how to make mathematical candy Jean-Luc Thiffeault Department of Mathematics University of Wisconsin { Madison Summer Program on Dynamics of Complex Systems International Centre for Theoretical Sciences Bangalore, 3 June 2016 Supported by NSF grant CMMI-1233935 1 / 45 We can assign a growth: length multiplier per period. the taffy puller Taffy is a type of candy. Needs to be pulled: this aerates it and makes it lighter and chewier. [movie by M. D. Finn] play movie 2 / 45 the taffy puller Taffy is a type of candy. Needs to be pulled: this aerates it and makes it lighter and chewier. We can assign a growth: length multiplier per period. [movie by M. D. Finn] play movie 2 / 45 making candy cane play movie [Wired: This Is How You Craft 16,000 Candy Canes in a Day] 3 / 45 four-pronged taffy puller play movie http://www.youtube.com/watch?v=Y7tlHDsquVM [MacKay (2001); Halbert & Yorke (2014)] 4 / 45 a simple taffy puller initial -1 �1 �1�2 �1 -1 -1 -1 1�2 �1�2 �1 �2 [Remark for later: each rod moves in a ‘figure-eight’ shape.] 5 / 45 the famous mural This is the same action as in the famous mural painted at Berkeley by Thurston and Sullivan in the Fall of 1971: 6 / 45 The sequence is #folds = 1; 1; 2; 3; 5; 8; 13; 21; 34;::: What is the rule? #foldsn = #foldsn−1 + #foldsn−2 This is the famous Fibonacci sequence, Fn. number of folds [Matlab: demo1] Let's count alternating left/right folds. 7 / 45 #foldsn = #foldsn−1 + #foldsn−2 This is the famous Fibonacci sequence, Fn. number of folds [Matlab: demo1] Let's count alternating left/right folds. The sequence is #folds = 1; 1; 2; 3; 5; 8; 13; 21; 34;::: What is the rule? 7 / 45 This is the famous Fibonacci sequence, Fn. number of folds [Matlab: demo1] Let's count alternating left/right folds. The sequence is #folds = 1; 1; 2; 3; 5; 8; 13; 21; 34;::: What is the rule? #foldsn = #foldsn−1 + #foldsn−2 7 / 45 number of folds [Matlab: demo1] Let's count alternating left/right folds. The sequence is #folds = 1; 1; 2; 3; 5; 8; 13; 21; 34;::: What is the rule? #foldsn = #foldsn−1 + #foldsn−2 This is the famous Fibonacci sequence, Fn. 7 / 45 So the ratio of lengths of the taffy between two successive steps is φ2, where the squared is due to the left/right alternation. Hence, the growth factor for this taffy puller is φ2 = φ + 1 = 2:6180 :::: how fast does the taffy grow? It is well-known that for large n, p F 1 + 5 n ! φ = = 1:6180 ::: Fn−1 2 where φ is the Golden Ratio, also called the Golden Mean. 8 / 45 Hence, the growth factor for this taffy puller is φ2 = φ + 1 = 2:6180 :::: how fast does the taffy grow? It is well-known that for large n, p F 1 + 5 n ! φ = = 1:6180 ::: Fn−1 2 where φ is the Golden Ratio, also called the Golden Mean. So the ratio of lengths of the taffy between two successive steps is φ2, where the squared is due to the left/right alternation. 8 / 45 how fast does the taffy grow? It is well-known that for large n, p F 1 + 5 n ! φ = = 1:6180 ::: Fn−1 2 where φ is the Golden Ratio, also called the Golden Mean. So the ratio of lengths of the taffy between two successive steps is φ2, where the squared is due to the left/right alternation. Hence, the growth factor for this taffy puller is φ2 = φ + 1 = 2:6180 :::: 8 / 45 the Golden Ratio, φ A rectangle has the proportions of the Golden Ratio if, after taking out a square, the remaining rectangle has the same proportions as the original: φ 1 = 1 φ − 1 9 / 45 a slightly more complex taffy puller [Matlab: demo2] Now let's swap our prongs twice each time. 10 / 45 This sequence is given by #foldsn =2#folds n−1 + #foldsn−2 For large n, #folds p n ! χ = 1 + 2 = 2:4142 ::: #foldsn−1 where χ is the Silver Ratio, a much less known number. Hence, the growth factor for this taffy puller is χ2 = 2χ + 1 = 5:8284 :::: number of folds We get for the number of left/right folds #folds = 1; 2; 5; 12; 29; 70; 169; 408 ::: 11 / 45 For large n, #folds p n ! χ = 1 + 2 = 2:4142 ::: #foldsn−1 where χ is the Silver Ratio, a much less known number. Hence, the growth factor for this taffy puller is χ2 = 2χ + 1 = 5:8284 :::: 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#folds n−1 + #foldsn−2 11 / 45 Hence, the growth factor for this taffy puller is χ2 = 2χ + 1 = 5:8284 :::: 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#folds n−1 + #foldsn−2 For large n, #folds p n ! χ = 1 + 2 = 2:4142 ::: #foldsn−1 where χ is the Silver Ratio, a much less known number. 11 / 45 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#folds n−1 + #foldsn−2 For large n, #folds p n ! χ = 1 + 2 = 2:4142 ::: #foldsn−1 where χ is the Silver Ratio, a much less known number. Hence, the growth factor for this taffy puller is χ2 = 2χ + 1 = 5:8284 :::: 11 / 45 Both major taffy puller designs (3- and 4-rod) have growth χ2. the Silver Ratio, χ A rectangle has the proportions of the Silver Ratio if, after taking out two squares, the remaining rectangle has the same proportions as the original. χ 1 = 1 χ − 2 12 / 45 the Silver Ratio, χ A rectangle has the proportions of the Silver Ratio if, after taking out two squares, the remaining rectangle has the same proportions as the original. χ 1 = 1 χ − 2 Both major taffy puller designs (3- and 4-rod) have growth χ2. 12 / 45 The largest eigenvalue of the matrix 2 1 M = 1 1 is φ2. What's the connection between taffy pullers and these maps? linear maps on the torus These quadratic numbers are reminiscent of invertible linear maps on the torus T 2, such as Arnold's Cat Map: x 2 1 x 7! mod 1; x; y 2 [0; 1]2 y 1 1 y 13 / 45 What's the connection between taffy pullers and these maps? linear maps on the torus These quadratic numbers are reminiscent of invertible linear maps on the torus T 2, such as Arnold's Cat Map: x 2 1 x 7! mod 1; x; y 2 [0; 1]2 y 1 1 y The largest eigenvalue of the matrix 2 1 M = 1 1 is φ2. 13 / 45 linear maps on the torus These quadratic numbers are reminiscent of invertible linear maps on the torus T 2, such as Arnold's Cat Map: x 2 1 x 7! mod 1; x; y 2 [0; 1]2 y 1 1 y The largest eigenvalue of the matrix 2 1 M = 1 1 is φ2. What's the connection between taffy pullers and these maps? 13 / 45 the torus The `standard model' for the torus is the biperiodic unit square: 14 / 45 This loop will stand in for a piece of taffy. action of map on the torus The Cat Map stretches loops exponentially: 15 / 45 action of map on the torus The Cat Map stretches loops exponentially: This loop will stand in for a piece of taffy. 15 / 45 Construct the quotient space S = T 2/ι B A p3 p2 =) A B A B p0 p1 Claim: the surface S (right) is a sphere with four punctures! hyperelliptic involution Consider the linear map ι(x) = −x mod 1. This map is called the hyperelliptic involution( ι2 = id). 16 / 45 B A p3 p2 =) A B A B p0 p1 Claim: the surface S (right) is a sphere with four punctures! hyperelliptic involution Consider the linear map ι(x) = −x mod 1. This map is called the hyperelliptic involution( ι2 = id). Construct the quotient space S = T 2/ι 16 / 45 Claim: the surface S (right) is a sphere with four punctures! hyperelliptic involution Consider the linear map ι(x) = −x mod 1. This map is called the hyperelliptic involution( ι2 = id). Construct the quotient space S = T 2/ι B A p3 p2 =) A B A B p0 p1 16 / 45 hyperelliptic involution Consider the linear map ι(x) = −x mod 1. This map is called the hyperelliptic involution( ι2 = id). Construct the quotient space S = T 2/ι B A p3 p2 =) A B A B p0 p1 Claim: the surface S (right) is a sphere with four punctures! 16 / 45 p3 p pacmanize 2 zip =) =) A B p0 p3 p1 p1 p2 p0 The punctures p1;2;3 are the rods of our taffy puller. The puncture p0 is like a point at infinity on the plane. Linear maps commute with ι, so all linear torus maps `descend' to taffy puller motions. sphere with four punctures Here's how we see that S is a sphere: p3 p2 A B p0 p1 17 / 45 zip =) p0 p3 p1 p2 The punctures p1;2;3 are the rods of our taffy puller.