Dance and Mathematics Karl Schaffer MATH Dr. Schaffer and Mr. Stern Dance Ensemble and De Anza College Mathdance.org karl_schaffer@yahoo.com Materials developed with Erik Stern and Scott Kim 2012 Joint Mathematics Meetings Boston Jan. 4 & 6, 2012 DANCE ( workshop with Leon Harkleroad Bowdoin College lharkler@bowdoin.edu)
Erik Stern Karl Schaffer Weber State Univ. Scott Kim Puzzle Designer
Clap Your Name Clap = Consonant Slap = Vowel
Roberta vs. Joan R o a b t e r Schlafli symbol {7/4} = {7/3} Ludwig Schlafli, 1814-1895
Joanne vs. Joan J o e a n n {6/2} = {6/4}
7 0 1 6 2 5 3 4 Mystic “heptagram” - for warding off evil - adopted for some Sherrif’s stars. {7/2}
Poinsot Stars Louis Poinsot (1777-1859) Thomas Bradwardine (1290-1349) {9/3}={9/6} Christian symbol 3 interlaced triangles.
7 0 6 1 2 5 3 4 18th century Netherlands {7/3}
Coat of arms of Azerbaijan {8/3}
Classroom questions {7/3} and {7/4} look alike, why?
{7/2}: one continuous strand {6/2}: two strands {7/2} {6/2} (1) When is { n / k } one strand? (2) For given n , how many { n / k } are one strand? (3) How many distinct strands in { n / k }? (4) How many edges in each strand of { n / k }?
Asteroids
Power of mathematics …and dance, and language, and culture! Sound, movement Language (vowels, Geometric consonants) representation Cultural connections
Finger tetrahedron Hand dances
Try these: (1) 2 person, 4 handed tetrahedron (thumb, 1st, 2nd fingers per hand) (2) 2 person, 4 handed cube (thumb, 1st, 2nd fingers per hand) (3) 4 person, 8 handed cube (thumb, 1st, 2nd fingers per hand) (4) 4 person, 8 handed interlocking tetrahedra (3 fingers per hand) (5) 1 person, 2 handed tetrahedron (thumb & 1st finger per hand) (6) 2 person, 3 handed tetrahedron (1st and 2nd fingers per hand) (7) 1 person trefoil know (1st and 2nd fingers per hand) (8) 3 person, 5 handed 5-pointed star (1st and 2nd fingers per hand) (9) 5 person, 10 armed 5-pointed star (hands or arms) (10) 3 person, 6 handed octahedron (1st, 2nd fingers per hand) (11) 3 person, 6 handed cube (1st, 2nd fingers per hand, reversible!)
Trefoil knot ... and its reflection
Figure eight knot
Borromean Rings Molecular rings 2004 - Stoddard Wikipedia, accessed 6/5/07
Make the trefoil using arms - each person is a “loop”
Make the figure 8 knot using arms - each person is a “loop”
Make the Borromean rings using arms - each person is a “loop”
Trefoil Figure 8 Borromean rings
1/2 Somersault 1/2 Cartwheel 1/2 Spin
1/2 Cartwheel then 1/2 Somersault then 1/2 Spin Upside down? Right side up? Facing forwards? Facing backwards? Use your hand to model!
How many ways can 1/2 Cartwheel 1/2 Somersault 1/2 Spin be put in order? Does each sequence leave the doll in the same place? Why or why not?
Facing Orientation Same Same • Translation (Slide) Opposite Opposite • Reflection (Mirror, Flip) Opposite Same 180 • Rotation ( , Turn) Same Opposite • Glide (Footsteps)
Combining Symmetries
Second symmetry R T G M M T First G symmetry M G R p q d gives p d Mirror Rotate Glide
The Klein four group Z 2 × Z 2 , T G M R T T G M R G G T R M M M R T G R R M G T
Bilateral symmetry Mirror or (op oppos osite direction ons) face to side face each other Translation on Rot Rotation on (same me (op oppos osite direction on) direction ons) face to side face same way Glide Glide (same me direction on)
Turn table Second turn 0 1 2 3 0 0 1 2 3 First 1 1 2 3 0 turn 2 2 3 0 1 3 3 0 1 2
Powers of i = − 1 1 i –1 – i 1 1 i –1 – i i i –1 – i 1 –1 –1 – i 1 i – i – i 1 i –1
Other Dance Symmetries Symmetries in time: one dancer or instrument repeats a phrase a certain number of beats after another (Canon). Reversals : Retrograde. Movement performed as if time were reversed. Inversion. Sequence of movements performed in reverse order.
Opposition: the resemblance in walking between the arms and legs in normal opposition, sometimes called “helical symmetry” or “screw rotation.” www.nordicwalker.com
Front/back silhouette
Resemblance in shape and www.jupiterimages.com motion between the arms and the legs displayed to the sides, as in a cartwheel (reflection in the horizontal or “transverse” plane - or also in the “sagittal” front-to- back plane). Momix
Reflection in y-axis (–2,3) (2,3) (–2,–3) (2,–3) 180 degree rotation Reflection in x-axis
(–3, (–3, (–2,3) (2,3) (–3,2) (3,2) (–3,–2) (3,–2) (–2,–3) (2,–3)
Eight Square Dancers Dihedral group D 4 of order 8 1 2 3 4 Quarter turn
± e (± x) and ± ln(± x) 3 2 1 - 3 - 2 - 1 1 2 3 - 1 - 2 - 3
Change-Ringing Ringing church bells, polyhedra, and DNA mutations
Change-Ringing: permuting the bells Ringing a set of tuned church bells in mathematical patterns called “changes,” which run through all the permutations of the bells.
Congress Bells of the Old Post Office Tower 10 bells, 581 to 2953 pounds each Each bell can be rung once every 2 seconds Rung on holidays, opening and closing of Congress
The Church of the Advent, Boston 8 Tower Bells: 19-1-17 (2173 pounds) Sunday , approx. 10:10 - 11:05 am. Wednesday , 7:00 - 9:00 pm. The Old North Church, Boston 8 Tower Bells: 14-1-0 (1596 pounds) Sunday , 12:00 pm - 1:00 pm. Service Ringing Sunday , 2:00 pm - 4:00 pm. Practice -- 2nd & 4th Sundays, beginning with July 11 Saturday , 11:00 am - 1:00 pm. Practice -- Every weekend through July 3rd; after that only when there is no Sunday practice
Change-Ringing Each bell swings freely… … the bells are unwieldy, each bell rung by one person, and organized by permutations, not melody
Change-Ringing 1,2,3,4 then 1,3,2,4 (single swap) 1,2,3,4 then 2,1,4,3 (double swap) 1,2,3,4 followed by 4,1,2,3 would not work - why not? • Is it possible to play all permutations once by doing a single swap each time?
1 2 3 6 permutations of 3 people standing in a line. Two people standing next to each other may switch. Move through all six permutations, return to the original order, 123, by switching two neighbors at a time?
1 2 3 4 What about four people?
123 2 and 3 switch places 213 132 Switch 1 and 3 231 312 321 The six switches for change-ringing 3 bells
2134 1243 Switch Switch 1 and 2 3 and 4 1234 Switch 2 and 3 1324
2143 Switch Switch 2314 3 and 4 1 and 2 Switch 1 and 3 2134 1243 Switch Switch Switch 2 and 4 1 and 2 1423 3 and 4 3124 1234 Switch Switch 2 and 3 1 and 3 1324 Each permutation connects to 3 others by Switch neighbor switches. 2 and 4 1342
The “permutahedron” (truncated octahedron) - all 24 permutations Connecting lines show “neighbor switches”
The permutahedron in 3-D (truncated octahedron)
Here is one way to do it.
Change 1,2,3,4 to A,C,T, and G. A=Adenine C=Cytosine G=Guanine T=Thymine Switches are like mutations in DNA sequences! Can you find the smallest number of switches (mutations) from ACGT to TGAC? This is the “evolutionary distance” between the sequnces.
Green: Switch first 2 elements Blue: Switch last 2 elements Red: Switch middle 2 elements
Start with 4 End with 1 Each hexagon starts Each hexagon or ends with same starts or ends Start with 3 element. End with 3 Start with 2 with the same element. End with 4 Start with 1
Change-Ringing “Plain Hunt Minimus” Using this swapping method, it is NOT possible to perform each of the 4! = 24 permutations exactly once and then return to the starting sequence. This method uses double swaps alternating with single swaps - can you see them?
Change-Ringing “Plain Bob Minor” What exactly is the method here? 6! = 720 permutations, for 6 bells. 1963: ringers in Loughborough played all 8! = 40,320 permutations of 8 bells…. … in 18 hours - no one has done it again! 12 bells would take over thirty years, as the factorial function increases faster than exponentially.
Tap-Clap-Bump-Snap Tap-Bump-Clap-Snap Tap-Bump-Snap-Clap Tap-Snap-Bump-Clap Tap-Snap-Clap-Bump Tap-Clap-Snap-Bump
Circles in Dance Bulgarian Women’s Dance
Circular floor patterns
Body circles pirouette: vertical axis somersaults or flip: cartwheel: left-to-right axis front-to-back axis
Circular Phrasing
Prop Circles
Poi fire dance Photographer : Jean-Romain PAC
Costumes
Arm Circles 72
Positive / Negative (-) (+) Turning around and walking backwards gets you to the same place as … walking forwards!
i –1 1 –i
k j i
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