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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.

SYMMETRY. 180 degree rotation. Turn this design upside down and it reads the same both ways. INSPIRATION. Created for a performance by the Dr. Schaffer and Mr. Stern Dance Ensemble, May 15, at Peery's Egyptian Theater, Ogden, Utah. STORY. The Egyptian Theater in Ogden is something of a landmark, a beautifully restored vintage theater. It is only one of a half dozen theaters in the United States that features an "atmospheric" ceiling: for three minutes before every performance the ceiling puts on a light show that makes it appear to change from a morning sky to an evening sky, complete with sunset and twinkling stars. Ogden is home to Weber State University, where Dr. Schaffer and Mr. Stern Dance Ensemble founder Erik Stern teaches dance. This was our first major public performance in Erik's home town. In addition to one public performance, we also gave four performances to schoolchildren bused in from surrounding areas. I took a look at some Egyptian writing before drawing this inversion. Some of the design elements I worked in include the snake on the E, the long triangular cuneiform crossbars of the E and T, and the pyramidal dot on the I.

SYMMETRY. Rotation by 180 degrees. This design looks the same when turned upside down. INSPIRATION. Created for the Stanford Computer Science Department's 25th Anniversary, November 9-11, 1990. Posted on the occasion of my 20th undergraduate reunion. STORY. Stanford is my alma mater, both for undergraduate schools, where I majored in music, and graduate school, where I did an interdisciplinary PhD called Computers and Graphic Design. Throughout my years at Stanford the Computer Science Department was my main home. This design was commissioned by Carolyn Tajnai, who worked first with the department and later with the computer science forum, which is a liaison between the school and industry. Schmoozing with Carolyn over the years was certainly a big part of what made the department feel like home. The lettering style imitates the bold geometric outlined lettering so often associated with university names. Notice how the crossbar of the F turns into a natural extension of the tail of the A. The trickiest part was turning the S into the D, which I rationalized partly by extending both strokes to make the border.

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. Rotation by 180 degrees. INSPIRATION. Commissioned by Michael Morton on the occasion of the wedding of his brother Evan Wakefield Morton to Rebecca Anne Derby. STORY. I first became acquainted with Michael Morton (a.k.a. "Mr. Machine Tool" and "Harmonic Motel") when I sent in my registration for his shareware anagram program Ars Magna ("great art" in Latin, also an anagram for "anagrams"). We maintained an amiable email exchange, occasionally reporting choice anagrams to each other. About a year ago Michael asked me if I could design an inversion for his brother's upcoming wedding. After many unsatisfying attempts to combine the couple's first names Evan and Becky, we decided to break the task into two steps. First I would give them a pair of inversions on just Evan and just Becky which they would use on their wedding invitation. Then at the wedding Michael would unveil an engraved glass plate with the first names in the middle and the full names around the edge. Inverting the first names was easy; inverting the full names took more effort but worked rather well. We settled on three repetitions of the names around the circumference, which makes it more obvious that the names are meant to be turned upside down than if there had been two or four repetitions. Michael lives in Hawaii and my glass engraver has since quit doing engraving, so I recommended that he contact a local high-end department store and see what engravers they used. Immediately after my suggestion, a story appeared in the local paper talking about how local craftspeople were having trouble finding business. The person they profiled just happened to be a glass engraver! Problem solved. As usual, I did my original lettering on paper, scanned them into the computer, and traced them in Adobe Illustrator. To curve the lettering around the edge I drew the names straight then distorted them with Letraset Envelopes, a plug-in for Illustrator. The little 6-lobed asterisk that separates copies of the names is made out of the top diagonal stroke in Becky.

SYMMETRY. 180 degree rotation. Turn this design upside down and it reads the same both ways. INSPIRATION. Created on the occasion of my father's 70th birthday. STORY. Dr. Lester Edward Kim, with whom I share a middle name as well as more than a few genes, was born in California of parents who came from Korea. Raised in the family restaurant with his brothers and sister, he found his calling in college to become a Presbyterian minister. Realizing that his race might limit his options, he steered his work toward the fledgling field of pastoral counseling, which is not as it may sound counseling in the pastures but rather counseling in the context of churches. He recently started the Good Samaritan Counseling Center in Southern California. He is a devoted social activist, having stirred things up in the areas of fair housing, human rights, and peace-making. "Comfort the afflicted and afflict the comfortable". The dot on the I, which is shared by both the right side up and upside down I's, is a symbol from the center of the Korean flag. I've continued the red and blue colors into the lettering. Like the circle divided into two parts, Korea remains a divided country, but as the reunification of Germany has shown, unthinkable gulfs can be bridged.

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.

Title lettering for book published by Byte Books, a division of McGraw Hill, currently out of print. This inversion has become my signature image, literally. For my book Inversions I wanted the title to turn into my name. Titles have to be very legible, so I worked hard and went through dozens of drafts before I arrived at this solution. I also use it on my business card and many things I do. A few things to notice. The calligraphic style, with its generous serifs, helps mask some of the compromises I had to make in the letter shapes. Notice how the serifs at the bottoms of the I and N at the beginning of "Inversions" are essential for making the M at the end of "Scott Kim". The O and N at the end of "Inversions" run together to make what is called in typeface design a ligature. All the capitalization is correct, an ideal I strive for but do not always achieve. Often inversions end up mixing upper and lower case.

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.)

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 disk plus 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 runs online class for educators on how to use multimedia and the internet in their teaching. Lucy had seen the TEACH/LEARN design and wanted to use it in her work. Her request prompted me to make this 3d color version, which I hope to include in a set of full-color Inversions posters later this year. 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. You can buy a poster of Teach-Learn at my Cafe Press store.