Two squares of side 12 inside one of side 17
descent is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
With nothing chosen
The descent from 99 and 70
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- the side of the big square is a whole number between 2 and 200 ×2
- the side of the small squares is a whole number between 1 and 200 ×2
- and the discrepancy never changes size ×1
- and the new pair is strictly smaller ×1
- every step is strictly smaller than the one before ×1
- the descent carries the discrepancy to minus itself ×1
- the descent runs out of room before the discrepancy runs out ×1
- the descent takes at least three steps ×1
- the pair is one the descent can start from ×1
- the two small squares overlap and still reach the corners ×1
- the two small squares, less their overlap and plus the corners, are the big one ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
Countable, and everywhere
The numbers a polynomial can catch arrive in finite batches, so they can be listed. They are also in every interval, however short. Being listable turns out to say nothing whatever about being sparse.
NumberThe square that cannot shrink
The usual proof that the square root of two is irrational is about even and odd numbers. There is a proof about squares instead, in which a supposed solution is folded into a smaller one — and the folding is a drawing.
NumberWhich roots refuse to be fractions
The square root of two is not a fraction, and neither is the square root of three, five, six or seven. The rule behind the list turns an infinite question into a search over the divisors of a single number — and the search finishes.