SYMMETRY. Reflection about a vertical axis. Looks the same in a mirror.
SYMMETRY. Rotation by 180 degrees.
|Meir Yedid 1998
SYMMETRY. Rotation by 180 degrees.
J. S. Bach 1981
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. 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.
Happy New Year 2001
SYMMETRY. Rotation by 90° turns the first letters of the family member names into HAPPY NEW YEAR.
INSPIRATION. Created for my parents as an annual greeting card.
STORY. Every year since I was in high school I have produced a greeting card for my parents. This card features the names of my parents (Lester and Pearl), my siblings and myself (Scott, Grant, Gail), our respective spouses (Amy, Chilju, Vaughn), and our respective children (Gabriel, Michael & Eliott, Kyra & Liana). Eliott usually goes by his Korean name Han Sol.
I’ve kept family members grouped together, with names listed from oldest to youngest within each family. The only exception is Kyra, who is older than Liana; I couldn’t find a way to list Kyra first. Note that families align neatly with the boundaries of the words HAPPY, NEW and YEAR. Also note that the letters in each name are drawn to match the style of the initial capital.
I produced an earlier version of this design in 1995, in honor of the births of Grant and Chilju’s son Eliott, and Gail and Vaughn’s daughter Kyra. Two years ago Gail had a second daughter Liana, and I had a son Gabriel, so I reworked the design to include two more names. The trickiest part is the overlapping G and L, which make EA.
SYMMETRY. Drawn with a single line on the surface of a cube. The line begins and ends at with the crossbars of the two E’s. Can you trace the entire line with your eye?
INSPIRATION. For Dick Termes, artist who paints on spheres.
STORY. I first met Dick Termes in 1992 at the Math Art conference. I had seen his work in books, but flat pictures hardly do justice to his work. You see, Termes paints images of landscapes and environments on the surface of a sphere, not on flat canvas. Instead of a narrow window onto a larger scene, you get the full panorama. And not just a 360° panorama like the walls of a room. You get absolutely everything in all directions, including sky above and earth below.
I find his borderless paintings wonderfully freeing. Instead of being locked into a single frame of reference, you are invited to look around. You can’t see the whole painting at once, so the act of seeing retains a sense of mystery and discovery. The seamless sphere restores a sense of wholeness lacking in a chopped off rectangular frame.
To produce his paintings Termes invented his own system of what he calls six-point perspective. He paints on tough light plastic spheres originally manufactured for light fixtures. After he roughs up the surface with sand paper he applies gesso and acrylic paint. The final spheres are hung from the ceiling on a motorized rotating mount. He has created over 140 Termespheres since 1969.
For pictures and stories about Termes’ work, visit his web site. He has many items for sale. The spheres themselves are expensive, but he also offers inexpensive paper models that fold into Platonic solids, as well as a book on his perspective system and a video.
When I met Dick Termes I knew I had to do something with his name. Naturally I wanted to draw his name on a sphere. His name has six letters, so I reshaped the sphere into a six-faced cube. I placed the letters in a zig-zag pattern so the whole name can be read by turning the cube around a single axis.
Doodling with his name I discovered that the letters connected well one to the next, so I searched for a way to draw the entire name with a single line. I found working with the entire surface of a cube a wonderfully three-dimensional experience. I was pleased to find that the two E’s provided natural beginning and ending points for the line. Only the R face has obviously superfluous lines. I drew the cube in Illustrator, popped it into three dimensions with Adobe Dimensions, and animated it with Flash.
If you would like to make your own “Termescube”, here is a model you can print, cut out, and fold together.
SYMMETRY. Visual canon by translation. Two copies of the same design, with one translated to the right, overlap to spell JOHN MAEDA. Move your mouse left and right to see how the design combines with itself.
INSPIRATION. For John Maeda, MIT Media Lab professor and pioneering visual artist. His piece Tap, Type, Write appears in the art exhibit 010101: Art in Technological Times.
STORY. With a background in both engineering and graphic design, John Maeda has pioneered a new direction in graphic design that takes full advantage of the algorithmic power of computers. Not content to use canned software packages, he writes his own compact code to create interactive masterpieces like Tap, Type, Write, an homage to the typewriter as seen through the eyes of the digital era. His work is both minimal and humane, sophisticated and child-like, a balance struck by Maeda’s graphic design hero Paul Rand. When I first met him five years ago at the Media Lab he seemed bothered that more artists were not pursuing this direction. By now he seems to have attracted quite a following.You can learn more about Maeda’s work through his web site, and his retrospective book maeda@media.
I’ve admired Maeda’s work ever since learning about him five years ago. When I saw Tap, Type, Write at the San Francisco Museum of Modern Art I wanted to create a design on his name. I could have created a simple rotation inversion, but that seemed inadequate. I wanted to do something more specific to Maeda’s propensity for digital media.
I wrote his name in low-resolution pixels. Then I noticed that the initial J was part of the second letter O. This led quickly to create a canon by translation, a thorny algorithmically-minded technique I have used only a few times in the past. For this sort of design to work, the last letter needs to be entirely contained within the second to last letter — I was pleased this works well with lowercase D and A.
Notice that the first five letters are perfectly split into two disjoint sets of pixels. The pixels of O that are not part of J become part of H, and so on. After M the division is not so clean: the parts that overlap to make the final A, E, and D share pixels. The overall design makes a satisfying arch, with an extra-wide capital M iin the middle, all created naturally by the demands of the symmetry.
Gathering for Gardner 5
SYMMETRY. Word cross with five fold rotational symmetry. Two overlap points: the initial G becomes the final R, and the first R becomes the N.
STORY. Through his writings, Martin Gardner has created a world-wide community that includes mathematicians, magicians, puzzle people, skeptics (pseudoscience critics). In 1992 Gardner fan and puzzle collector Tom Rodgers started the Gathering for Gardner as a way to bring together people who had corresponded with him but never met him. Although Gardner himself was able to attend only the first two G4G installments, the event continues as a wonderfully fertile meeting of minds.
The conference itself is a closed invitational event, but fortunately many of the fruits of the conference are available online at the Gathering for Gardner web site. Here you’ll find links to work by the G4G attendees on such topics as puzzles, recreational math, geometry, mazes, illusions, magic and the fourth dimension. You’ll also find a free downloadable PDF version of The Mathemagician and the Pied Puzzler, a book of articles based on the first G4G event. Every G4G attendee is asked to bring 150 copies of something to share with every other attendee; you can find some of the G4G4 exchange items on the site, including matchstick puzzles, ancient chinese mazes and a mechanical impossibility.
My contributions to G4G include helping to rally the papers for the first book, and designing the cover of the seoncd book, entitled Puzzler’s Tribute, edited by David Wolfe and Tom Rodgers. I also contributed a chapter to Puzzler’s Tribute that displays ambigrams on mathematical and magical names.
I created the logo above for the fifth Gathering for Gardner. Naturally I wanted to incorporate the number five. I thought immediately of a star with overlapping words. Although I have created many such “word crosses”, such as one on “M. C. Escher”, I had never created one where letters overlapped at such peculiar angles.
The challenge in this design was to make the letters appear to be all tilted at the same angle instead of leaning left and right willy nilly. The solution required blobby lettershapes and strokes that could be bent flexibly at different angles.The R/N combination was easy enough, but G/R took some wrangling…and the G still appears to be at a different angle from the other letters.
David Blaine / Mysterious
SYMMETRY. 180° rotational symmetry. DAVID BLAINE upside down becomes MYSTERIOUS.
INSPIRATION. Created for use on David Blaine’s recent DVD/Video Fearless, a compilation of three of his TV specials that is the best introduction to his work. The lettering appears printed on the DVD itself.
STORY. David Blaine is to magic as Yo-Yo Ma is to classical music or Tiger Woods is to golf — a superb performer who takes a venerable art form and makes it hip again, especially for younger audiences. David’s style takes magic off the stage and onto the street, performing closeup magic for innocent bystanders accosted by an ever-present video camera. Some of the tricks are classics, but his streetwise presentation, with t-shirt and menacing eyes, makes it all new again.
David’s hero is Houdini, who also took magic to the streets, creating major public spectacles such as escaping from a straitjacket while suspended upside down many stories above a public street. On the Fearless DVD you can see one of David’s recent public spectacles — standing inside a block of ice for a couple days. In our increasingly digital society such raw physical stunts have even more impact.
David originally contacted me about a puzzle book he was creating, in the spirit of the classic treasure hunt book Masquerade. I was pleased to pass the job of puzzle creation on to Cliff Johnson, creator of the superb computer story-puzzle game The Fool’s Errand. You can download copies of Cliff’s older games on his site, fools-errand.com. David Blaine’s book Mysterious Stranger will be available Oct 29, 2002, and offers a $100,000 prize to the first person to decypher the clues and find the hidden treasure. His fan site is davidblaine.com.
Although most of this inversion works very well, I had to cheat a bit on the initial D of DAVID, which does not correspond to anything in MYSTERIOUS. On the CD I printed it in a paler color. In this animation I have the letter fade out and in as it shuttles between top and bottom.