20 points, 2 gap lengths
The first 20 multiples of 377/610, marked on a circle of circumference one, together with the point at zero. The 21 gaps between neighbours take 2 distinct lengths — 21/610 (8 of them), 34/610 (13 of them). Two lengths, which happens exactly when the number of points is one short of a continued-fraction denominator. Every quantity here is a whole number over the denominator, so nothing is measured.
2 essays call
three-gaps. The drawing above is what it returns with no arguments at all; every
call below passes it something, because a placement that passes nothing draws whichever member
of the family the generator happens to default to rather than the one its essay argues about.
Every one of this site's 393 essays names its parameters at the
call site, which the standard pass of 2026-08-09 established and param-floor
holds.
Where it is called
Changing this generator changes every one of these figures.
Order without repetition
Three gaps, and never four
Mark the points α, 2α, 3α round a circle of circumference one. They look scattered. The gaps between neighbouring points do not: for every angle and every number of points there are at most three distinct gap lengths, and when there are three the largest is the sum of the other two. That is where a chain with exactly two tile lengths comes from.
Order without repetition
How often each patch occurs
That a patch has a frequency at all is the ergodic theorem. What the frequency is turns out to be an eigenvector: the substitution acts on blocks as well as on letters, the block matrix has a Perron vector, and its entries are the frequencies exactly. For the Fibonacci chain those entries take three values at every length, and the three values are the three gaps of a rotation.