mediant
mediant 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.
At its defaults
show: "farey"
show: "ford"
What it checks while it draws
Collected by running the family and listening to lib/verify.js, not written
here. The count is how many separate times this build put that claim to the test.
- the circles on 0/1 and 1/7 touch ×62
- 0/1 and 1/7 are Farey neighbours ×24
- 1/1 arrives already in lowest terms ×15
- the sequence has one entry per coprime pair up to 7 ×2
- circles that are not neighbours stay clear of each other ×1
- no fraction is produced twice ×1
- reading the drawing left to right reads the fractions in increasing order ×1
- the depth is a whole number between 2 and 5 ×1
- the drawing scale is a whole number between 200 and 400 ×1
- the order is a whole number between 2 and 12 ×1
- the tree is full to its stated depth ×1
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
Every fraction, exactly once
Take two fractions, add the tops and add the bottoms. That is not how fractions are added, it is not an average, and repeating it produces every positive rational exactly once, already in lowest terms.
NumberHow close a fraction can get
Drop eight points into seven boxes and two of them share. That one line, applied to the multiples of an irrational number, proves that every irrational has infinitely many astonishingly good rational approximations — and no construction is needed anywhere.