a general quadrilateral tiles
A general quadrilateral — convex, with no equal sides and no parallel edges — with copies placed by half-turns about edge midpoints. The patch was checked by sampling 2000 points inside a disc: every one of them lies in exactly one tile, so there is no gap and no overlap anywhere in the region tested.
2 essays call
one-tile. 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.
The classification
Which shapes tile by themselves
Every triangle tiles the plane. So does every quadrilateral, convex or not. Six sides admits three families, seven sides admits nothing at all — and the five-sided case took a hundred years and finished with a computer search. The bound at seven needs no search: it is Euler's relation with the curvature set to zero.
The classification
Nothing decides whether a set of tiles tiles the plane
This collection rests on decidability — generate a pattern, forget the group, rediscover it, compare. One question in the same subject has no procedure at all: given a finite set of tiles, whether they cover the plane cannot be decided by any algorithm whatever. What can be done is two half-searches, and measuring what they leave behind.