Enumerating all 3-tuples

[bartc]

On 10/03/2018 01:13, Steven D'Aprano wrote:
> I am trying to enumerate all the three-tuples (x, y, z) where each of x,
> y, z can range from 1 to ∞ (infinity).
> 
> This is clearly unhelpful:
> 
> for x in itertools.count(1):
>      for y in itertools.count(1):
>          for z in itertools.count(1):
>              print(x, y, z)
> 
> as it never advances beyond x=1, y=1 since the innermost loop never
> finishes.
> 
> Georg Cantor to the rescue! (Well, almost...)
> 
> https://en.wikipedia.org/wiki/Pairing_function
> 
> The Russian mathematician Cantor came up with a *pairing function* that
> encodes a pair of integers into a single one. For example, he maps the
> coordinate pairs to integers as follows:
> 
> 1,1  ->  1
> 2,1  ->  2
> 1,2  ->  3
> 3,1  ->  4
> 2,2  ->  5
> 
> and so forth. He does this by writing out the coordinates in a grid:
> 
> 1,1  1,2  1,3  1,4  ...
> 2,1  2,2  2,3  2,4  ...
> 3,1  3,2  3,3  3,4  ...
> 4,1  4,2  4,3  4,4  ...
> ...
...
> But I've stared at this for an hour and I can't see how to extend the
> result to three coordinates. I can lay out a grid in the order I want:
> 
> 1,1,1   1,1,2   1,1,3   1,1,4   ...
> 2,1,1   2,1,2   2,1,3   2,1,4   ...
> 1,2,1   1,2,2   1,2,3   1,2,4   ...
> 3,1,1   3,1,2   3,1,3   3,1,4   ...
> 2,2,1   2,2,2   2,2,3   2,2,4   ...
> ...
> 

I can't see the patterns here that I can see in the 2-D grid (where the 
first number in each pair in the n'th row is n, and the second number in 
the n'th column is n).

Maybe it needs to be 3-D? (Eg if the 3rd number in the triple is the 
Plane number, then plane 1 looks like:

   1,1,1   1,2,1   1,3,1
   2,1,1   2,2,1   2,3,1
   3,1,1   3,2,1   3,3,1 ...
   ...

But whether that has an equivalent traversal path like the diagonals of 
the 2-D, I don't know. I'm just guessing.)

-- 
bartc