In August of 1950, when I was just a little tyke and my sister Laura but a babe in arms, our family set out in our 1947 Kaiser from Princeton, New Jersey, for parts West. We were moving out to California so that my dad could take up a new position in physics at the almost-unknown institution named Leland Stanford Jr. University. En route, we passed through many states and innumerable gas stations. I loved the smell of gasoline when we filled up, and was fascinated by the logos of the many different brands of gas. One day, as we were passing through Ohio, my dad pointed at the sign of the Standard Oil of Ohio station where we had stopped:
The “SOHIO” logo that my dad showed me could be read upside down.
He offhandedly commented that if you twisted your head around, you could read it upside down. He even said it out loud: “Oy-hose.” What a silly-sounding, meaningless word! I practically split my sides. “OIHOS” was the funniest thing my 5-year-old self had ever heard. It was also the first ambigram I had ever seen.
This essay is an adapted excerpt from Hofstadter’s new book.
Well, actually, it wasn’t a true ambigram, but it was a close cousin to one. Let me explain. By ambigram, I mean a piece of writing expressly designed to squeeze in more than one reading. The etymology combines Latin’s ambi, meaning “two-sided,” with Greek’s gram, meaning “piece of writing”—and thus, if you look at an ambigram one way, it says one thing, and if you look at it another way, it says another thing (or possibly the same thing)—and deliberately so. A true ambigram is intentionally designed so as to have that Janus-like property. Since it’s unlikely that the creator of the SOHIO logo had the nonword OIHOS in mind as a rotated reading, I hesitate to call it a true ambigram, although, as my dad keenly observed, it had two pronounceable readings.
When you engage in “ambigrammia” (the act or art of producing an ambigram), you are not so much creating something new as discovering something old—or rather, something timeless, something that already (sort of) existed, something that could have been found by someone else, at least in principle. Ambigrammia is thus neither fish nor fowl, in that it floats somewhere between creation and discovery.
Let me spell this out a bit. Some ambigrams, when you see them, make you think, Oh, that was such an obvious find. A triviality! Anybody would have seen that possibility a mile away. Those are discovery-type ambigrams. Other ambigrams, though, make you wonder, How on earth did anyone ever dream this up? What kind of a mind could have created this? Those are creation-type ambigrams. And then there are ones that lie in between those two extremes.
Over the years, my passion for ambigrammia has given me many insights into creativity and what I call “discoverativity” (a proneness to making discoveries) and how they are linked. Aside from those two “-ivities,” ambigrammia also involves “explorativity” (a passion for probing unfamiliar terrain), “manipulativity” (a bent for tweaking things), and “projectivity” (the ability to imagine how others see things).
Once in a while I write Ambigrammia with a capital A, using it as a proper noun, almost as if it were a place, a realm, a territory, a world—for indeed, Ambigrammia is a microcosm inhabited by, well, Ambigrammists, of course. It’s time that I exhibited some more ambigrams to make all of this more concrete.
Like SOHIO, this design has a second reading if you rotate it by 180 degrees. Do so, and you’ll see that it still says the same thing. It’s not that the word ambigram is a palindrome, of course, nor that it is intrinsically symmetrical. Rather, it was forced to be symmetrical by the act of distorting its letters.
Here’s another example:
This one doesn’t have rotational symmetry; it has mirror symmetry. Reflect it, and it will look identical. Once again, the word doesn’t naturally have this property; it was forced to be symmetrical by that sadistic letter-abuser, Douglas R. Hofstadter.
My first ambigrams were drawn in the mid-1960s (but that name for them only came along 20 years later). I was following in the footsteps of my friend Peter Jones, but neither of us tried to make our ambigrams graceful; for us, they were just awkward-looking jokes with letters, and we only did a few dozen each and then ran out of gas. Below is Peter’s rendering of the university that we were both attending at that time:
It’s pretty gawky, verging on the illegible, but this kind of thing amused us to no end.
In the mid-1970s, I met Scott Kim, who had independently come up with the same idea as Peter had (but roughly 10 years later), and his ambigrams (or “inversions,” as Scott called them) were incredibly graceful. A simple example is shown below.
I was amazed and even intimidated by the beauty of Scott’s creations (or discoveries), and it took me several years to overcome my shock and to start drawing my own artistic ambigrams. (It was in late 1983 that I coined the word ambigram.)
Usually, an ambigram takes me about an hour from the moment of tackling the challenge with crude pencil sketches until the final artistic product has been rendered in felt-tip pen, in full color. Here’s a typical example, based on the name of my daughter. Producing it took about an hour altogether.
But some ambigrams come far more quickly, and many come far more slowly. Sometimes it takes a week or two before I find a satisfying solution to a difficult challenge! Altogether, I’ve designed about 5,000 ambigrams, and I’ve probably devoted 10,000 hours of my life to ambigrammia.
I very much enjoy the ambigram below, which I designed on the name of a great Russian composer, as it’s a visual pun—namely, the three orange circles can be seen as representing his suite entitled The Love for Three Oranges.
In case you couldn’t make it out, the composer in question is Sergei Prokofiev. And I’ve made ambigrams on the names of dozens of other composers as well.
Below is a 90-degree-rotation ambigram in honor of a great Baroque composer, known particularly for his fugues.
When I first met Scott Kim, he introduced me to the recherché musical notion of “canon by retrograde inversion,” which means that if you turn the score around by 180 degrees (thus reversing time while also turning upward melodic jumps into downward ones, and downward ones into upward ones), it remains unchanged—thus it is a musical ambigram. Scott showed me such a canon for two violins that is attributed to Mozart, and I, thereby inspired, wrote my own musical ambigram for piano, of which the first line is given below (so if you rotate it by 180 degrees, you get its last line).
One line of a four-line ambigrammatical piece of music written by the author in 1976.
A few years ago, I yearned to draw a map of my natal town. I don’t recall the spark that lit the fire, but it launched me on an epic trek, riding five horses at the same time! Actually, the horses were merely burros, and to be honest, the burros were just boroughs. But no matter! Off I set on an epic trek upon five boroughs at once. And good grief—before I could even finish saying “The Bronx is up and the Battery’s down!,” my map was already complete! And it had all been drawn in capital letters, to boot! And all using 180-degree rotations! Will wonders never cease?
Occasionally, when I show people ambigrammatical stunts like the five boroughs, they ask me, with eyes full of wonder, “Can you do this for any name?” I usually reply, “Well, it all depends on what you mean by ‘do this.’” I think what people generally mean by their question is: “Can you take any old word or name and make it turn into itself via 180-degree rotation?”
They don’t take into account such crucial questions as how legible the end result might be and who the intended audience is, nor the possibility of my resorting to other symmetries, such as wall reflections, lake reflections, quarter turns (clockwise or counterclockwise), and so forth—nor does it occur to them that I even have the option of reframing the challenge itself (meaning that, instead of doing “Elizabeth,” I might try to do “ELIZABETH” or “Betty” or “BETTY” or “Liz” or “LIZ” or “Lizzie” or “LIZZIE,” depending on Elizabeth’s range of nicknames). Like legibility and audience, such dodgings or tweakings of the challenge are out of sight, out of mind for them.
People tend to believe that, whatever curveball they might throw at me, I can hit it smack out of the park. I certainly cannot always do that, but maybe it’s just as well for me and other ambigrammists that people have such a belief; it makes what we do seem more like a set of magically inexplicable tricks than like hard work.
The first year I taught a class on ambigrammia was 1985; since then I’ve done it several times more. In a recent ambigrams class, the first assignment I gave involved just single words, such as the students’ first names, but after that, I wanted them to tackle sets of words, so I asked them for suggestions, and one student suggested the 12 canonical birthstones (one for each month of the year). This was a very difficult assignment, and not all the students were able to complete it, but we all had fun tackling it.
Since I was more advanced in ambigrammia than my students were, I threw in an extra element to the challenge when I myself tackled it: I aimed for doing all the birthstones as 180-degree rotations, and on top of that, in capital letters. Such a feat was not a priori doable, but after a lot of blood, sweat, and tears, I managed to come up with a 12-member “unigram” (meaning a design consisting of numerous ambigrams realized under the same set of constraints) that satisfied me.
One of my best unigrams is a set of mirror-reflection ambigrams on the seven colors of the rainbow (associated with the mnemonic name ROY G BIV). The rainbow is displayed below, with, at the bottom, a mirror-reflection version of my signature, DOUG, which, by some miracle, is simultaneously a mirror-image reflection of the year in which it was made (2006).
This “DOUG/2006” ambigram, aside from being mirror-symmetric, is also an “oscillation ambigram.” It’s called that because, like the famous Necker cube, it oscillates back and forth in the viewer’s mind between two readings without any need for rotation or reflection. Such ambigrams, being very difficult to carry off, are very rare. One of the best oscillations I’ve produced is shown below.
An oscillation ambigram describing the strange quantum nature of light.
It describes the dual nature of light—both wave and particle simultaneously. The letters in WAVE are wide capitals, while the letters in Particle are narrow and lowercase (except that the P is a capital).
Why, you might wonder (and “you” includes me), have I devoted such a large portion of my life to the obscure and esoteric art form of ambigrammia? Is ambigrammia just a hobby or pastime for me? No, it’s far more than that. It’s what I would call a passionate binge—one among dozens scattered down the decades of my life, ever in pursuit of some elusive form of beauty. Indeed, when I look back, I see my life as a relentless search for beauty—a quest that I recently took to calling “My Wild Grace Chase.” (In fact, I’m working on a book by that title right now.)
What is it that makes creating ambigrams so compelling for me? It’s not just their double-readability or their sometime symmetry. The hope of coming up with a design with those alluring traits is merely a launching pad that sets me off on a quest. What fires me up during the quest are those exciting moments of discovery, those sudden jolts of insight—and later, when it’s done and polished, the repeated savoring of unanticipated small pieces of visual magic that I found along the way. I often gaze and wonder, How did I ever think that up? Yes, strangely enough, my peak achievements in ambigrammia continue to gratify me, their onetime maker, with their surprising internal harmonies.
My joy at creating a new ambigram and then savoring it many times over reminds me a bit of my reactions, when I was a teenager, to hearing the fugues in Bach’s Well-Tempered Clavier the very first time, and then over and over again. It wasn’t the intellectual side of these highly complex pieces that thrilled me; it was the powerful emotions that they evoked. But the fact that these pieces were fugues—compositions having subtle and intricate contrapuntal structures, compositions having voices interweaving in a way that I had never imagined—was inseparable from the feelings of awe and reverence that coursed up and down my spine as I listened to them. I think that something comparable, though diluted, could be said about the pleasure that ambigrams give me, both when I make them and when I look at them. Each one started out as a mystery shrouded in total fog—but I persevered, had some bad luck and some good luck, eventually found a hidden pathway, and finally, with hard work, wound up with a polished gem that retains charm for me long after the fact.
When facing a fresh new ambigram challenge, I can never anticipate what I’ll wind up doing with it. Will the end product be a 180-degree rotation, a wall reflection, a lake reflection, a quarter turn, or something else? Will it be in capitals, smalls, cursive, or some mixture thereof ? I don’t know in advance. Will it be curvy or angular, jagged or jaunty, ornate or minimal, swashbuckling or spare? No idea! And sometimes it takes me many hours—even days—to discover the secret that, all the time, was lurking hidden inside it. Because it was always potentially findable, it’s probably better to call it a discovery rather than a creation.
What gives me the greatest pleasure is the fact that here is a brand-new miniature piece of art, born from the yoking-together of a challenge and a constraint (or set of multiple constraints). What a joyful experience it is to be able to convert a novel challenge into a tiny visual gem! In the end, it’s not the fact that I’ve discovered or created a doubly readable piece of writing that transports me. In the end, it’s the beauty—the fact that I’ve somehow come up with a totally unforeseen embodiment of beauty—that matters.
I have never been inclined to let machines do my thinking for me, which is why I have no interest in seeing AI applied to ambigrammia. I have my doubts as to whether current AI systems could come up with great ambigrams, but whether they could or could not do so, I would be loath to use such an approach. I have a reverence for the creative/discoverative human mind and want to use my own mind as much as I can in all facets of life. This may strike some readers as an old-fashioned attitude, but that’s me all over.
A few months ago, in talking with my friend Joshua Cynamon, I explained that I’ve recently felt plagued by waves of troubled confusion over putting so much energy and care into a book of doubly readable pieces of calligraphy—just frilly little colorful baubles. What a crazy thing to spend one’s later years on, especially in the frightening times that we’re living through. I said to Josh, “I’m working on this book during a period when many of my dearest friends are growing older and confronting great sadness, when democracy seems teetering on the brink of destruction, when the very concept of truth seems to be going down the drain, when artificial intelligence may soon overtake human intelligence, when global warming is threatening the survival of all sentient life on Earth. At this scary moment in history, doesn’t it seem weird and self-indulgent of me to be working on something so frivolous and flighty as a collection of ambigrams?”
Josh replied, “I can well imagine your doubts, Doug, but actually I feel that it’s very sensible for you to be working so hard on your ambigrams book—in fact, it’s a crucial thing for you to be doing—because bringing beauty into the world is so important to you, and to us all, most especially in dark times.”
Josh’s thoughtful reaction helped me to feel once again that my artistic project was worthwhile, and it brought back a memory of a remark that my daughter Monica made to me late one night over the phone, just as the results of the 2016 presidential election were becoming clear. She asked me, “How can this nightmare be happening?” I said, “I don’t know. I don’t know.” She asked, “What is humanity coming to?” All I could think of to say was, “The world is unpredictable. Nobody can know where things will go.” And then, out of the blue, Monica said, “In the end, only art can save the world.” It was such a powerful and idealistic statement of faith that I’ve never forgotten it, and I dearly hope that she is right.
This essay is an adapted excerpt from Hofstadter’s brand-new book, Ambigrammia: Between Creation and Discovery (Yale University Press, 2025).