SYMMETRY. Reflection about a vertical axis. Looks the same in a mirror. INSPIRATION. Appears in my book Inversions as part of a trio of inversions in tribute to the book Gödel Escher Bach.
STORY. As a pianist, I’ve always been drawn to Bach’s music. I am particularly fond of the canons and fugues in the Well-Tempered Clavier, Musical Offering and Art of Fugue. Canons are similar to inversions — the goal in both cases is to compose an aesthetically pleasing result by following a mathematically precise rule. I have composed a number of canons over the years. Here is a Canon by Augmentation I composed on the theme of the Musical Offering. There are two voices, which start an octave apart. Both voices play the same notes, but the higher voice plays twice as fast as the lower voice. Notice that the higher voice completes two repetitions in the time it takes the lower voice to complete one. The exact symmetry is broken only on the last note. I wrote this canon as a gift to Douglas Hofstadter when I was helping him teach a course based on the then forthcoming book Gödel Escher Bach.
SYMMETRY. Rotation by 180 degrees. INSPIRATION. Created at the Gathering for Gardner, January 1998. STORY. I first heard about magician Meir Yedid when I was a student at Stanford. One of my interests is creating mathematical figures with nothing but your hands. I’ve figured out ways to create tetrahedra, cubes, hypercubes, knots and other shapes with anywhere from two to twelve hands. Stanford mathematician Persi Diaconis told me about a magician in New York who had developed all sorts of tricks using his hands. I wrote to him and got his catalog. I learned about an act he had created called Finger Fantasy, in which he makes fingers vanish from his hand one by one. Years later I met Meir at the Gathering for Gardner, an occasional invitation-only event honoring Martin Gardner. Knowing that he would be attending, I created this design for him. Notice that not only does MEIR turn upside down to become YEDID, both figures appear to be human hands! This is one of the few cases where I’ve been able to work three meanings into one design. You can read more about Meir Yedid’s magic on his web site appropriately named mymagic.com. This is a site for magicians, only, so SHHH, don’t tell how it’s done.
SYMMETRY. Rotation by 180 degrees. INSPIRATION. Created for a dance performance in Montgomery, Alabama. STORY. In November 1999 I traveled with the Dr. Schaffer and Mr. Stern Dance Ensemble to Montgomery, Alabama, where we did a week of performances and talks about mathematics, dance and creativity in local high schools, culminating in a performance for the community. We were inspired by the tremendous support for the arts we saw in schools, not just for privileged kids, but for everyone. We stayed with at the Governor’s mansion, warmly hosted by Governor Don Siegelman and First Lady Lori Siegelman. Lori is a major supporter of the arts in education, and a childhood friend of Karl Schaffer, co-founder of Dr. Schaffer and Mr. Stern. I presented this design in our performance to the Siegelmans. This design worked out rather easily, despite the length of the name. Every part of the design is a combination of letters I have used before. You can use pieces of this design to help you make inversions on the following words and names: MARY, MINT, GO, ONE. By the way, the logo for the computer company GO is a clever inversion using uppercase letters. (No, I did not work on the GO logo.)
America Online 180 degree rotation, 1996 The largest internet online service.
When I first joined America Online in 1991 it was a fledgling community dwarfed by oldtimers like CompuServe and Genie. Now it has grown larger than all its competitors and is the largest internet online service. Despite its size it maintains a notably human-scale, folksy feel, reinforced by founder Steve Case’s honest monthly newsletters.
My relation to America Online shifted from user to developer when I started developing the online board game MetaSquares in 1995, working with Kai Krause and Ian Gilman at the software company MetaTools. While there are many other more specialized online gaming services, MetaTools wanted to reach a broader audience that would appreciate a nonviolent, thoughtful game. Furthermore, MetaTools chief inventor Kai Krause launched his career posting Photoshop tips on America Online. MetaSquares now runs on both Windows 95 and Macintosh versions of America Online, and attracts thousands of players every day. In the future MetaTools plans to release Go, Hex, and other online games.
I created this inversion as a present to the folks at America Online Games for the Electronic Entertainment Expo in May 1996 in Los Angeles. The inversion worked rather easily; the letter combinations are all ones I’ve used before.
SYMMETRY. Reflection about a vertical axis. Looks the same in a mirror. INSPIRATION. Commissioned by Key Curriculum Press, publisher of my book Inversions, for use at the 1999 annual meeting of the National Council of Teachers of Mathematics. STORY. Key Curriculum Press is a leading publisher of innovative high school mathematics books, videos, software and other materials. Outstanding titles include the textbook Discovering Geometry, the software Geometer’s Sketchpad, and the kit Exploring Math Through Puzzles, which includes enough pieces to make 54 puzzles. Every year they dress up their product catalog with a different wacky theme. This year the theme is “Be a Math Action Hero”. Scattered throughout the catalog are such mathematical superheroes as Captain Chaos, Deductiva and Calculadora, drawn in comic book style. For their trade show booth at the annual conference of the National Council of Teachers of Mathematics, they asked me to create a mirror symmetrical version of the word “Superteacher” as a card to give away to conference attendees. Only the left half of the word is printed on the folding card; the right half is a mirror that lets you view the other half of the word, and see the real math action hero: yourself. I usually prefer rotational over reflective symmetry, but this word works well, with the split nicely placed in the middle of the capital T. If you were to wear a shirt with this word emblazoned across the chest, it would read the same in a mirror.
Bach / Faure 180 degree rotation, 1996 Title lettering for a piano concert performed by Margaret Fabrizio in Bombay, December 1996.
Margaret Fabrizio is a wonderful harpsichordist, composer and artist in San Francisco. I had the good fortune to study harpsichord with her when I was at Stanford University and have enjoyed her concerts ever since. My favorite of her compositions are the Holograms: mesmerizing shifting patterns performed on double-manual harpsichord (which lets the two hands play in the same range). In the visual realm, she assembles dense mysterious collages, which she layers in slowly dissolving slide shows. This design was created for an intriguing piano concert she performed in Bombay, India, while staying at the home of an art dealer. The first half of the concert was all Bach and the second half was all Faure. Turn this design upside down and Bach becomes Faure. The two could hardly be more different: Bach, the intricately precise baroque composer, and Faure, the florid lyrical impressionistic French composer. And yet there are connections…both are concerned with texture in a way that might intrigue an Indian ear. To play up the contrast, Margaret dressed in stark black and white for the first half of the concert, and flowing brightly-colored scarves for the second half. Bach is my favorite composer. I enjoy playing his music on the piano, and have composed many pieces in his style, which I hope to post to this site soon. Faure I rarely hear performed, but every time I do I want to hear more. If you want to hear Faure’s music, Margaret recommends Alicia Stott’s recordings of Faure’s complete piano music.
SYMMETRY. 180 degree rotation. Turn this design upside down and each name reads the same both ways.
INSPIRATION. Created during a week-long dance residency in Thousand Oaks, California, April 21-25. STORY. I improvised these two designs on the first names of the daughter and son of the organizer of our residency. Since the two names are related, I tried to make the lettering styles related.
These two inversions illustrate two of the most common liberties I take with letterforms. If there is more than one copy of a letter in the same word, I prefer to make the shapes as similar as possible. In BRIANNA, however, the two A’s are fundamentally different. With names, I prefer correct capitalization: either all capitals, initial capital and the rest lowercase, or if necessary. In JORDAN, however the J is uppercase and ODAN are lowercase, but R is uppercase.
INSPIRATION. In the 1970s I was a big fan of the emerging field of computer graphics, and spent many memorable evenings watching demo reels at Information International courtesy Richard Taylor, who has gone on to a spectacular career in special effects. This was the early days, when computer graphics was just starting to appear in TV commercials. When TRON was in the works — the first feature film to include extensive computer animation — I drew this ambigram and sent it to Richard. (The original was monochorme; this version has been enhanced with the metallic effect of the film’s logo.) My ambigram didn’t make it into the movie, but it did appear on the special effects crew jackets.
(Move the cursor across the image to control its speed and direction.)
Whitney 1996
SYMMETRY. Interactive animation. INSPIRATION. In tribute to computer animation pioneer John Whitney.
STORY. From 1940s to the 1990s John Whitney, sometimes working with his brother James Whitney, created a large body of work as an independent animator. Some of my favorites include Five Film Exercises (1943), which featured mechanically synthesized images and sounds, Celery Stalks at Midnight (1952), a hilariously low tech and very effective visualization of jazz, and Arabesque (1975), his most mature computer animated work. John was quite an inspiration to me. I first saw his movie Permutations when I was in high school. One of the first things I did at college was to program Whitney-inspired kinetic patterns on an old Adage vector graphics computer. After my undergraduate years I worked for a summer writing 3d graphics programs for John to use on a commercial job. (John never did his own programming, he was more tinkering in the analog world.) When I wrote my first book Inversions in 1981, publisher Byte Books also published John’s manifesto Digital Harmony, which put forward his ideas on creating visual compositions with the same sense of tension and resolution present in music. I later learned that Digital Harmonyinspired my friend, virtual reality pioneer Jaron Lanier, to write his first computer program. He later incorporated Whitneyesque imagery into his computer game Moondust (forCommodore 64). I continued to visit John occasionally when I was in Los Angeles. Late in his life he was able to realize his dream of composing sound and image directly together, using a custom PC graphics program driving both a computer monitor and synthesizer. He intended the works to be viewed directly on computer screen. I had the pleasure of seeing the works in his home on one of my visits. Several images stand out when I remember the lovely home he shared with his artist wife Jackie: light reflecting off their backyard pool onto their living room ceiling making patterns much like his art, a huge circular flip book of his computer animation, childhood drawings by his sons (who continue to do pioneering work in film), and his workroom with computer, Tektronix storage tube monitor, 16mm movie camera and movieola.
This animation uses one John’s most basic compositional elements: a simple visual pattern that goes in an out of phase, creating shifting displays of visual dissonance and consonance. Here the H travels at twice the velocity of W, I travels at three times the velocity of the W, T travels at four times the velocity and so on. When letters reach the edge of the frame, they wrap around to the opposite edge. Eventually, when W has traveled the full width of the screen, the letters come back into alignment. Halfway through the cycle the letters coincide in two columns, a third the way through the cycle they coincide in three columns, and so on. These visual harmonies have the simple fractional relationships as the frequencies of harmonically related musical pitches. Similar coincidences happen vertically; I’ve drawn the letters of his name so they superimpose to make a square with four crossing lines. Some of John’s works were programmed by Larry Cuba, an accomplished independent animator in his own right, who has turned his lifelong interest in the history of “visual music” into a nonprofit institution called the Iota Center. The center is currently touring a film program around the country devoted to Oskar Fischinger, the prolific visual music animator who played an important but uncredited role in the creation of Walt Disney’s original edition of Fantasia. To learn more about visual music, visit the Iota Center web site, which includes a video store, links to artists, notices of events, and a mailing list you can subscribe to.
SYMMETRY. Reflection about a horizontal axis. TEACH reflects to become LEARN. INSPIRATION. For the companion book to The Visual Almanac, a multimedia publication from the Apple Multimedia Lab.
STORY. In 1988 I visited the Apple Multimedia Lab in San Francisco. Researchers there were just finishing up the Visual Almanac, a pioneering video diskplus HyperCard stack for teachers that demonstrated some of the ways multimedia could be used in education. Commercial products that have spun out of the Visual Almanac include Wacky Jacks, Countdown, and Planetary Taxi. The second chapter of the companion book to the Visual Almanac talks about the big idea of using multimedia in education. One of the key points is that multimedia can encourage students to become creators of information, not just passive consumers, smudging the line between teaching and learning. Someone familiar with my work asked me to provide a typographic illustration for that concept. I submitted several ideas, and this one was chosen.
The version of TEACH/LEARN published in the Visual Almanac is monochromatic and two-dimensional. The three-dimensional color rendering and animations were done just recently, prompted by Oregon educator Lucy MacDonald, who is an expert at making online education work. The 2d art for this image was produced using Adobe Illustrator. I used KPT Vector Effects to give the letters dimension, and Adobe Photoshop to layer the elements and give them transparency.
180 degree rotation, 1996
