Rational polygons database

What is this?

A searchable database of about 500 million rational polygons and their invariants: interior and boundary lattice points, area, Ehrhart quasipolynomial, automorphism group and more. Every polygon has its own page with a drawing, its properties, and where it appears in the literature.

What is a rational polygon?

The convex hull of finitely many points with rational coordinates. Its denominator is the smallest k such that kP has integral vertices; polygons of denominator 1 are lattice polygons. Polygons are considered up to affine unimodular equivalence.

Where do the polygons come from?

Mainly from the classification [BS26], whose data is on Zenodo, see the Coverage page. All computations were done with RationalPolygons.jl [Spr24]. Each polygon's page lists additional references.

Which polygons are included?

The Coverage page lists for which denominators and numbers of interior lattice points the database is complete. Every search also says whether its results are complete.

Why are thin polygons left out?

Polygons that fit into ℝ × [0, 1] have no interior lattice points and come in infinite families for every denominator, so classifications leave them out. Tick "include thin polygons" to see the few the database contains.

How do I look up a polygon I have?

Enter its vertices on the Identify page. It finds the polygon in whatever position you give it.

Can I use the data in my own programs?

The API returns polygons as JSON. For whole classifications, download the datasets from Zenodo, which RationalPolygons.jl can read.

How do I cite this?

Please cite [BS26], and for particular polygons the classifications listed under their sources.

Found a mistake or a missing reference?

Please write to j.springer@imperial.ac.uk.

References