Footless Chair – Saxon’s Poly-Universe

by Jánosz Szász Saxon

“János Szász Saxon has created a coherent theory, that of ‘poly-dimensional universe’. This is a unified world explanation in which the point, the (straight) line and the various planar and spatial figures all play a crucial role and which does not contradict the scientific world view of our time either, albeit it is full of idiosyncratic formulations and attempts at a subjective expression. The artist tries to express the basic experience that an enthralling order governs the structures of the cosmos (Figure 1), whose system encompasses both the infinitesimal (nano) and the infinitely huge (giga) dimension structures. (This is how he relates to the considerations of fractal and chaos theory present in science today.) We all are the distinguished parties in one and the same infinite process and the works are concretisations thereof as well. Our anticipation of the system of the world is, at the same time, a notion of the emergence of the world, which inspires artistic creation. (‘Creation’ is a keyword here, both as a theological ‘genesis’ and – among 20th-century non-figurative and constructivist artists – as aesthetic ‘creation’.) As opposed to other geometric artists who chose the technique of construction from geometric elements for fear that they should imitate theological creation and servile copying, Saxon sought and found geometry as a language to express a sense of the world” (see László Beke [1] ).

The Poly-Dimensional Field

As is generally known, Constructivist geometric artists, including me, work with geometric forms. While working, it often happens that if we place geometrical elements of varying size or proportion, but of similar form, on a sheet of paper, our eyes will perceive the connections between large, small and even smaller elements in perspective. We perceive the starry sky, the plane projection of the Cosmos perceptible for us, in a similar way, where we see the nearer celestial bodies bigger, the further ones smaller. In reality the bodies that look bigger may not be bigger than the others. In our present experiment, however, the plane forms, i.e. those trapped in two dimensions, possess the parameters in correspondence with their actual scale. What looks the biggest is the biggest and what looks the smallest is the smallest (see Saxon [2]).

The question arises, what happens if we connect and combine the same forms’ Take the square – the most abstract geometric form – as a starting point (Figure 2). Let us choose outward building as direction of progress (exterior = adding to the area), marking the corners as connecting points. We attach smaller squares obtained from the previous form in 1:3 proportion to each corner. Let us repeat the process several times. We can see that it is possible to attach four smaller squares to the first one, and three squares to the free poles of the four squares, and so on to infinity?

In the meantime the area of the original square (T₀=1) has been expanded T₃= 1 + [4/9] + [4/9 × 3/9] + [4/9 × 3/9 × 3/9] = 1,64197… times in three steps, while the number of squares has increased to D₃ = 76. We can get the further number of pieces by the simple formula D_n+2 = 5 + 4 × [3 + 3² + 3³ +… 3^n-1 + 3ⁿ]. If a means the segmentation of sides, that is 2, 3, 4, 5 etc., and n means the number of connection rings, then we can use the formula T_n = T₀ + [4/a²] × [1 + 1/a + 1/a² + 1/a³ +… 1/a^n-1 + 1/aⁿ]. We can point out that if (n = infinity), T_n < 2, that is, much as our new form tends to multiply itself up to infinity, it cannot double itself.

However, we can also see that it is a system creating itself on the basis of its own laws – perspective ceases to be effective, and we arrive at new structures constituted by the different forms attached to one another. During the past thirty years, studying these basic geometrical shapes (the square, the circle, the triangle) I have named these image structures “poly-dimensional fields”. Now I had the analogy of my childhood observations in nature, since the “poly-dimensional fields” thus emerging are able to model the abundance of nature (trees, blood and water systems, crystals, cell division, etc.) and the infrastructural growth of human civilization (networks of roads, pipe systems, networks of communication, etc.). On the other hand, they can represent the dimension structures of atomic and stellar systems, which have a similar structure, but are realized on extreme scales.

The systems based on their own laws queried individual creative principle, therefore I gave up the didactics of mathematics since as an artist, I had not only logical but aesthetic construction requirements as well. After this my works of art became so-called condensed pictures, universal event-figures (the last square in Figure 6 and Figures 9 and 10), since it is physically impossible to represent all the stations in the infinite process. With proper respect, I can emphasize or rearrange certain parts without causing harm to the essence; thought will then glide out anyway, skipping on the biggest or smallest element of the open system.

It was art historian Géza Perneczky who first discovered in the middle of the 1990s that my works of art were fractals by nature, and he devoted a book to the presentation of this revelation (see Perneczky [3]). During the fifteen years of artistic work, isolated from the world, living in my inward solitude I had not had the faintest idea about this. That was why I could establish my own specific way of expression and world of images in this area. One of the most intriguing formal elements which explain why my poly-dimensional structures never turn into mere illustrations in spite of their similarity to fractals is the “auxiliary plane” technique I use (Figures 2 and 10). For when I have iterated a geometric form and grouped the emerging shapes, I may still not consider my work complete; I as a constructor take the opportunity to create auxiliary planes to my composition by linking the corner points and bridging the gaps. This procedure is an arbitrary move from the point of view of mathematics, but aesthetically speaking it is something extraordinary from the vantage point of the cultural background of these paintings, since this is how the work is turned into an icon, an image of symbolic meaning, and this is how it receives its rich symbolic aura that enables it to take on an additional sacred function. In my works this quasi-sacred element is a utopian meaning – it emphasises my aim that my works should define norms and laws and serve as symbolic models for possible worlds.

The Poly-Dimensional Space

There is only one step towards the creation of the poly-dimensional space from the poly-dimensional field taking shape from the squares as seen in the previous chapter. There is nothing else to do but replace the plane figure with a corresponding cube, then attach 1, 1/27, 1/729, 1/19683… [1/(a³)ⁿ] size cubes to each possible corner point, the sizes deriving from the 1:3 proportion of the previous scale, and continue the process to infinity (Figure 3).

This poly-dimensional space-construction will fire your imagination. At first sight, it looks like an unimaginable crystal structure, in which microscopic systems are connected step by step, in an indirect way to macroscopic worlds. In order to have a clearer insight into the interconnectedness of spaces or dimension structures on various levels, I placed identically formed chess pieces on a poly-dimensional field consisting of squares (Figure 2). The proportion of the pieces should follow the different sizes of the planes. Then I constructed two wooden stools of sixteen legs each, continuing the splitting of the legs mentally (4, 16, 64, 256, 1024, 4096, 16384… 4^n-1, 4ⁿ ) up to infinity (Figure 5).

After the completion of the Dimension Chess (Figure 4), I sank onto an infinite-legged dimension chair, and, after having taken a short rest, it occurred to me that this game does not follow the usual stereotypes. The chess table lying in front of me is a poly-dimensional field, practically the horizontal projection of the micro- and macro-world’s vertical texture. One of the pieces lined up is me, and I can move about in the unfolding Poly-universe freely, by disposing of the parameters of the actual dimensions at every single move.

The Footless Chair

The infinite-legged chair is a concomitant of the poly-dimensional chess. It would not be possible to break out of the parameters of our present world unless our thoughts rested on such an object. In a physical sense, the infinite number of legs is not a very reassuring idea, since the segmentation of the plane involves a smaller and smaller surface for the legs to support themselves on, and reaching infinity (n = infinity) the plane crumbles (T_n = [4/9]ⁿ) and the legs rest on an infinite number of points with no dimension. Thus, getting at the infinite-legged chair we can speak about the singularity of the chair, that is, about a legless chair, or as I named it, footless chair (Figure 5).

Naturally, if we venture rather far, we cannot only lose the legs of the chair on the poly-dimensional chess board but our legs too! No wonder that during my work, whenever I have concentrated on one point, the crucial fact that the point was actually an entity without extension, the tiniest unit, an axiom in the mathematical sense, overwhelmed me. This infinitesimal point of no size, a paradox of dimensional statuses, constructs the lines, the planes and the space, our world, and even the infinitely large universe, in a way that resembles a hierarchic world model, in which systems of lower levels continually connect to each other forming further developed structures, ad infinitum.

Hence the point partakes of every dimension; either as the intersection of two lines or as the basic unit of the plane or as the indivisible entity of space. It is as a matter of fact the border or dimensional gate of the black hole and later the white hole effect, where all the space-time dimensions of any given dimensional structure collapse irreversibly (see Saxon [4]). In other words time would stop and space would collapse – the pieces fall into singularity passing through each connecting point of the poly-dimensional chess board.

Idea of Immaterialisation

It is a paradox indeed that the desire for immaterialisation is present in my art. My thoughts germinating while observing nature took the object form in my first work of art very early, at the end of the 1970s when I was 15. I called it “Universe” (Figure 1). The image is made up very clearly by the possible permutation of halving the diagonals of the square. In order to get an idea of immaterialisation, we may set up a logical experiment: If there is a set of planes made up by at least two other sets of planes that in turn include two further sets of planes each, and so forth ad infinitum, then we may witness the termination of the plane as a form, as it becomes a set of points. If, on the other hand, we take space, then the same process leads to the depletion of space or an object, and the substance, after reaching a density of infinite fineness, is immaterialized, is transformed in our mind definitively.

Let us examine the plane sections of the footless chair in our logical experiment, and let us work with the square again (Figure 6). The direction of progress is inward-building (interior = taking some of the area away, leaving a gap), that is, diminution of the plane, since the aim is to decompose the form. We mark the corners as connecting points again, in each of which we leave the smaller black squares obtained from the 1:3 proportion of the sides of the previous scale. We follow the same procedure several times. It is obvious that in the first square there are four smaller T₁ = 4/9 elements left, and four more elements in each of them, up to infinity… In the meantime the area of the starting square (T₀ = 1) has been diminished T₃ = 1 – (5/9) – (5/9) × (4/9) – (5/9) × [(4/9) × (4/9)] = 0.087792… times in three steps, while the number of squares will be D₃ = 64. Further number of squares can be calculated by the formula D_n = 4ⁿ and provided n = infinity, the remaining form will be a cloud of dust made up of infinitesimal granules, invisible to the naked eye. In this fierce fight our black square has whitened, losing the last bits of its area.

This complete transfiguration, this absolutely transparent state, I could only model in painting by using such elements as even in themselves represent the supremacy of pure sensation. Thus two basic Suprematist elements, the square and the cross through which the square is divided into four parts (see Malevich [5]), have served as points of departure. In this case, the square bears a yellow colour symbolising existence, whereas its opposite, the cross is characterised by a white tone that creates an impression of emptiness. I must mention that to me the yellow colour in relation to white reflects the sensations of being and non-being, something and nothing, in a more vivid contrast than, say, black and white would do. During the construction of the picture, i.e. the deconstruction of the yellow square, I came to sense total depletion, or, more precisely, to set up a poly-dimensional net. The net that connects micro- and macro-worlds, is the virtualisation of the absolute mind, which, stretched in infinite dimension structures as a hyper-filter, incessantly attempts to jettison the imperfect objects (yellow squares) of existence from its “body” (Saxon, Immaterial Transit 1997, the last graphic in Figure 6).

Up Suprematism to the “supreMADIsm”

In the previous chapter I have referred to the Suprematism of Malevich. Generally it very seldom happens that a geometric form capable of iteration should be suitable as an icon on its own. If we find any, it is because it is formally related to the genre of the icon. One example can be the square, as it has the shape of the wooden board. On one occasion, studying the borderlines of the shading I had made on the shapes drawn in graphite, I did indeed take Malevich’s Black Square as a starting point, and gave my work the title “Poly-dimensional Black Square” (Figure 7). The sides of this square are divided in a 1:5 proportion, and this is the scale-shift that leads to the creation of the “fringes” surrounding the central shape. I applied this division to the picture three times. In this case let us calculate the fractal dimension of the outlines. Here we have eleven steps for a change of 5 length units; the result is hence log [11/5], that is, 1.4898… By the deliberate fusion of the black tone of the different-scale squares our eyes are stimulated to see one poly-dimensional square, in contradiction to mathematical laws. Indeed, this is what we experience in the Poly-universe. I also made the negative, white version of this work. In addition, I animated the clinging of the fringes on the edges of the base form using gradually thinning sounds according to the scale-shift.

I did my next experiment during the “supreMADIsm” festival organized in 2006 in Moscow. I embedded the white cross, one of Malevich’s basic Suprematist elements into the other basic Suprematist element, the black square, the former trying to deconstruct the latter (Figure 8). The confrontation of these two forms can be found in my earlier works of art, but in the present case transcending the geometric shaping did not take place in terms of some “Russian spiritualism”, but rather pragmatically. Before that we had been able to understand the scientific nature of my works in their fractal character described by the “dimension shifting”. Now, the main field of interest of mine included dividing the plane surfaces with the help of geometric figures and rearranging Malevich’s cross in a poly-dimensional way (Figure 9). During this work I created horizontal and diagonal constructions, poly-dimensional cross-icons, but in this case, as a result of the closed system of the form, there arose finite, only about a dozen of variations for each. Strict monochrome, or more unambiguously, the black and white contrasts produce a powerful psychological effect besides the variations of visual logical structures.

The Poly-dimensional Triangle

In the ’90s the square comprised my field of research almost exclusively. However, in the past ten years my attention has focused on discovering the triangle as the form of the “Absolute”. Rearranging the triangle in a poly-dimensional way did not cause a problem, since marking the directions (interior-exterior) similarly to those in case of the square, I attached the smaller forms to the corner points, and using auxiliary planes I could create icon-like panel paintings full of gaps or empty spaces. However, I discovered an image-formation, a geometrical form of multiple concentration, the spiralling poly-dimensional Star of David (Figure 10), which, similarly to supreMADIsm icons, is poly-universal in the strictest sense. This mathematical synthesis, formal reduction, sacred element, icon-like work of art occupies an honoured position in my activity (See Saxon [6]).

The same geometrical formula can be obtained, as a demonstration, by the following object: I retained the common parameters of Figure 10 to be able to see the differences in size within the plane. Turning the centre, I placed two equilateral triangles on top of the other, which, after summing up the free corner points, gave a non-poly-dimensional figure, similar to a Star of David. In the subsequent step, retaining the centre as a point of connection, I took the half of the original form, then the half of this latter one, and so forth six times. The result is seen in the figure below, a crystal-like geometrical object gained from the iteration of scale-shifting rings of triangles (Figure 11).

Comparing the two objects studied above, there is an obvious difference between them, one being a construction of artistic intention, a sacred work of art (Figure 10) and the other a didactic figure following the lines of mathematics consistently (Figure 11). Poly-universe, where as an artist I spend my days wandering about, is full of such or similarly compound infinite structures. Meeting them, having a “dialogue” with them, these structures let me make a reduction of form. We can see the same process in the case of my painting objects (circle, triangle, square) called Modules of Poly-Universe, which could be presented in a further paper or lecture.

János Szász Saxon

References

• [1] László Beke, Polydimensionen in den werken von János Szász SAXON, exhibition catalogue, Galerie Emilia Suciu, Ettlingen, Germany. 2007.
• [2] Saxon, Dimension Crayon, Espace de l’Art Concrete, Mouans-Sartoux, France. 2000.
• [3] Géza Perneczky, The Polydimensional Fields of Szász Saxon, Mobile MADI Museum, Budapest. 2002.
• [4] János Szász Saxon, The Might of the Point or the Punctuality of Space and Mind 1979-96, Shadow Weavers, copy art, fax art, computer art (1989-2004), edited by Árnyékkötők Foundation, Budapest pp. 294-300. 2005.
• [5] Kazimir Malevics, A tárgynélküli világ (The World without Objects), Corvina, Budapest. 1986.
• [6] Saxon, Etoile de Poly-D, Salon Réalités Nouvelles, exhibition catalogue, Paris. 2008.