The days this site can compute, and the ones it cannot
Zero on the first day, ±1 on the second, and thereafter the simplest number in every remaining gap — the construction run by the game recursion, which produces only fractions with a power of two underneath however long it goes on. Below it, three objects the same recursion reaches when the stopping rule is removed, each written with its option set and the exact reason this site's machinery cannot hold it. They are named rather than drawn, which is the honest half of a figure-first collection.
3 essays call
birthday-limit. 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.
The positions it draws
57 distinct positions, harvested by running this generator again at the options each essay passed it.
Where it is called
Changing this generator changes every one of these figures.
How it was found
The numbers came out of the game
The construction is always taught numbers first and games second, and the discovery ran the other way. Conway arrived at the number system from positions, which is why the definition quantifies over sets of previously built objects rather than over cuts — and why it produces a genuinely different collection at every finite stage.
Values
The recursion this site cannot run
Remove the stopping condition from the construction and it reaches ω, its reciprocal, and one third — none of which this site's evaluator can represent, because it interns a position from a finite list of options. The figure draws what it computes and names what it cannot, which is where the boundary belongs.
Values
The simplest game above both
Values sit in a partial order, and a partial order is entitled to be ragged: two things with no least thing above them. The 22 values born by day two are not ragged at all. Every one of their 253 pairs has a least upper bound and a greatest lower bound among the same 22, and the order is distributive on all 10,648 triples — so it is a lattice, and the join of zero and star is one half.
The whole library · The position index · The figures that play back