Using Origami to Find Rational Approximations of Irrational Roots Jeremy Lee Amherst Regional High School Hudson River Undergraduate Mathematics Conference Williams College April 6 th , 2013
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Using Origami to Find Rational Approximations of Irrational Roots
Jeremy LeeAmherst Regional High School
Hudson River Undergraduate Mathematics ConferenceWilliams College
April 6th, 2013
Origami Instructions
• Folding paper allows us to find rational approximations of the square root of two.
Origami InstructionsSide Unit fits Diagonal Once with remainder.
Origami InstructionsSide Unit fits Diagonal Once with remainder.
Remainder fits side unit twice with remainder2 space
Origami InstructionsSide Unit fits Diagonal Once with remainder.
Remainder fits side unit twice with remainder2 space
Remainder2 fits Remainder unit twice with Remainder3 left over.
Summary of Origami Exercise
• As we keep doing the exercise, you always get the number two after the first step. The next remainder will fit into previous remainder twice again.
Summary of Origami Exercise
• All these approximations of the square root of two come from the continued fractions representation of the square root of two.
Continued Fractions Method
1
1
1
12
3
2
1
0
a
a
a
a
Continued Fractions Method
1
1
1
112
3
2
1
a
a
a
Continued Fractions Method
1
1
12
112
3
2
a
a
Continued Fractions Method
1
12
12
112
3
a
Continued Fractions Method
12
12
12
112
There will be twos continuing throughout the fraction after one.
How to Use Continued Fractions to Build Approximations
12
2
3
2
112
5
7
2
12
112
12
17
2
12
12
112
hi ki
Rational Approximations Of the Square Root of Twohi ki
Denoting Parts of the Square Root of Two Continued Fraction
12
12
12
112
Let Big Box equal x
What does the big box converge to?
Algebra that Continued FractionsConverges to the Square Root of Two
212
222
2
82
012
12
2
1
2
2
xx
xx
xx
Why does the answer have to be ?21
Assuming that this continued fraction converges, that is what x converges to.Therefore the continued fraction equals the square root of 2.
)21(12
The Babylonian Method
n
nnx
Sxx
xS
2
11
0
408
577
12
17
2
12
17
2
1
12
17
2
3
2
2
3
2
1
2
3
1
21
2
1
1
3
2
1
0
x
x
x
x
Let’s try it with the square root of 2Have x0 equal 1, our first approximate.
Algebra that Babylonian MethodConverges to the Square Root of Two
2
2
22
12
1
2
2
2
1
2
22
22
x
x
xx
xx
x
xx
xxx
There is a wonderful blog entry that is focused on the Babylonian Method and how it works with the square root of two.
http://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-the-square-root-of-2/
The reason why x is positive is because if the Babylonian Method converges , the result is what it will converge to.
If x is negative then the result is the negative square root of two.
There is almost a common ratio between consecutive numerators of the rational approximations. There is almost a common ratio between consecutive denominators of the rational approximations. The ratio is around 1 + sqrt(2) for both of these relationships.
Maybe, hi and ki are the sums of two geometric sequences with a common ratio of 1 + sqrt(2)
What are the explicit formulas for each the numerator and the denominator of all rational approximations of the square root of two?
hi ki hi+1/hi ki+1/ki
Rational Approximations Numerator
ii
ih 212
121
2
1
21
120
20
725721
55255215
21
222222
3223
121
2
23
2
3
2
2
23
2
2
21
2
21
211
2
212
211
r
rr
rrrc
rcrrch
rcrrch
rrcc
rcch
rcch
rcch
1
16
186212186
1721217
21
42
21
2
212
4
214
cc
rrcc
rcch
rcch
0
1221
122
12121
21
21
2121
211
cc
cc
cccc
cch
i hi
Rational Approximations Denominator
02
22)222(2
2)223(
1)21(
2
2
21
211
2
212
211
rcrcc
rcck
rcck
rcck
21
012
0422
0522
55)255(5
5)257(
2
2
2
21
2
3
21
211
3
213
r
rr
rcrcc
rcrcc
rcck
rcck
ii
ik
c
c
c
cc
cc
ccc
cc
214
221
4
2
4
2
4
2
122
1)21()21(
0212223
1)21()21(
1
2
2
22
21
221
21
i hi ki
Connection between Continued Fractionsand Babylonian Method
What happens when the continued fraction approximation is put into the Babylonian Method?
ii
ii
i
i
i
i
kh
kh
h
k
k
h 22 )(2)(
2
12
2
1
i
i
ii
ii
i
ii
ii
i
ii
ii
ii
ii
iii
i
iii
i
iii
i
k
h
kh
kh
hkh
kkh
kh
k
k
h
2
2
22
2
2222
2
22
22
222
222
222
)(2)(
2
1
212
121
2
12
214
221
4
22
218
221
8
2
214
11
2
121
4
12
218
11
4
121
8
1
214
11
2
121
4
1
This is the Babylonian Method by taking a continued fraction approximation and average it with twice its reciprocal.
i
i
k
h
2
2
The result is the continued fraction approximation of the square root of two, it’s just the number of steps of the continued fractions approximation is doubled for the Babylonian Method. You only did one step of the Babylonian Method.
Conclusion
http://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-the-square-root-of-2/
The Blog Entry
•We did an origami experiment that finds the same rational approximations as the continued fractions method.
•We found that the continued fraction and Babylonian Methods produced the same rational approximations for the square root of two.
Conclusion
http://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-the-square-root-of-2/
The Blog Entry
•We did an origami experiment that finds the same rational approximations as the continued fractions method.
•We found that the continued fraction and Babylonian Methods produced the same rational approximations for the square root of two.
•Could we do something similar with Fibonacci Numbers and the Golden Ratio which has the square root of five in it?
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