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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. Rotation by 180 degrees. Turn this design upside down and you will see that it reads the same both ways. INSPIRATION. For Ultima, fantasy role-playing computer game series. STORY. In the late 90s my wife Amy Jo Kim was obsessed with Ultima Online, an online fantasy role-playing game developed by Origin Systems and published by Electronic Arts. Ultima is an elaborate graphical online world where thousands of players meet, go on adventures together and live out alternative lives in a mythical land of sword and sorcery. Ultima Online is based on the best-selling series of Ultima games which started on the Apple II and has grown into a veritable cult. Ultima Online is the first online multiplayer installment of Ultima. Amy began playing Ultima as part of her research for a book she was writing, Community Building on the Web. The book lays out design principles synthesized from her years of online strategic design consulting for sites like Ebay. Although many people have built particular online communities, Amy has a uniquely broad perspective, having worked with dozens of major clients in both entertainment and business. When Amy began playing Ultima Online I knew nothing about the game other than its name. Now I have heard her reports on hours of conversations with the folks at Origin, including Origin founder Richard Garriott, dozens of players, and assorted industry pundits. Although the Ultima universe is not my alternate reality of choice, I am fascinated by the thought that went into building it. Besides the usual difficulties of building software and maintaining an online service, Origin has essentially taken on the social challenges of running a city. The relationships that players form in Ultima are very real: for at least one player who moves cities often, her Ultima play partners are the most stable friendships in her life.

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.

SYMMETRY. Reflection about a vertical axis.INSPIRATION. Commissioned by high school mathematics teacher David Masunaga for a talk about origami, and published in Peter Engel's book Origami from Angelfish to Zen.STORY. In April 1988 I attended the 25th anniversary reunion of the Design Science program at Harvard University's Carpenter Center for the Visual Arts. Although I hadn't actually attended the program, let alone Harvard, I felt such kinship with this merry band of artist-mathematicians that I showed up anyway. The program, founded by Harvard professor Arthur Loeb, is a uniquely interdisciplinary of 3d polyhedral geometry and sculpture. Equal parts art, mathematics and engineering, the program has produced one of the most interesting collection of alumnae I've ever seen. Talks at the event included: -- Peggy Weil talking about her vision of the Weatherium: a multi-story inverted polyhedral globe that you view from a platform floating on the inside, with a live projected image of earth's surface as seen from weather satellites. -- Amy Edmundson, who wrote a most illuminating book about Buckminster Fuller's geometry called A Fuller Explanation, giving a hands-on workshop on building structures out of toothpicks and marshmallows. You quickly learn that in order to build anything stable you have to use triangulated structures like tetrahedra. -- A talk about radomes, which are geodesic domes built to protect radar dishes from high winds. Geodesic domes are strong, light, easy to assemble and require only a few distinct shapes. Unfortunately, the nearly parallel bands of creases across the surface of a geomesic dome tend to interfere with the waves being received. A more randomly constructed dome would not have that problem, but would be much harder to build and require manufacturing many more different shapes. To solve this problem, the various pentagonal nodes spaced around the surface were all twisted slightly in whorls, an operation that Buckminster Fuller dubbed "jitterbugging." The complexity of assembly and number of distinct shapes stayed the same, but the added irregularity cleared up the reception problem. As a lover of polyhedra, I've always been fond of the jitterbug transformation, but never suspected that it had a practical application. To learn more about Buckminster Fuller's Geometry, check out the Design Science Consortium, which was started by the same people that for many years ran the Buckminster Fuller Institute. -- High school mathematics teacher David Masunaga and mathematician-architect-writer Peter Engel talking about the mathematics of origami. I already knew David because he had brought me to Hawaii to give talks about inversions to mathematics students throughout the islands. Peter, whose mathematically virtuosic constructions put him at the forefront of origami, I met for the first time at this event. David Masunaga teaches at Iolani School in Honolulu, Hawaii. He continues the design science tradition by filling his classrooms with stunning polyhedral models built by students. Ever the resourceful teacher, David challenged me to create an inversion on "origami". Since origami is the art of paper folding, I created a design that illustrates the symmetry of folding. I also created a bilingual version that imbeds "origami" in Chinese characters inside "origami" in English (this inversion will be posted next week!). David showed these inversions in his talk, taking advantage of the way an overhead transparency can be manipulated. After the talk, Peter Engel asked if he could use the designs in his forthcoming book "Folding the Universe", which was re-released by Dover Publications as Origami from Angelfish to Zen" I cleaned up the designs a bit and created the version you see here. The Folding Universe is a unique combination of origami instruction and mind-expanding philosophy...sort of the Goedel, Escher, Bach of paper folding. For a wonderful portrait of Peter and his art, see Discover magazine, June 1988. For photographs of origami folds by Peter Engel and others, see John Paulsen's origami menagerie. For more information about Peter Engel's book and other origami supplies, visit Fasinating Folds: Origami and the Paper Arts. I generally prefer rotational symmetry to reflective symmetry, but in this case reflection was clearly appropriate. The hardest part of this design was figuring out how to turn the two letters I, with their dots, into parts of oher letters. The G/A combination in the middle is a slam-dunk, except that the two letters are of different cases.

SYMMETRY. Tessellation with two 90° centers of rotation. INSPIRATION. Inspired by the work of R. Buckminster Fuller, STORY. R. Buckminster Fuller, best known as the inventor of the geodesic dome, was an advocate of doing more with less. His watch word was “synergy” — the behavior of a whole not predicted by the behavior of its parts. I am particularly fond of his interest in tensegrity figures (available in toy form as Tensegritoy), gravity defying constructions of sticks and strings which were originally developed by sculptor Kenneth Snelson. You can find out more about Fuller’s work through the Buckminster Fuller Institute. This design practices synergy in two ways. First, the word crosses itself four times at two different types of junctions: S becomes Y and E becomes R. Second, letters are joined in pairs, reducing the number of modules to just three. I was pleased to find that this pattern fills a grid without leaving any gaps.

SYMMETRY. 180 degree rotation. Turn this design upside down and it reads the same both ways. INSPIRATION. Created for a week of performances April 21-25, 1997, by the Dr. Schaffer and Mr. Stern Dance Ensemble, at the Civic Arts Plaza's Forum Theater in Thousand Oaks, California in Ventura County. STORY. This week we will be performing nine times at the same theater: two shows a day for school children Tuesday through Friday, and a public evening performance Thursday night at 8pm. For details, see our performance schedule. This is one of the easiest inversions I've done in a long time. All the letter combinations are old standards. You can use the same letterforms to make inversions on the words RAVE, VERA or RUNE. I frequently use this lettering style, which is based on the italic handwriting most often taught in calligraphy classes, because it is elegant, legible, and very malleable.