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Pick theorem — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A lattice polygon of area 22.5. A polygon with all its corners on the integer grid, with the 20 grid points strictly inside and the 7 on its boundary marked; its area is the first count plus half the second, less one.

    Area by counting dots

    Draw a polygon with every corner on a grid of dots. Count the dots strictly inside, add half the dots on the edge, subtract one — and the answer is the area, exactly, with no measuring anywhere.

    part 1 · discrete
  2. 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.

    part 2 · discrete
  3. 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.

    part 3 · discrete
  4. 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.

    part 4 · discrete

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