**Natural observations. **Even young children can observe that there are similarities between the growth and physical composition of totally different living and lifeless “forms of life”, such as crystals, plants, animals and the man. This fact was known in the early period of man, since for example the same word, trunk, was invented for the body of a tree, an animal or the man when languages were formed, and we still use that. We can also observe similarities in the design of crystals, the structure of trees and bushes, the branches of rivers on the surface of the earth or the blood vessels in our body. This similarity results from the fact that the process of growth is self-repetitive in nature, that is, one big part breaks into two smaller ones, then the two smaller branches divide into two smaller ones again. The process never stops but from time to time it shifts its scales, for example the tree and its leaf: we can trace the outline of the whole structure of the tree in the leaf.

**Mathematical systems. **In mathematics the systems working as self-similar organizations are called scale-shifting symmetry or fractal geometry. Fractal geometry is interested in the objects revealing a peculiar version of symmetry. Every type of symmetry results in a kind of invariable behaviour; axial symmetry, for example, shows equal extensions as regards the original forms and their mirror images. Fractal forms are invariable as far as changes of scale or dimension are concerned. In other words, after random scale alterations, the characteristic details of the original form will eventually recur. Early use of scale shifting symmetry can be found in medieval architecture in a church mosaic (13th century) of Anagni (Italy), in the mathematical objects of a Polish mathematician Sierpinski (19th century), who later developed the theory of fractals, and can also be found in 20th century abstract-geometrical art, namely in the Saxon poly-dimensional triangle.

**The poly-dimensional point.** From a mathematical perspective, we can say about a point that it is the smallest unit, an axiom. On the other hand, these infinitesimal points which do not even have an extension constitute lines, planes, space, our physical world, and our infinitely large Universe as well. This is the real dimension paradox. From artistic correlation we could define the point as a multidimensional phenomenon, as the condensation of all dimensions and dimension structures. The point, indeed, “remembers” all dimensions and dimension structures: it is the intersection of lines, the micro-plane element of planes, the basic particle of space and in fact the impression of worlds on various scales.

**The poly-dimensional line.** The trunk of a tree branches in two or three directions, the thicker branches in turn divide into boughs of smaller circumference, down to the thinnest twigs at the end of which we can find the leaves. If we continue our observation, we may see that the capillary vessels within the leaves reflect the image of a small tree. Taking our contemplation even further, we might conclude that the divisions of our own body resemble those of the tree – the limbs (boughs) extending from the trunk end in fingers (twigs). Moreover, the network of veins in our bodies (or, for that matter, the network of fountains, streams and rivers all over the earth) is characterised by the same divisions.

**The poly-dimensional field and space.** If we place geometrical elements of varying size or proportion, but of similar form, on a plane or on a space, our eyes will perceive the connections between large, small and even smaller elements in perspective. If, however, we connect and combine the same forms, perspective ceases to be effective, and we arrive at new structures constituted by the different forms attached to one another. The “poly-dimensional objects” 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 civilisation (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 realised on extreme scales.

**Seeking proportions.** If we are to create poly-dimensional objects from basic geometrical forms, we could at first choose the simple proportions of halving or doubling, or, alternatively, splitting into three or trebling. Splitting or multiplying the square or the triangle, we should take the sides as a point of departure. Halving the sides divides the area of the square into four equal parts which in turn can also be divided in a similar way, and this leads to the series of 1:4, 1:16, 1:64, 1:256, etc. Splitting the sides into three will divide the area of the square into nine equal parts, and repeating this action will lead to the series of 1:9, 1:81, 1:729, etc. Interestingly, the area division of circles will show the same proportions as that of the square and the triangle, though here we have to choose the radius, and not the circumference, as the basis for scale change.

**Compositional borders.** Even halving the forms more than three or four times might make it impossible to use the derived elements for composition, as their areas will become too small (1:256 of the original, for instance). They may even vanish from our sight. In the opposite case, increasing the areas more than three or four times will lead to forms not only unsuitable as compositional elements, but also too huge to store in our home, or even in gallery spaces. Just imagine an exhibition opening where the visitors are looking for the works of art on the walls, while the edges of those works might reach as far as the border of the town. Nevertheless, we need not despair: in their physical realisation, three or four such manageable scale changes prove more than sufficient to incite in our minds the poly-dimensional movement leading to an infinity of similar leaps.

### **Géza Perneczky: About the objects of the Poly-dimensional Universe**

János Szász SAXON’s nearly twenty-five-year-old artistic work focuses on panel-like compositions which he calls poly-dimensional fields. In this context ‘poly-dimensional’ does not mean transgressing the usual three-dimensional world and entering a space of four or five or even more dimensions. It simply means that in SAXON’s paintings one form is repeated in a smaller or bigger scale – as he defines it, in a smaller or bigger dimension – what is more, the composition is made up of repetitions of the basic form in smaller or bigger scales.However, SAXON’s paintings have more than a simple repetition: the strict logic of it. When, for instance, we start to diminish the basic shape in a 1:3 proportion, we must keep to this scale in the further process as well. Moreover, we have to observe the rule of theoretically maintaining the 1:3 proportion in both downwards, towards the smaller sizes and upwards, when we increase the patterns towards the bigger scales. Other proportions to be observed are also possible, for example 1:4 or 1:5, or those described with fractions, powers or logarithmic scale-shifting. It is important to assert the basic principle of the composition, the similarity despite scale-shifting.

Since the process of scale shifting should be done to infinity to both directions, amplification and diminution, this formal system with everlasting iteration could theoretically increase until it fills up the whole universe. This is, however, not possible in everyday life. Thus, with his works the artist must settle for starting the process of iteration growing towards infinity and finally crowding the universe. His pictures can only be models and examples of creating a world of self-similar structure that expands to infinity.The systems modelled in this way are conceivable to us, since not only its forms but the distances stretching between them and the set-up parameters also diminish or increase in the initial order and proportion. Due to this, three or four steps are enough to see what this expanding new world would be like if a god accepted a painting by SAXON to use it as a model in creating a new universe.

I must admit that it is more than a witty stylistic device, since physicists and astronomers dealing with cosmology claim that the order of smaller groups of bodies in one galaxy is repeated again and again in a bigger proportion, in the form of aggregation of stars. What is more, galaxies themselves form groups and aggregations that repeat the set-up observed in the smaller units, thus organizing even bigger units in the depths of universe. It means that the distribution of matter in the world is not continuous but periodically iterated, or condensing and rarefying in a regular way, and that the universe in its bigger proportions is similar to its smaller parts. In other words, it does not only have gaps but maintains the initial arrangement in its forms and proportions as well, that is, has the character of self-similarity.

Most of SAXON’s works vary this universe of gappy and self-similar structure in forms of panels and models that you can even hold in your hand and which are easily conceivable and adjusted to our scale. Mathematicians describe these self-similar organisations as a special case of symmetry. They mean symmetry that is realised through scale-shifting, and describe its laws with the help of fractal geometry.

7 October 2010, Gallery B55 Budapest