Dance and Mathematics Karl Schaffer MATH Dr. Schaffer and Mr. - - PowerPoint PPT Presentation

Dance and Mathematics

Karl Schaffer

• 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

(workshop with Leon Harkleroad Bowdoin College lharkler@bowdoin.edu)

MATH DANCE

Erik Stern

Weber State Univ.

Scott Kim

Puzzle Designer

Karl Schaffer

Clap Your Name

Clap = Consonant Slap = Vowel

R

• b

e r t a

Schlafli symbol {7/4} = {7/3}

Ludwig Schlafli, 1814-1895

Roberta vs. Joan

J

• a

n n e

{6/2} = {6/4} Joanne vs. Joan

Mystic “heptagram”

• for warding off evil
• adopted for some Sherrif’s stars.

{7/2} 1 2 3 4 6 7 5

Poinsot Stars Louis Poinsot (1777-1859) Thomas Bradwardine (1290-1349) Christian symbol 3 interlaced triangles. {9/3}={9/6}

18th century Netherlands

{7/3}

1 2 4 6 7 3 5

Coat of arms of Azerbaijan

{8/3}

{7/3} and {7/4} look alike, why?

Classroom questions

{6/2}

{7/2}: one continuous strand {6/2}: two strands

{7/2}

(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

Language (vowels, consonants) Sound, movement Geometric representation

Power of mathematics

Cultural connections

…and dance, and language, and culture!

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 Cartwheel 1/2 Somersault 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 Opposite Opposite Opposite Same Same Opposite

• Translation (Slide)
• Reflection (Mirror, Flip)
• Rotation ( , Turn)
• Glide (Footsteps)

180

Combining Symmetries

T G M

R

T G

M G

R

First symmetry Second symmetry

p q d gives p d

M

Mirror Rotate Glide

T G M R G T R M M R T G R M G T

The Klein four group Z2 × Z2,

T G M R T G M R

Bilateral symmetry

Mirror

• r

(op

• ppos
• site

direction

• ns)

Translation

• n

(same me direction

• n)

Rot Rotation

• n

(op

• ppos
• site

direction

• ns)

Glide Glide (same me direction

• n)

face to side face to side face each other face same way

1 2 3 1 2 3 1 1 2 3 2 2 3 1 3 3 1 2

First turn Second turn

Turn table

1 i –1 – i 1 1 i –1 – i i i –1 – i 1 –1 –1 – i 1 i – i – i 1 i –1

−1 Powers of i =

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

• pposition, sometimes

called “helical symmetry”

• r “screw rotation.”

www.nordicwalker.com

Front/back silhouette

www.jupiterimages.com

Momix

Resemblance in shape and motion between the arms and the legs displayed to the sides, as in a cartwheel (reflection in the horizontal or “transverse” plane -

• r also in the “sagittal” front-to-

back plane).

(2,3) (2,–3) (–2,3) (–2,–3)

Reflection in x-axis Reflection in y-axis 180 degree rotation

(2,3) (2,–3) (–2,3) (–2,–3) (–3,–2) (3,–2) (3,2) (–3, (–3, (–3,2)

Eight Square Dancers

Dihedral group D4 of order 8

1 2 3 4

Quarter turn

• 3
• 2
• 1

1 2 3

• 3
• 2
• 1

1 2 3

± e(± x) and ± ln(± x)

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

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? Change-Ringing

1 3 2

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

• rder, 123, by switching two neighbors at a time?

1 3 2 4 What about four people?

123 132 312 321

2 and 3 switch places Switch 1 and 3

231 213

The six switches for change-ringing 3 bells

1234 1243 2134 1324

Switch 3 and 4 Switch 1 and 2 Switch 2 and 3

1234 1243 2134 1324

Switch 3 and 4 Switch 1 and 2 Switch 2 and 3

2143 2314 1423 3124 1342

Each permutation connects to 3 others by neighbor switches.

Switch 1 and 3 Switch 1 and 3 Switch 2 and 4 Switch 2 and 4 Switch 3 and 4 Switch 1 and 2

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

• f 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 1 End with 3 End with 1 End with 4 Start with 3 Start with 4 Start with 2

Each hexagon starts

• r ends with same

element.

Each hexagon starts or ends with the same element.

“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. Change-Ringing

Tap-Clap-Bump-Snap Tap-Bump-Clap-Snap Tap-Bump-Snap-Clap Tap-Snap-Bump-Clap Tap-Snap-Clap-Bump Tap-Clap-Snap-Bump

Bulgarian Women’s Dance

Circles in Dance

Circular floor patterns

Body circles

pirouette: vertical axis somersaults or flip: left-to-right axis cartwheel: front-to-back axis

Circular Phrasing

Prop Circles

Poi fire dance

Photographer : Jean-Romain PAC

Costumes

72

Arm Circles

Positive / Negative

(+) (-)

Turning around and walking backwards gets you to the same place as … walking forwards!

i –i 1 –1

i k j

Quaternion Group

• f order 8

1 i j k – i ... 1 1 i j k i i

• 1

k

• j

j j

• k
• 1

i k k j

• i
• 1

– i ...

An object fixed by a belt to a central point takes two rotations to return to its starting position. Philippine Wine Dance Dirac Belt Trick Feynman Plate Trick Louis Kauffman’s Quaternionic Handshake Air on the Dirac Strings (film, 1993) The special unitary group of unit quaternions SU(2) “double covers” the special orthogonal group of rotations SO(3)

Twirl Excerpt (2005)

• Short diversions
• Choreography and

performance

• Extended mathematical

explorations In the classroom:

Dance

• Body mechanics
• Aesthetics
• Performance
• Culture and history

Available at Mathdance.org

An invitation: Fall 2012 or Fall 2013 Make a mathematical dance for yourselves or your students