k,m,n):\eq \frac{\text{Mod}(k,mn)-\text{Mod}\big(\text{Mod}(k,mn),m\big)}{m}$$

Then

• Points of the form:
  • $$\big(\text{Mix}_x(k,m,n),\text{Mix}_y(k,m,n)\big)$$ span an $m\times n$ grid, for $k$ from $0$ to $mn-1$

Notice that these coordinates return to the start for $\text{count}\eq 21$ or $k\eq 20$, this is critical to allowing this to be used in higher dimensions.

3D grid

Here we see $3\times 8\times 6$ using the formulas below

Notice that after the count is $\ell\times m\times n$ (here, $3\times 8\times 6$) it goes back to the starting point, this is key for using in higher dimensions. Direct link

Suppose we have an $\ell\times m\times n$ grid, we can build this using $\text{Mix}_x$ and $\text{Mix}_y$:

• for $k\in\mathbb{N}_0$ with $0\le k\le \ell\times m\times n-1$
  • $x$-coordinate: $\text{Mix}_x(k,\ell,m\times n)$
  • $y$-coordinate: $\text{Mix}_x\big(\text{Mix}_y(k,\ell,m\times n),m,n\big)$
  • $z$-coordinate: $\text{Mix}_y\big(\text{Mix}_y(k,\ell,m\times n),m,n\big)$

In the picture on the right we show a $3\times 8\times 6$ example, notice that these multiplied give $144$ ($k\eq 144-1\eq 143$), we do $145$ ($k\eq 144$) instead, this gives the diagonal returning back to the 1st point, and is the key property in allowing this concept to extend to higher dimensions (returning to the start, where we would increase the 4th dimension to create a distinct point, if we did so here)

Generalisation

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TODO: Explain method, generalise to arbitrary dimensions

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