by Aditya Dev Sood
It was sunny but cool as we drove into Budapest, and I had that kind of new city buzz that you can only get from having been in transit all night. We cut across one of the many bridges over the Danube that interconnect the Buda and Pest sides of the city, and at a traffic light something caught my eye. It was a sign for a taxi bank, bright yellow, vertical like a post and uncannily designed so as to be legible from almost all sides. There’s something about that sign, I said to Nita, that captures a lot about this culture. Later on, as we walked about the city I took multiple photographs of the thing.
In its formal inventiveness it vaguely reminded me of Bauhaus signage, of El Lessitsky’s posters and perhaps Russian constructivist and Czech avant-garde designs from the early part of the last century. But in its sculptural, volumetric, 3D-ness, there was something very particular about it. The design made the most of this set of letters, which are all bilaterally symmetrical, and therefore capable of being rendered vertically. I also saw the sign as an attempt at rendering the text legible from all directions, in a way that flat signage can never be, and it was interesting that this need or desire is intimately linked to the very idea of the sign for a taxi bank. What made this possible was the rendering the forms of the letters as cylinders, cones and discs with suitable cut-outs. The letters now seemed abstracted, as if they represented fundamental mathematical operations, or else belonged in a Chinese puzzle or a game of some kind. The configuration of the arbitrary forms of the letters of the Latin script into this elegant and meaningful three-dimensional sign haunted me as we walked around the city, and I kept looking for clues as to where and how this kind of thinking had come into being.
On the way home from dinner and we came upon a giant jagged apparition at some distance, which we wondered at, in that half-light, on the diagonal. It was really hard to tell if that was a giant ball or some kind of urban sculpture or what. It was only when we came closer, and were standing right across the building that it became clear that this was an intentional play of color and form, an invitation to indulge one’s pleasure in the visual experience of geometry. I came back later and took the photograph below.
As I learned later, this is a piece of trompe d’oeil inspired by the Hungarian modernist painter Viktor Vasarely, who had so fully captured the zeitgeist by about 1973, that you would know his work by his influence on everything from poster design to album art, even if you never knew him by name.
Vasarely was associated with many of the modernists, and part of his career was spent in Paris, yet Cubism and its aftermath does not fully explain his peculiar preoccupation with the illusionistic creation of three-dimensional optical effects using color upon simple compositions of cubes. He makes paintings that are carefully crafted grids that deform at critical junctures to break into the third dimension, or Escheresque, dissolve into unresolvable contradictions. He seems haunted by riddles of dimensionality, by the folding contradiction of the three dimensions and the pull of gravity. He makes, for instance, a composition of folding chess pieces, now straight, now flat, now folding into a cube-like chess board, whose uncertain post-Euclidean space captures the observer’s mind with force, eliciting complex emotions and wonder.
Budapest is also, of course, the home of Ernő Rubik, the father of the Rubik’s Cube and sundry similar three-dimensional puzzles. Quiet and retiring, Rubik has worked as an architect, designer, and teacher of descriptive geometry in a long and singular career loudly punctuated by his sudden global fame thanks to his Cube. He is now staging a major retrospective exhibition on the occasion of the thirtieth anniversary of his famous Cube, which is traveling around the world. Rubik’s original Cube itself has become such a ubiquitous token among children, geeks and gamers around the world, that it is hard, after all these years, to bring it back into focus: What faculties of mind and imagination does it actually represent?
Molded in black plastic with colored stickers in bright primary colors, the Cube is an abstraction of spatial configurations and possibilities. Unlike most objects of material culture, it is the rare thing that arose entirely within the mind of its creator, and then took shape in order to illustrate and instantiate, as it were, the riddles its inventor was posing to himself. To contemplate the movement of little abstract cublets of different colors moving around hidden sides of the Cube is to stimulate the mind to develop a kind of visual-spatial and kinesthetic sensibility that is not necessarily innate to all of us, and which is either deeply pleasurable or else chokingly difficult and ultimately defeating. I had a Rubik’s Cube as a child, and I derived the most pleasure from breaking the entire think apart and reassembling it in its solved form.
Rubik has spoken of his work as a kind of synthesis of design-thinking and space-research: “Space always intrigued me, with its incredibly rich possibilities, space alteration by (architectural) objects, objects transformation in space (sculpture, design), movement in space and in time, their correlation, their repercussion on mankind, the relation between man and space, the object and time.” His approach appears to pull together scientific and artistic thoughts around form and space-making, an approach that seems rare and necessary to the development of such an odd thing as the Rubik's Cube. Rubik designed several other puzzles in his time, including Cubes of ever more numbers of component cublets, but none of them achieved the iconic status of the original cube. All of them, though, share a similar fascination for how two and three dimensional shapes can fold and twist and intertwine into one another, suddenly creating meaning or else always remaining twisted into a kind of noise or inelegance, inviting the gamer to intervene and so create the order of a solution.
The Dinner Party
These were the thoughts playing on my mind when I came over to visit Adam, a friend of mine of several years standing from the media art and information design circuit. When last I’d visited him in Budapest, we’d been discussing ways for my organization, the Center for Knowledge Societies, to collaborate with a research lab he’d recently cofounded, called Kitchen Budapest. He’d then gifted me an odd little sculpture plucked from a media art work that hadn’t worked quite as he’d intended. As seen in the image to the right, it is a kind of three-dimentional pixel, which shows a different mix of Red, Green or Blue depending on the side from which it is seen.
We were at Adam’s place for dinner, on the Buda side of the Danube. It was a celebration of sorts, for it was the twentieth of August, Hungary’s National Day, and a holiday weekend. Adam had promised a view of the fireworks from the roof of his apartment building, and he was also going introduce us to Hungary’s different wine regions, beginning with this fine Kadarka, light bodied but full-tasting and spicy and warm to the heart. His wife Anita had made a beef stew in paprika gulyás-style, so in many ways the evening really was shaping out to be a tribute to Hungary.
Up on the roof I ended up buttonholing Peter, Adam’s business partner, about the visual ideas I’d had over the course of the last few days on the visual culture of Budapest. I told him I thought there was a particular kind of approach to visuality that I was noticing in the city, a kind of preoccupation with geometric shapes and their transformation in various ways, for example from 2D to 3D. So what is your thesis again, Peter asked, that there are some funky taxi signs in the city where Erno Rubik lives? I told him about the Vasarely-inspired building we'd seen and offered up my still-nascent theory. There is a kind of spatial imagination alive here, in this city, which is unique, I said. There is a pleasure in spatial transformations as in Rubik’s cube, and from 2D to 3D, as in the taxi bank signage, and in the confusion or questioning between 2 and 3D, as in Vasarely’s visual constructions. That there is a kind of geometric imaginaire active here, which makes that kind of expression and communication possible.
Well, I buy that, he said. And I might be able to think of a couple more data points for your argument — when we get back downstairs you should ask Adam to show you his Gömböc. After we’d all traipsed back down to the apartment, Adam fished out a small presentation box that had within it a white metal mass, something like a paper-weight. He placed it on the dining table and it began to dance unsteadily for a few seconds, before careening over and falling into a long-term oscillation from one end to another. It behaves like a Russian doll, in that it however unlikely it seems to the eye, it comes back to a single point of stable equilibrium. But unlike the Russian doll, its volume is of uniform density. The Gömböc is a special class of convex shapes, with particular properties, the existence of which was hypothesized only in 1995, and which were proven to exist a few years ago. It has only one point of unstable and one point of stable equilibrium. This particular shape is the most interesting, visually and formally, for most of the other solutions are rather similar to a sphere. The tactile and kinetic joy of playing with a Gömböc cannot entirely be disconnected from the knowledge that it is its shape alone, and not some hidden artifice, motor or internal configuration which is responsible for it behavior.
The Gömböc was ‘invented’ by Gábor Domokos along with Péter Várkonyi. Adam said he’d enjoyed studying Descriptive Geometry with Gábor Domokos back at University. The very idea of studying such a discipline as part of an Architectural or Engineering education was novel for me, for it combined Engineering Drawing with Mathematics in a highly specialized way, not entirely disconnected from Aesthetics or from the cultural appreciation of form, just as one might encounter for example in Art and Architectural History.
I’ve got something else for you to look at, said Peter. On Adam's computer, he’d opened up an article in Hungarian on Gábor Dénes, the Hungarian-origin scientist who is credited with inventing Holography. Gábor’s approach to Holography involved an attempt to capture light information emanating from an uneven surface in a predictable way, so as to be able to record the variable features of that surface. His work well pre-dated the laser, so in his time it was not yet possible to reconstruct the information he had recorded, but that was what he was trying to achieve. It was only in the later part of his life, when Gábor’s scientific life was largely over, that interest in his work began to grow, on account of advances in the visual reconstruction of holographic information, partially on account of lasers. Gábor became a kind of all-purpose public speaker and ambassador for Holography, envisioning Holographic films, and other innovative means for social communication that involved three dimensional imagery. Does Gábor's work and the entire field of Holography really involve geometry? So far as I can understand it, his work involved attempts at comprehending spatial information at minute scale, and innovative approaches to reconstituting lost elements of data so as to reconstruct the original. So his challenge did not necessarily involve geometry per se, but rather a geometric imagination, which could imagine the relationships between surfaces and light information, and an intuitive-inductive ability which allowed him to arrive at a means for reconstructing that information in such a way that it might be experienced again. There have been important developments in Hungary since Gábor’s time, and it may yet turn out that your first Holographic Television will be Hungarian, at least in some way.
Perhaps the foundation stone for any argument around Hungarian approaches to Geometry would have to be the shadowy figure of János Bolyai, the mathematician and geometer who lived through the first half of the nineteenth century. First taught by his father, Bolyai apparently began questioning Euclid's parallel postulate, resulting in his discovery — or creation — of the “strange new universe” of non-Euclidean Geometry. Bolyai's work proceeded in parallel with that of the Russian Nikolai Lobachevsky, but it is a matter of some pride for Hungarians that one from among them is considered to have founded the discipline. If one thinks of Geometry as a kind of language, logic and notational system to describe the spaces, shapes and form that we normally inhabit and experience with our bodies, Bolyai was the first to use that language to describe spaces and objects that can only be experienced in the mind. He opened up the human geometric faculties to possibilities that are beyond visual experience, but not beyond conceptualization and even visualization, albeit in the analogical or suggestive manner of Vasarely's paintings.
Finally, and most recently, there is Prezi.com, the software-service venture that Adam and Peter cofounded. Prezi is a new kind of presentation software, which allows you to place and organize different kinds of media formats in an infinitely scaling virtual space, into which you can zoom in and out, so as to be able to explain to your audience how those different kinds of text or data interrelate with one another. Prezi is hard to describe and necessary to experience. It allows a kind of multimedia wall-painting that we have not been capable of since Lascaux, when I imagine the presenter used his torch to call our attention to the features and attributes of the different animals drawn and painted on the cave wall. Prezi uses a two-dimensional surface, but an approach to information that is profoundly non-linear and multidimensional. While it obviously operates within Euclidean space, its approach to information is scalar, allowing swooping descents in and out of the informational space. The software allows you to assemble different kinds of media formats, dealing with them through a new kind of grammar of visuality based on bracketing the gaze, a feature which allows the presenter to carve a path through the series of assemblages that she might zoom in and out of our visual field, and therefore our attention and shared focus.
The technology of vector drawing may be trivial, but the conceptual and cognitive virtuosity required to derive meaning from image, text, and spatial relationships is not. The underlying thinking or drive behind the conceptualization of Prezi seem to me spatial and kinetic in nature, perhaps as an attempt to be able to bring to realization and shared experience the inner experience of thought organized in the complex spaces and folds of the mind. It is in this spatialization of thought, I believe, the common thread of Hungarian geometric imagination is revealed. The geometry of Prezi involves a kind of approach to form and space that perceives them not as mute phenomena, but as critical means for encoding and deriving information.
Prezi is a potentially radical cultural development, because the experience of use of it decenters and recenters our conception of topos. The battlefield of ideas may still be planar, but it is not bounded, it has become scale-independent. This one shift makes it possible for our footing to shift radically, within an instant, and for important relationships to be subsumed under ever larger intersections of forces and ideas. While the genius behind Prezi may be quite local, as I have been arguing above, if Adam and Peter's venture is to achieve any standing and success in the world, this may be because most people in the business world experience their everyday life and organizational and intellectual challenges to operate within just such a flux-ridden, volatile, dynamic and uncertain ground.
A Magyar Conception of Geometry?
The diverse cultural, commercial, scientific and technical phenomena discussed above would suggest that there does exist something like a Hungarian geometric imaginaire, a shared horizon of thinking, making, doing research and training, that is relatively stable across at least a couple of hundred years, and which reasonably interlinks different areas of specialized knowledge practices. At minimum, there are some shared proclivities for representing and experiencing folding, transforming, and zooming spaces, and consequently important technological and cultural artifacts that themselves become milestones for the further development of Hungarian visual, spatial and geometric knowledge and culture.
Howsoever it be founded, the Hungarian affinity for Geometry and geometrical thinking, appears to have been reinforced through institutions of higher learning in which a certain ideology of visual-spatial possibility came to be ensconced and reproduced generationally. The fact that Ernő Rubik and Gábor Domokos both taught Descriptive Geometry, a course of study that Adam, Peter and subsequent generations of bright young people might also pursue, effectively reproduces a live current of cultural and scientific thought that is still unfolding. These inquiries are informed by a knowledge of other major intellectual figures, who are worthy of emulation, and whose thought may still color or inform the new projects, cultural or technological or scientific that these new minds might undertake. In all these ways, and for these reasons, it is possible — and possibly useful — to speak of a Hungarian geometric imaginaire.
I doubt I would have pursued any of the above conversations, or indeed this entire line of questioning, were it not for the writings of Benjamin Lee Whorf. A sentence fragment from his posthumous book, Language, Thought and Reality has been echoing in my mind, to the effect that if there were scientists from Native American tribes — I think he had in mind the Hopi — they might bring into being entirely new conceptions of reality and therefore of technological possibility than those which are known or current today. I went looking for the quote the other day, and was surprised to discover that he uses the example of Non-Euclidean Geometry as an example for how reality can be constructed theoretically in ways which are only glancingly related to what we may call Standard Average Indo-European models. In some ways, one might say, the Hungarian case might have been exactly the example he was looking for. Given this data and an argument around a geometric imaginaire, I believe Whorf would have wanted to look further, into the foundations of Hungarian geometric imagination. That is to say, why do Hungarians especially indulge and enjoy visual-spatial representations, transformations and manipulations in the first place? Is it perhaps that there is some predisposition that the Magyars enjoy –- or suffer -– which makes them especially susceptible to visual thoughts, on account of which they have paid so much attention to the spatial-visual dimensions of their experience?
The first thing to say is that we have not demonstrated in any way that this is the case. Further research inquiry might do so, but that is not the data we have today. If, however, it were possible to show this, one may conjecture as to some of the factors contributing. Hungarian is a Finno-Ugric language, and it is likely that along with the Finns and the Estonians, these semi-nomadic peoples migrated into western Europe, perhaps on horseback, from the Ural mountains, now in Russia. Whorf would have liked to have known whether there was something about the Magyar language, which predisposes its users to understand their experience in spatial, geometric, notational terms. My limited and preliminary inquiries into the logic of the Hungarian language has not revealed anything telling, although the tendency for the language towards agglutination and compounding, the concatenation of words into indefinitely long sequences with a determinate syntactic meaning could have some bearing on the matter.
There is, perhaps, another factor around the Magyar language at play, which derives not from its linguistic structure per se, but perhaps from the phenomenology of inhabiting an island language, that is, a language all but unintelligible to others all around, unlike most other languages that are by degrees similar and glancingly intelligible to speakers of related proximal languages. All native speakers seek, to some extent, validation and ratification of the experience that they have, of the direct relationship between their words and the world around them. Speakers of vernacular languages with strong links to classical language enjoy the etymological relationship that governs the numerous words of power and moment, which invoke authority at particular ceremonial occasions. This particular experience of language would be denied to native speakers of Magyar, who would see rather that their own language captures a reality, which they cannot easily share with neighbors, even in degrees or gradations. In the absence of strong and repeated reinforcement for one's symbolic experience of reality, appeal may be made by the mind, to the deictic, to the realm of meaning that C. S. Peirce termed 'secondness,' the world of pointing, space-making, form and shape. If, indeed, this tendency towards secondness, towards the directionality of arrows and the parallelness of lines is indeed the impetus behind Hungarian explorations of Geometry, it is a happy accident that those inquiries have eventually led to new and increasingly imaginative conceptualizations and manipulations of reality, in which we can all now share.
I'm very grateful to Adam Somlai-Fischer for his several invitations to Budapest and for the conversations we've had together. So also to Peter Halacsy, for early interest and encouragement with this line of thought. Nita Soans Sood humored me in various iteractions of this argument and helped me organize the images for this article.
Some images here have been taken by me, while the copyright may be held by others in other cases. Sources for images in this article may include: