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Concept

Pick theorem

The rule giving a lattice polygon's area as the interior dot count plus half the boundary count minus one. It is exact, it has no analogue in three dimensions, and its exactness comes from the boundary being made of segments between lattice points.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Four solids with the same counts and every volume. The Reeve tetrahedra at heights 1, 2, 3, 5, drawn in wireframe with a table of their lattice-point counts and volumes. All have four boundary points and none inside; their volumes run from 0.17 to 0.83.

The theorem that has no version in space

A lattice polygon's area is decided completely by two counts of dots. The obvious guess is that a lattice solid's volume is decided by the same two counts in three dimensions, and there is a family of tetrahedra with identical counts and every volume that says otherwise.

discrete · Pick theorem The sixteen lattice polygons with a single point inside. A grid of sixteen small lattice polygons, each drawn on its own patch of grid with the single interior point marked, labelled with its number of boundary points.

Sixteen polygons with one dot inside

Fix one of Pick's two counts at one and ask what is left. The answer is a finite list, the list has exactly sixteen entries, each one is its own kind of object with a dual that is another entry, and the whole classification is a search a page can carry out.

discrete · Pick theorem The lattice-point count inside a circle, less its area, out to radius 160. A plot of the difference between the number of lattice points in a disc and the disc's area, against radius, with envelopes proportional to the square root and the two-thirds power drawn.

The dots a circle catches

Pick's theorem gives a lattice polygon's area exactly, with no error term anywhere. Ask a circle the same question and the exactness is gone: the count is the area plus something, the something has been measured for two centuries, and nobody knows how big it is.

discrete · Pick theorem

Named alongside it

The objects these essays reach for when they reach for this one.

LatticeExhaustive searchAsymptoticsClassificationConvex hullCounterexampleCounting two waysDimensionDivisor functionDualityEhrhart polynomialEquivalence

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