Experiencing Symmetry:

Geometric Transformations in Art, Music and Dance

By Scott Kim, Apr 15, 2011, for the 2011 National Council of Teachers of Mathematics annual meeting, Indianapolis, IN. Also presented as Symmetry, Art and Illusion for the Museum of Mathematics, Apr 7, 2011.


For all you teachers who want to use bits of my talk in your classroom, here are links to the handouts, visuals, music and movies.


Ambigram handout. Superimpose two copies of this handout to complete the half-words. Make two transparencies of this handout to demonstrate the solutions on an overhead projector.

John Langdon's ambigrams

Scott Kim's ambigrams



Palindromic 3rd movement from Haydn's Symphony 47. Score for the Haydn movement. The orchestra plays the first part twice forwards, twice backwards, the second part twice forwards, twice backwards, and finally the first part twice forwards, twice backwards.

Crab Canon from the Musical Offering by J. S. Bach. Didn't show this in my talk. Another good example of a musical palindrome.

Sugar Water. Music video by Michel Gondry for the Japanese singing group Cibo Matto. Not in my talk either. An incredibly intricate visual crab canon that deals with time reversal, analogous in structure to the Bach crab canon. Mildly risque, inappropriate for some students.


Table Music. Duet for two violinsts: one score read from two different sides. Often attributed to Mozart, who almost certainly did not write it. And here's the written out score.

Invertible faces by Rex Whistler.

Upside Downs. Invertible comic strip by Gustave Verbeek.


Can-can (actually called the Infernal Galop) from Orpheus in the Underworld, by Jacques Offenbach, 1858.

The Tortoise from Carnival of the Animals, by Camille Saint-Saens


Binary Hand Dance. One of a series of brilliantly entertaining quirky rapid-paced math-art videos by recreational mathemusician Vi Hart. Not part of my NCTM talk, but was shown in my Museum of Math talk — Vi Hart introduced me by showing this video.

Math Dance. Book and exercises. By Karl Schaffer, Erik Stern, and Scott Kim.

Dr. Schaffer and Mr. Stern Dance Ensemble. Excerpts of choreography.

Canon (movie) by Norman McLaren. You'll see that I only showed the last third of the film. The first two parts build up the ideas that are fully exploited in the last part. By the way, when a character enters upside down, the music is also inverted (high pitches becomes low pitches). McLaren himself wrote the music for the second and third parts.

Norman McLaren (most of his films are also on YouTube, but this is the official site). I particularly like Spheres (spheres in space choreographed to Bach) and Synchromy (lively colorful animation where the picture IS the soundtrack (bands of dark and light with different frequencies and widths make tones of different pitches and loudness).

Come Into My World. A brilliant 4-part visual canon music video by film genius Michel Gondry. I cut it from my talk because of time, but it's really worth seeing, and probably more interesting to today's kids than Canon. Also see the Making of Come Into My World

At the NCTM conference I met the leaders of three interesting math+dance ventures:

Malke Rosenfeld is a professional dancer and teacher who has been doing math/dance integration since 1996, emphasizing percussive dance — rhythmic and spatial footwork that she calls jump patterns.

GeoMotion Group, founded by Dr. Debby Mitchell, creates a variety of dance / math / fitness programs using colorful mats reminiscent of Dance Dance Revolution, but without the electronics. High energy, definitely aerobic.

Math & Movement, by math educator Suzy Koontz, also uses large floor mats to make basic number patterns danceable for young students. Math & Movement mats are large, emphasizing skip counting across large number arrays, whereas GeoMotion mats are more compact, intended to be used in large groups where each student has a mat.

In addition, I should mention that renowned mathematician John Conway, who has done much important work on symmetry, gave dance-step names to the seven "frieze patterns" (repeating patterns on an infinite strip). When he gives talks on the subject he gleefully demonstrates by hopping, skipping and twirling on stage.

And to connect these visual and dancical patterns back to sound, Vi Hart wrote an interesting paper showing how the frieze patterns appear in music.

-- Scott