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July 17 2020

Origami

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. This design was created at the request of David Masunaga, who used it in a talk about origami to illustrate the concept of grafting one construction onto part of another construction. See the previous inversion of the week for the complete story.
The four letters RIGA in the middle of ORIGAMI form the two Chinese characters which are called ORI and GAMI in Japanese. ORI means “to fold,” and is composed of the characters for hand and ax. GAMI means “paper,” and is composed the characters for silk and family.
For comparison, here is the original calligraphic version of origami on which I based my design, written by calligrapher and seal carver Xu Yunshu.

The G and A work rather easily. The biggest liberty was turning the substantial right angled roof on the ax character into the more liquid melting dot on the I. Notice that I repeated the same sort of droopy dot on the other I.
The letters O, M and I are written in the same visual style. I could have written them in purely Roman letters, as I did with Elise Diamond. M and I don’t particularly look like Chinese characters, but O looks strongly like the the word “ko” (“mouth” or “entrance”) in Japanese, so maybe this design looks like someone is eating a piece of origami, or an entry sign at an origami exhibition.

June 24 2020

Superteacher

Superteacher 1998

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.

June 24 2020

Montgomery

Montgomery 1999

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.)

June 24 2020

Meir Yedid

Meir Yedid 1998

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.

June 24 2020

J.S. Bach

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.

June 24 2020

Synergy

Synergy 1981

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.

June 24 2020

Happy New Year 2001

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.

June 24 2020

Termes

Termes

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.

June 24 2020

John Maeda

John Maeda

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.

June 24 2020

Gathering for Gardner 5

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.

INSPIRATION. Created for the fifth Gathering for Gardner, organized by Tom Rodgers. See my other ambigrammatic tributes to Martin Gardner.

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.