Tinker Toy numbers

[Marc LeBrun]

Speaking of isoceles right triangles, call numbers of the form

	i + r sqrt(2)

"Tinker Toy" numbers, with "integral part" i and "radical part" r, both
integers.

(Because each successively larger Tinker Toy rod size braces the hypotenuse
formed by its predecessor--and besides, it's more fun than calling them Q2).

It seems that if you have Tinker Toy numbers you can develop a lot of ideas
without ever explicitly introducing rationals.  For example, I think the
unique prime factorization of integers allows you to prove that (i,r)=0
implies i=0 and r=0, which is equivalent to the irrationality of sqrt(2),
except you never need to do any division.

1. You'll want to compare Tinker Toy numbers.  Is there a reasonably
elegant algorithm?  The best I've come up with is this:  Form the component
differences di and dr.  Then if (di)^2 > 2(dr)^2 use the sign of di,
otherwise follow dr's sign.  Can anyone come up with a prettier procedure?

2. Given comparison, we can define the sequence T[n] of the ordered Tinker
Toy numbers with non-negative components.  Is there a good algorithm for
generating the successive (i,r)?  I've resorted to this clunky method: the
Tinker Toys between (k,0) and (k+1,0) are the translates of those between
(k-1,0) and (k,0), with possibly at most one new number of the form (0,j)
inserted (there are ceiling(k/sqrt(2)) Tinker Toys for unit interval k).
So, to compute T[n] I generate them in unit batches, until I have enough.
Is there a less kludgey way to, say, just compute T[n], given n?

3. The way those translates snowball makes it clear that if we also include
the Tinker Toys with negative components there must be (countably)
infinitely many in each unit interval.  In fact, we can always find two as
close together as we please.  But can we show that *every* sub-interval
contains infinitely many?  (I think so, but some kind of Cantorian
clustering might countervene.)  On the other hand--assuming such always
exists--given two arbitrary Tinker Toy numbers, can we give an algorithm to
construct one in between them?

Of course if we can do *that* then we can construct the reals (or something
very like them<;-) as the limits of Tinker Toys--without ever introducing
the rationals.

Can we?


(In any case this also suggests all sorts of sequences, including some neat
numbrals, which I'll try to follow up later on seqfan, when I'm not so tired).




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