What Should Adorn Next Year’s Math Contest T-shirt? By Kevin Ferland.

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Presentation transcript:

What Should Adorn Next Year’s Math Contest T-shirt? By Kevin Ferland

Last Year’s T-shirt total area = area of inner square + area of 4 triangles b a a b a b ab c c c c

Proof Pythagoras B.C.E. The t-shirt proof is believed to be the type used by the Pythagoreans.

Pythagorean Theorem Given a right triangle we have b c a

James Garfield 1876 total area = area of inner half-square + area of 2 triangles b a b a c c

Euclid B.C.E. (modernized) proof from Elements I II I ← c → ab xc-x

The result was known by Babylonian mathematicians circa 1800 B.C.E.

The oldest known proof is found in a Chinese text circa 600 B.C.E.

January 2010

Last Year’s T-shirt total area = area of inner square + area of 4 triangles b a a b a b ab c c c c

Generalizing the t-shirt total area = area of inner hexagon + area of 6 triangles ba b a b a ba b a b a c c c c c c 120°

FACT : The area of a regular hexagon with side length s is FACT : The area of a 120°-triangle with (short) sides a and b is

Proof

Result Given a 120°-triangle we have 120° a b c

b a ba b c c a c

STOP?

Clearly, this argument extends to any regular 2k-gon for k ≥ 2. DON’T STOP

Result (n = 2k) Given a θ-triangle we have θ a b c

θnEquation c 2 = 90°4 120°6 135°8 144°10 150°12 ׃׃׃

FACT : The area of a regular n-gon with side length s is s s s s s s s s θ/2 θ θ θ

h s/2

FACT : The area of a θ-triangle with (short) sides a and b is θ a b c h

Trig Identity : Proof

Generalized Pythagorean Theorem total n-gon area = area of inner n-gon + area of n θ-triangles b a ba b c c a c

Proof

Result Given a θ-triangle we have θ a b c The LAW OF COSINES in these cases.

Pythagorean Triples A triple (a, b, c) of positive integers such that a 2 + b 2 = c 2 is called a Pythagorean triple. It is called primitive if a, b, and c are relatively prime.

E.g. Whereas, (6, 8, 10) is a Pythagorean triple that is not primitive

Theorem : All primitive solutions to a 2 + b 2 = c 2 (satisfying a even and b consequently odd) are given by where

stabc ׃׃׃׃׃

E.g. n = 6, θ = 120° What is the characterization of all such primitive triples? 120° 5 3 7

Other 120°-triples (7, 8, 13), (5, 16, 19), … There does exist a characterization of these. What can you find?

Next Year’s t-shirt ba b a b a ba b a b a c c c c c c 120°