Pick’s Theorem
Asher Auel
Department of MathematicsYale University
Pathways to Science CaféOctober 5th, 2014
Area = 3
1 1
1 1
12
Area = 412
1 2
Area = 3
2
3
Area = 3 = 3×22
Area = 3 = 3×22
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
12
A = i + b2 − 1 0 +82 − 1 = 3
A = i + b2 − 1 2 +72 − 1 = 4
12
A = i + b2 − 1 0 +32 − 1 =
12
A = i + b2 − 1 0 +32 − 1 =
12
A = i + b2 − 1
= 2 + 62 − 1
THE
E
D
Ford circles
Farey sequences
01,
11
01,
12,
11
01,
13,
12,
23,
11
01,
14,
13,
12,
23,
34,
11
01,
15,
14,
13,
25,
12,
35,
23,
34,
45,
11
01,
16,
15,
14,
13,
25,
12,
35,
23,
34,
45,
56,
11
01,
17,
16,
15,
14,
27,
13,
25,
37,
12,
47,
35,
23,
57,
34,
45,
56,
67,
11
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