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Pick’s Theorem
Asher Auel
Department of MathematicsYale University
Pathways to Science CaféOctober 5th, 2014
Area = 3
1 1
12
Area = 412
1 2
2
3
Area = 3 = 3×22
?
1 12
1 3
Area = 9 − 512 = 312
Georg Alexander Pick 1859–1942
A lattice polygon has all of its vertices on lattice points.
i = the number of lattice points strictly inside
b = the number of lattice points on the perimeter
Pick’s TheoremThe area of a lattice polygon is
A = i +b2− 1
A = i + b2 − 1 3 +32 − 1 = 3
A = i + b2 − 1 0 +82 − 1 = 3
A = i + b2 − 1 2 +72 − 1 = 4
A = i + b2 − 1 0 +32 − 1 =
A = i + b2 − 1
= 2 + 62 − 1
THE
E
D
Ford circles
Farey sequences
01,
11
12,
13,
23,
14,
34,
15,
25,
35,
45,
16,
56,
17,
27,
37,
47,
57,
67,
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