Pick’s Theorem - Mathematics at Dartmouthauel/papers/docs/sciencecafe.pdf · Pick’s Theorem...
Transcript of Pick’s Theorem - Mathematics at Dartmouthauel/papers/docs/sciencecafe.pdf · Pick’s Theorem...
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Pick’s Theorem
Asher Auel
Department of MathematicsYale University
Pathways to Science CaféOctober 5th, 2014
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Area = 3
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1 1
1 1
12
Area = 412
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1 2
Area = 3
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2
3
Area = 3 = 3×22
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Area = 3 = 3×22
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Area = 3 = 3×22
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?
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1 12
1 3
Area = 9 − 512 = 312
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Georg Alexander Pick 1859–1942
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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
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A = i + b2 − 1 3 +32 − 1 = 3
12
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A = i + b2 − 1 0 +82 − 1 = 3
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A = i + b2 − 1 2 +72 − 1 = 4
12
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A = i + b2 − 1 0 +32 − 1 =
12
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A = i + b2 − 1 0 +32 − 1 =
12
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A = i + b2 − 1
= 2 + 62 − 1
THE
E
D
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Ford circles
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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