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For Input/Output, an exhibition of computer art curated by Lucia Grossberger Morales for the Siggraph '85 computer graphics conference. What goes in, must go out! In 1985 the Macintosh computer was new and I was just starting to explore what you could do with short animation loops on the computer screen. The images were drawn in MacPaint and animated in a prototype animation program called Fellini.

SYMMETRY. Reflection about a vertical axis. This image looks the same when seen in a mirror. INSPIRATION. Created for the Nueva School, a private prekindergarten through eighth grade school, founded in 1967 in Hillsborough, California. STORY. I have long been a fan of this outstanding school. Having been appalled by my own grade school education -- although excellent by conventional standards -- I like being around a school that shows me a better model. The various parts of the school feel more like university departments, with devoted teachers, remarkable curriculum, and a strong sense of making every aspect of the school better every year. Very high parent involvement is certainly a big part of the school's success, as it is with most outstanding schools. A high school teacher of mine who received students from many different intermediate schools including Nueva reported that students from Nueva were more self-confident and much less likely to take a statement as true just because an adult said it. Reports from friends who went to Nueva are that it can be hard to enter conventional high schools after attending Nueva. I am especially interested in the math teachers at Nueva, who include Mary Laycock and Peggy MacLean. Their well-developed methods around using physical manipulatives in the classroom result in an entire school population that takes math confidence for granted. I hope to bring some of their thinking to software one day. This mirror image design, inspired by one of my first visits to Nueva, has been popular on t-shirts, tote bags and other items sold by the school.

For Solomon Golomb, mathematician and inventor of pentaminoes. Created for a presentation to MathCounts, a national junior high school mathematics competition, May 10, 1996. If two squares side by side is a "domino", then n squares joined side by side to make a shape is a "polyomino", an idea invented by mathematician Solomon Golomb of USC. There are two distinct "trominoes" (three squares): a straight line and an L. There are five distinct "tetraminoes" (four squares), popularized in the computer game Tetris, which was inspired by Golomb's polyominoes. Shown above are the twelve distinct pentaminoes -- shapes made of five squares. There are dozens of games you can play with pentaminoes, like trying to arrange them into a 5 by 12 rectangle, 6 by 10 rectangle, or 3 by 4 by 5 solid. A two-person pentamino game was filmed for the movie 2001, but was cut in favor of chess. 2001 author Arthur C. Clarke later incorporated pentaminoes in his novel The Fountains of Paradise. If you are interested in purchasing a set of pentaminoes to play with, check out the online puzzle store Puzzletts. My favorite pentamino sets are ones made of 3-d cubes, not just flat squares, since they can be stacked into three-dimensional shapes as well as flat shapes. There are twelve pentaminoes and six letters in "Golomb", which leads to the nice challenge of spelling "Golomb" using just two letters to make each letter shape. First I worked on the "O"s. I wanted both shapes to be the same, and to have at least mirror symmetry since they couldn't both have rectangular or square symmetry. The long piece obviously belonged with the "L", and the zig-zag piece with the "B". Making a convincing "M" was rather difficult. Finally I used the remaining pieces to make a "G", probably the weakest letter. Solomon Golomb is a prolific inventor of interesting bits of recreational mathematics, including Rep-tiles (shapes that can be dissected into several smaller copies of the original shape) and Golomb's Ruler (if a ruler has markings only at 1, 3, 6 and 7 inches, it can still measure every integer distance from 1 to 6 inches in length). You can read more about polyominoes in Golomb's book Polyominoes, published by Princeton University Press.

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. Reflection about a vertical axis. INSPIRATION. Created for my book Inversions. STORY. When I wrote Inversions, I needed to fill out my quota of sixty words. One of the subjects I chose was symmetry. Besides MIRROR, I also did inversions on UPSIDE DOWN and SYMMETRY.     Naturally I wanted to write MIRROR in mirror symmetry. Note that the M reflects into the three letters ROR, and that the centrally placed dot bonds with either of the two I's. Also notice that the place where one stroke passes under another at the top of the O separates the second R from the O. The lettering, influenced by the calligraphic style called Fraktur, helps rationalize the odd shapes.

SYMMETRY. Rotation by 180 degrees. INSPIRATION. Designed as part of my book Inversions. STORY. I studied classical piano for as a child and majored in music in college, so when I wrote my book Inversions I naturally thought about using composer names as subjects. Mozart seemed particularly appropriate because I had seen in Martin Gardner's writings a copy of a piece of music he wrote that was intended to be turned upside down. The full story, including a copy of the musical score, is described in Inversions. This name works rather easily. My main goal in executing the lettering was to capture something of the delicate elegance of Mozart's music in the way I drew the curved, tapered lines.

SYMMETRY. Reflection about a vertical axis. This image looks the same when seen in a mirror. INSPIRATION. Created as part of a puzzle for the March 1993 issue of NewMedia magazine. STORY. For the past seven years I have written and illustrated the puzzle on the back page of NewMedia magazine, the largest trade magazine for the computers and multimedia industry. The puzzles are great fun to create, and give me a chance to learn about different aspects of the industry, try new pieces of software, make up puzzles and art, and correspond with readers. In March 1993 I created a puzzle called "Filter Fantasies," which you can find reprinted in my book the NewMedia Magazine Puzzle Workout. The puzzle involved applying filters from Kai's Power Tools, a plug-in for Adobe Photoshop, to an image. Applying the filters in different orders created different results; the puzzle was to figure out the filter order for each of six resulting images. The idea for this inversion had been brewing in my mind for many years when I finally decided to include it as part of the Filter Fantasies puzzle. I refined the design the way I usually do: make many sketches in pencil, scan my favorite into Adobe Illustrator, and tweak the design in Illustrator until the curves are just right. The shapes are influenced by the work of fantasy artist Roger Dean, who has a special flare for inventive organic lettering. Dean is a favorite of KPT's creator and namesake Kai Krause, who I had met a couple years earlier. Bits of fantasy symbolism crept into the lettering: the central T strong suggests a sword, while the initial F rears back like a flaming dragon in order to make the final Y.

Created for the Gathering for Gardner, January 1996, Atlanta Georgia. Martin Gardner books and magazine articles about mathematics and science have charmed several generations of readers into careers with a mathematical twist. His Scientific American column Mathematical Games, which ran for 25 years, inspired my own career as a puzzle designer. In January 1996 the second Gathering for Gardner took place in Atlanta Georgia. Over a hundred magicians, mathematicians, skeptics of psychic phenomena (Gardner is a major player in the Committee for the Scientific Investigation of Claims of the Paranormal) and other mischief makers entertained each other for several days with lectures, performances, and heated shmoozing. One of the events was a round robin exchange of small gifts made of paper; mine included a series of four inversions on "Martin Gardner". The first inversion reads the same upside down. Shown below is an earlier version of the design, which appeared in my book Inversions in 1981. Notice that the central letter of the design is G. In the new improved version, I used an R with one vertical stroke instead of two, which shifts the visual rhythm of the lettering over by one stroke, so that G is no longer the central letter. Not only are the three R's now consistent, MARTIN can now turn cleanly into GARDNER. Revisiting the name was interesting challenge. Hold a mirror horizontally just below the second inversion and you will be able to read both MARTIN and GARDNER at the same time. The third inversion turns Gardner into his adventurous alter ego Doctor Matrix, who would sometimes invade Mathematical Games with tales of mathematical intrigue. Finally, the fourth inversion takes advantage of the fact that both Martin Gardner and Mathematical Games have the same initials to create what Douglas Hofstadter an "oscillation": the oscillation between one reading and the other takes place in your mind.

SYMMETRY. 180 degree rotation. Turn this design upside down and it reads the same both ways. INSPIRATION. Inspired by a performance by Luanne Warner, marimba, and Mary Chun, conductor with the San Francisco Concerto Orchestra of Tomas Svoboda's Concerto for Marimba and Orchestra. February 16, 1997, Angelico Hall at Dominican College, San Rafael, California. Marimba by Ron Samuels of Marimba One. STORY. There certainly have been a lot of marimbas in my life recently. My wife Amy Jo Kim, who designs online environments, recently joined the techno-tribal music group D'Cuckoo, which performs on electronic marimbas that trigger prerecorded samples. A. J. is subbing for marimba virtuoso Luanne Warner, who also plays percussion with the San Francisco Opera Orchestra and the Women's Philharmonic in San Francisco. Tonight we drove to San Rafael to hear Luanne perform a modern marimba concerto which proved to be a movingly beautiful performance. The piece features an unusual quintet within the orchestra of piano, harp, celeste, orchestra bells and crotales, most of which are tuned percussive instruments like the marimba. Afterward we chatted with Ron Samuels, a marimba maker from Arcata, California, who made Luanne's fabulous five-octave instrument. His company Marimba One builds custom marimbas for people all over the world. Recently two west African marimba masters moved into Ron's home town of Arcata. Although popularized in Latin American music, the marimba and the word "marimba" originally come from west Africa. Marimba music seems to be enjoying a surge in popularity. Marimba bands are becoming popular in the Pacific Northwest. Then there's the Java software company Marimba, founded by Kim Polese from A. J.'s alma mater Sun Microsystems. I met Kim through Amy as Java was gaining popularity and Kim was making plans to leave Sun and launch a new company. A canny marketer, Kim came up with the names Java and Marimba.

For Ned Kahn, an artist who works with fluid flow, clouds of gas, and other natural phenomena. Ned Kahn has created some of the most intriguing art/science exhibitions at the Exploratorium science museum in San Francisco, including a tornado vortex of swirling mist that you can put your hand into. His recent installation at the Yerba Buena Gardens Center for the Arts in San Francisco, called Breathing Sky, created a fog that hung in the air outside the center, which responded to the whims of the weather. For this design I wanted to show the amorphous flowing quality of his work.

Animated at the Geometry Center, University of Minneapolis, Minnesota. I have always been interested in making images that help people understand mathematics. During a residency in the art department at Princeton University I met the charmingly wild mathematician John Conway and the visual topologist Bill Thurston, who now heads the Mathematical Science Research Institute in Berkeley, California. They were teaching an inspired mathematics course for undergraduates that taught cutting edge geometric and topological ideas through the sorts of hands-on experiences usually found only in grade schools. The course was wildly popular, with students who might otherwise shun mathematics clamoring to get in. Bill was at the time also working on a movie called "Not Knot" at the Geometry Center at the University of Minnesota, Minneapolis. Later that year I talked my way into doing the title sequence for Not Knot. The title comes from the idea of visualizing the structure of the negative space around three interlocked loops, in other words, the space that is not the knot. The alliterative title suggested that I make the first word be the shadow of the last three letters of the second word. Every letter here is a knotted closed loop. In fact they are all topologically the same basic overhand knot. I originally wanted the letters to morph into identical knots to make the similarity apparent, but found it was hard enough to get the letters to rotate rigidly into position. The 15-minute movie visualizes some mind-bending recent ideas in the topology of knots. The movie ends with a positively trippy flight through a wildly fisheyed gridwork animted by mathematician Charlie Gunn, whose high animation standards had been shaped by time spent at Pixar. The movie is a treat to watch whether or not you follow the mathematics. While there are holes in the logic of the film, it is certainly a courageous first attempt to bring current mathematics to a wider audience. I animated the sequence in SoftImage on a Silicon Graphics workstation, coached by veteran computer animator Delle Maxwell, who I had met years earlier when she had worked at Pacific Data Images in Sunnyvale, California. Under her guidance I learned just how much work goes into lighting and animating 3-d models. My experience with then new SoftImage would prove useful later that year when I produced images for the George Coates Performance Works theater piece "Invisible Site", and years later designing the intricately 3-dimensional opening space of the computer game Obsidian.

Created at Mathcounts 1996, a mathematics competition for junior high school students. "Onomatopoeia" refers to a word like "boom" or "cuckoo" that sounds like what it means. The word comes from "onoma-" for "name", and "poi-" for "make". This inversion has 180 degree rotational symmetry: turn it upside down and it looks exactly the same. I created this inversion on the spot in response to a challenge from the audience at a talk I gave. Other long words I have improvised as inversions during talks include humuhumunukunukuapuaa (the state fish of Hawaii), and supercalifragilisticexpialidocious (the longest word you ever heard, from the movie Mary Poppins). This particular inversion works rather easily: the round O's and A's coincide nicely. The extra loop on the first O is optional, but necessary on the third O, to keep it from reading as a U.