Jean-Luc Thi eault
Transcript of Jean-Luc Thi eault
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Making taffy with the Golden mean
Jean-Luc Thiffeault
Department of MathematicsUniversity of Wisconsin – Madison
Madison Math Circle, 7 November 2011
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Making candy by hand
[movie 1] http://www.youtube.com/watch?v=pCLYieehzGs
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Taffy pullers
[movie 2] [movie 3] http://www.youtube.com/watch?v=YPP2_ZfOIVU
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Four-pronged taffy puller
[movie 4] http://www.youtube.com/watch?v=Y7tlHDsquVM
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A simple taffy puller
�1
�1�2-1
�1�2-1�1
�2-1�1�2
-1�1
initial
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Number of folds
[Matlab: demo1]
The number of left/right folds satisfies:
#foldsn = #foldsn−1 + #foldsn−2
So we get#folds = 1, 1, 2, 3, 5, 8, 13, 21, 34, . . .
This is the famous Fibonacci sequence, Fn.
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How fast does the taffy grow?
It is well-known that for large n,
FnFn−1
→ φ =1 +√
5
2= 1.6180 . . .
where φ is the Golden Ratio, also called the Golden Mean.
Along with π, φ is probably the best known number in mathematics. Itseems to pop up everywhere. . .
So the ratio of lengths of the taffy between two successive steps is φ2,where the squared is due to the left/right alternation.
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The Golden Ratio, φ
A rectangle has the proportions of
the Golden Ratio if, after taking
out a square, the remaining rect-
angle has the same proportions as
the original:
φ
1=
1
φ− 1
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A slightly more complex taffy puller
Now let’s swap our rods twice each time.
[Matlab: demo2]
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Number of folds
We get for the number of left/right folds
#folds = 1, 2, 5, 12, 29, 70, 169, 408 . . .
This sequence is given by
#foldsn = 2#foldsn−1 + #foldsn−2
For large n,
#foldsn#foldsn−1
→ χ = 1 +√
2 = 2.4142 . . .
where χ is the Silver Ratio, a much less known number.
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Number of folds
We get for the number of left/right folds
#folds = 1, 2, 5, 12, 29, 70, 169, 408 . . .
This sequence is given by
#foldsn = 2#foldsn−1 + #foldsn−2
For large n,
#foldsn#foldsn−1
→ χ = 1 +√
2 = 2.4142 . . .
where χ is the Silver Ratio, a much less known number.
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Number of folds
We get for the number of left/right folds
#folds = 1, 2, 5, 12, 29, 70, 169, 408 . . .
This sequence is given by
#foldsn = 2#foldsn−1 + #foldsn−2
For large n,
#foldsn#foldsn−1
→ χ = 1 +√
2 = 2.4142 . . .
where χ is the Silver Ratio, a much less known number.
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The Silver Ratio, χ
A rectangle has the proportions of the Silver Ratio if, after taking out twosquares, the remaining rectangle has the same proportions as the original.
χ
1=
1
χ− 2
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The original taffy puller
The taffy puller we originallypresented stretches the taffyby χ2 at each ‘period’.
It’s a special case of what wecall Silver Mixers: devices thatstretch by a power of the SilverRatio.
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Build it with Legos!
[movie 6] [movie 7] (Right-hand picture appearing on the cover of a math journal!)
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Some final thoughts
Is there a Bronze Ratio? Can we make such a taffy puller?
What about the taffy puller with four prongs?
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