Honors Capstone Project Proposal - Online Version

8/9/2019 Honors Capstone Project Proposal - Online Version

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How it all started


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Initial Goaly Find the most efficient packing for a 5x5 city tile

where only corners can overlap


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For Example


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Columns


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Offset Columns


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Modular Arithmeticy Mathematics of remainders: Clock math

y a = b (mod n) means there is a

remainder of b when you dividea by n. Or n divides a-b.

y 13 = 1 (mod 12)

y 28 = 4 (mod 12)


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an (mod p)y Look at powers of some number modulo a prime

y Ex. 2n (mod 5)

21 = 2 = 2 (mod 5) 41 = 4 = 4 (mod 5)22 = 4 = 4 (mod 5) 42 = 16 = 1 (mod 5)23 = 8 = 3 (mod 5) 43 = 64 = 4 (mod 5)24 = 16 = 1 (mod 5) 44 = 256 = 1 (mod 5)25= 32 = 2 (mod 5)26 = 64 = 4 (mod 5)27 = 128 = 3 (mod 5)28 = 256 = 1 (mod 5)

y 2 is called a primitive root of 5. 4 is not.


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Less-boringColumns UsingModulo2

43

1


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Noticed Patterns


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Sinusoidal Waves


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Non-touching lines


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Bigger PxP Tiles


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Same Pattern


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Same Pattern


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13x13 Tiles


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Same Pattern


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Gathering Points

X Y

0 27 -1

14 -3

21 -2

28 1

35 3

42 2

3n (mod 7)


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Finding a Formulayy = sin(x), add parameters

y = asin(bx + c)

y

a controls the amplitude (how high wave gets)y b controls the period (how long cycle is)

y c controls the phase shift (horizontal movement)

y=2sin(2x-2)

y = sin(x)


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Regression Analysisy Used http://www.xuru.org/rt/NLR.asp

to find decimal approximations for a, b, and c

y Fed these into WolframAlphato get guesses for exact answers,then tested equation X Y

0 2

7 -1

14 -3

21 -2

28 1

35 3

42 2

Points from3n (mod 7)


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Finding a Formula AlgebraicallyyAny linear combination of cosine and sine functions of

equal period is equal to a single sine function with the

same period but with a phase shift and a differentamplitude.

y Find equation of form mcos(bx) + nsin(bx) and turn itinto one of form y = asin(bx + c)

y Less work to solve for parameters


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Finding a Formula Algebraicallyy Choose a period to get b. b = 2/period

f(x) = mcos(bx) + nsin(bx)

has two unknowns: m and n.y Choose integers p and q so thatf(0) = p andf(1) = q,

set up two equations and solve with matrices:

Which, when solved, shows

=

m = pn = qcsc(b) pcot(b)


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Integer-Valued FunctionyAny time you put in a whole number you get a whole

number answer

y

y= 2x + 1y Easy for algebraic functions, difficult (until now?) for

some transcendental functions

y Especially for sinusoidal functions: sin(x), cos(x), etc.

y

Want integer outputs for all inputs, not just 0 and 1


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For integers p and q so thatf(0) = p andf(1) = q


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Integer-Valued Functionsy So far only periods of 3, 4, and 6 have this property

p = 5

q = 2


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Summaryy Tried to efficiently pack city tiles

y Noticed patterns in tiles arranged using modular

arithmeticy Gathered points and found an equation going through

them

y Discovered a class of integer-valued sinusoidal

functions for periods 3, 4, and 6.


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Questions to ExploreyAre these the first integer-valued sinusoidal functions?

y For what other periods do integer-valued formulas

exist?y Does it matter how I choose my points?

yAre there any integer-valued functions that dont fitinto the classes I found?

yWhy does this work?


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Acknowledgements/Referencesy Gilles Cazelais, Camosun College

y http://pages.pacificcoast.net/~cazelais/252/lc-trig.pdf

ywww.xuru.orgywww.wolframalpha.com


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Thank you!

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Honors Capstone Project Proposal - Online Version