Between mathematics and art
Janusz Kapusta occupies a unique place in the panorama
of individual artistic phenomena in the contemporary period.
Many artists of today and of the immediate past use and have used geometrical forms in the construction of their compositions. Most use them intuitively, employing a painter’s
instinct in creating a perfect harmony of shapes. The opposite
pole is occupied by those who use rigorous calculations or
mathematical formulas in their work, basing their construc-
tion of forms on the sinusoid, the cosine curve, the golden
section, Fibonacci’s series, or fractals. Between these poles
lies a whole range of differentiations and nuances.
Nevertheless, if we were to divide this profusion into cate-
gories, taking as a criterion the artists’ views, one would
have to consider the creative reasons that motivated their
decision to use the language of geometry in their art.
Beginning with the pioneers of the geometrical abstraction
movement - such artists from the beginning of the century as
Francis Picabia, Frantisek Kupka or the Futurists, through the great mystics of that movement, such as Piet Mondrian and
Kasimir Malevich, and the equally great rationalists, Joseph
Albers and Richard Paul Lohse or Henryk Stazewski - we find four basic approaches. The first shows the tendency for
ordering and harmony similar to that in music. The second
ascribes to geometry magical, symbolic and mystical senses.
The third looks for associations with mathematics, seeing
the only guarantee of perfection in its laws and shapes
drawn directly from it. In the fourth, it is the analytical ele-
ment that comes to the foreground.
Kapusta does not belong to any of these four categories; he is mentally situated both inside mathematics and inside art. His way of thinking is not that of an artist making use of mathematics. Neither is he a mathematician making
use of art. When Kapusta thinks in abstract concepts, they sometimes become shapes in space. He knows how to render
them artistically and how to describe them mathematically.
For him, they are neither mere concepts (e.g. of infinity),
mere attempts at their visual representation, or mere mathe-
matically describable and geometrically motivated forms. They
are all these at the same time, and only as such do they have
meaning and value. It is only by being able to move freely among philosophy, mathematics, and art that he is satisfied in his imperative for creation.
The fact that Janusz Kapusta graduated from a Polytechnic
and before that, from a Lyceum of Fine Arts, has not really had
any great influence on his work. It only contributed to his
learning the alphabet of mathematics and art and let him feel
at ease in both these disciplines. But it was neither his aca-
demic studies, his self-education, nor his study of philosophy
that led to the discovery which - like his way of thinking -
belongs simultaneously to both mathematics and art.
The essential condition for any important discovery either in science, art or any other domain of human activity is the extraordinary nature of the discoverer's personality. Only the individuals breaking the commonly accepted conventions and norms of thinking and behavior are able to cross the borderlines that had not been previously crossed. This is because they perceive the world differently than others; they have different standards of value with which they approach the observed phenomena; they choose a different purpose in life; devote their time and money to different objectives; treat different categories of things as most essential. Thus they lead a different existence. Their everyday behavior, decisions and choices are usually not understood and are regarded as weird by their community. It is precisely that separateness and weirdness that provide unconventional minds with a ticket to enter regions
not accessible to others.
Janusz Kapusta has such qualifications and necessary char-
acteristics. He has an analytical mind, geared at universal
concepts and questions. The objective world and the events
in the immediate or more distant reality, or the questions of
practical life are situated low in his adopted hierarchy of
things, in the plane of necessity and not of choice. His chosen
domain is that of intellectual questions and fascinations. The
artist wonders for example, whether the occurrences and events in the life of individuals, nations and the cosmos are predeter-
mined, or ruled by chance? Or, how tangible is the borderline between chance and determination? Between order and chaos?
The theoretical reflection on these questions has been accom- panied by attempts to resolve them graphically in long series of drawings, treated as a tool for conducting theoretical analysis. The artist recalls one exceptionally consequential event from the time of his studies at the Academy of Catholic Theology. One day while bicycling to a lecture, he pondered about how to draw infinity. Aware that this was an impossible task, he nevertheless embarked on it, hoping to learn what this idea may lead to. As a result, he created four works with illusive perspectives receding into the drawing, and delusive three-dimensionality of the seemingly material space, which seems to be coming out of the picture towards the viewer.
Another significant example will let us see how the artist’s
theoretical reflection and its practical consequences differ
from the conventional mode of thinking and action. “As memory is responsible for the past”, writes Kapusta, “imagination is responsible for the future, and only it is able to create something that didn’t exist before. Einstein said that imagination is more important than knowledge. I have created my own technique of expanding my imagination. An opera soloist rehearses his voice, a sportsman does his push-ups, and I wanted to create similar exercises for the imagination”.(1) Putting this ambition into practice, Kapusta decided to describe all possible aspects of the point. On the walls of his room he hung up square sheets of paper (40 x 40 cm) with a dot in the center, and defined it in various ways. “I chose the point, because it is the smallest spot which has position but no size; it is like a door to some world whose existence we can sense, though we don’t know anything more about it... I gave more than 400 descriptions; the difficulty consisted in finding the greatest number of them, so
that no one could add anything: a mental periodic table. The
first ten, hundred, two hundred, are relatively easy to find,
but after that it grows more difficult”.(2) Janusz Kapusta worked on this project for six months. Some of the project's descriptions include: “According to Euclid, only one straight line parallel to the base of the square may pass through this point”; “According to Riemann, no straight line parallel to the base of the square may pass through this point”; “It ends a sentence”; “Above the letter i”; “The size of the Earth as seen from some point in the Universe”; “The size of the Universe as seen from some point. In what?”; or simply, “Point”.
The artist presented the results of his arduous and prolonged
exercise at an exhibition held in a students’ club in Warsaw.
He entitled it The Divisions of the Point. As the exhibition
took place in the late 1970s, most of the viewers were know-
ledgeable about quickly changing movements in contempo-
rary art, and probably classified it as a typical post-conceptual
presentation. However, the artist does not identify with any
movement in art. He remains an outsider.
The unique character of Kapusta’s imagination may be illus-
trated with yet another example. “One day, while walking in New
York”, recalls the artist, “I thought of a scale. It wasn’t pre-
cisely defined, but I saw it very clearly. It was a kind of lad-
der with eight steps, which I had to climb. The tricky part
was that I had to build the steps myself while climbing.”(3)
The first of these steps, “survive”, as Kapusta put it, “was about
the most primitive instinct of securing one’s livelihood.”(4)
The second step, “let them know”, enjoined the artist to leave
behind the anonymous crowd and announce his presence.
The third step, “amaze”, expressed the necessity of doing
something that had not been done before. The subsequent
steps have not been built yet, they are still waiting to be
named. One will have to wait and see where the eight-step
ladder will lead the artist in the future. Kapusta’s kind of imagination is difficult to pinpoint and categorize. In his mostly figurative posters, book illustrations, and drawings for magazines, or in the 92 portraits of professions from the artist’s book Almost Everybody, it is largely surrealist and metaphorical. However, these works have commercial motivation. Kapusta’s art begins when his works are motivated exclusively by some internal need. When a philosophical problem needs to be solved, a visual experiment helps in this undertaking, as in the case of the drawings Inside Out and Outside In. This work's point of departure is the problem of two directions that are seemingly opposite, but in actuality, they are complementary. Another example is the series of studies on introducing order into chaos, where a sufficient number of similar elements dispersed chaotically in space eventually come to behave like a structure, i.e. embodied order.
The need for order seems to dominate in nature, and conse-
quently in the artist's work. He acknowledges his fascination
with geometry, which also manifested itself, soon after his
arrival in New York in 1981, in color structural compositions
consisting of pyramids, cones, or various stereometric forms,
densely filling up the surface of the canvasses.
The highest point so far in Janusz Kapusta’s work was a
1985 discovery on the borderline of art and mathematics. It is
a new shape, previously unknown: an eleven-faced solid,
which the artist named the K-dron. (The suffix -dron is short
for -hedron, denoting a face of a solid, cf. polyhedron). “Since
the solid has eleven faces, and the eleventh letter of the
English alphabet is K, we have called it the K-dron”, explains
Like most discoveries, the K-dron is a product of chance.
In 1985 Kapusta and another Polish artist, Andrzej Czeczot,
were preparing an exhibition of their works in New York.
The catalogue was to include - among others - two of
Kapusta’s drawings from the series Plus Minus Infinity, made
when he still lived in Poland. The thousand copies of the cat-
alogue turned out to be so badly printed that the printers
themselves offered to do the job again. Nevertheless, Kapusta
took home the botched edition with reproductions of his
works. He later wrote: “I sat at the table and mused: if life has
any meaning, then it must have it everywhere and in every
moment of time. So how mentally challenging, how paradox-
ical it is that I am sitting looking at a pile of totally useless
publications which provoke me to wonder about the meaning
of things. And at that very moment I noticed that if I clipped
the Plus Minus Infinity drawings in a right way, I would obtain
an intriguing image, in which the sides of a square inscribed
within a square stretch out and recede into mysterious space.
That night I cut out of the catalogues several hundred squares
inscribed within squares and I noticed that they allow for an
amazing number of different arrangements.”(6)
Two weeks later, invited to take part in a competition to
design a three-dimensional ceramic tile, he noticed that the
drawing Plus Minus Infinity can be transformed into a solid
figure. With his characteristic passion of an explorer eager to
engage in intellectual and artistic pursuit, he worked all
night, building the models. As he recalls, “I was particularly
interested in the fact that choosing a certain angle I could
construct an object which allows for such transformation
that the same elements may be arranged in the plan of both
a square and a hexagon. I knew from the book by Hugo
Steinhaus, Mathematical Kaleidoscope, that there are only
three regular nets which make it possible to cover any plane
infinitely. They are a triangle, a square, and a hexagon. The
most fascinating thing was that with the shape I had discov-
ered, I could cover two of those nets. I made ten modules. I
was fascinated by the number of possible combinations and
their amazing reaction to light. After a whole night of experi-
ments, at dawn, something strange happened: one of the
modules accidentally fell on top of another and disappeared.
I held my breath. I had a shape that offered such a huge num-
ber of combinations, unlike the other solids.”(7)
Kapusta later patented his K-dron.
Talking and writing about it, he often insists that he did not
create or invent it. It had always existed, like the cube or the
sphere; it only needed to be noticed, discovered in the net-
work of divisions of three cubes put together. This brings to
mind the famous statement of Michelangelo, who maintained
that sculptures already exist a priori, contained in blocks of
marble. The sculptor only has to take away the superfluous
material which prevents us from seeing them.
Kapusta noticed not only that the K-dron is extremely easy to
transport, because two K-drons put together form a paral-
lelepiped; he also found a great number of possible product
applications for the K-dron. It may be used e.g. as tiling on the
facades of buildings (the first K-dron blocks were produced
at Alias Studio in Hollywood, Los Angeles, in 1995), as roof
tiles, as a box with eight K-drons which one may rearrange
to create various patterns (used by the New York company
AGE to construct a game), as a projection of the Earth’s sur-
face on two K-drons, instead of a globe, etc.
These utilitarian properties of the new shape discovered by
the artist do not seem to be the most important, however. Jay
Kappraff, a professor of mathematics at New Jersey Institute
of Technology, the author of the book Connections - The
Geometric Bridge between Art and Science writes: “Janusz
Kapusta is an unusual designer, who succeeded in trans-
forming his passion for philosophy into a fascinating and ver-
satile structure . However, I believe that the K-dron is more
of a mathematical discovery than a design... In the sense of
G. Spencer-Brown, Kapusta made his mark of distinction
upon the undifferentiated space of Euclidean geometry, and
the result was the K-dron, a new form of fundamental impor-
tance that breaks the symmetry in the infinite space stretch-
ing both above and below it.”
With his technical education and interest in mathematics,
Kapusta appreciates this aspect of the K-dron. However, as
an artist he views the K-dron as an important aesthetic phe-
nomenon. The evidence is provided by the present exhibition
devoted exclusively to this shape, in its many aspects and
meanings, including its philosophical implications. According
to the artist, “If one imagined the world filled up with cubes,
then the K-dron already existed in this network of solids,
intersected by all possible axes, shifted by half a module.”(8)
This statement contains not only a trace of the surprising
turns of the artist's imagination, but also some ontological
The K-dron exhibition shows this new shape discovered by
Kapusta in its different aspects, formats and materials. The
K-dron is shown as a sculpture, as two-color tiling on both
sides of a wall; it is also shown appearing out of the network
of standard divisions of four cubes put together. One can see
how two black and two white K-drons make it possible to
produce 38,416 visually different combinations, how two
K-drons put together form a parallelepiped, how profiles of
the K-dron blown up on a wall produce a three-dimensional
image. The exhibition also shows high K-dron spikes, K-dron
pyramids, a wall with K-dron tiling made of gray polychrome
wood, creating a mobile structure, undergoing constant
changes depending on the angle of falling light and its inten-
sity. This is not only entertainment for the eye, but a chal-
lenge for the mind. The exhibition presents the K-dron in the
form of sculptures, bas-relief, paintings and photographs, or
as made up of soft ropes hanging down from the ceiling of
the room, or conversely, of vertical rods of different height
rising up from the floor. The K-dron is also rendered as a
drawing on many sheets of transparent foil hung at equal dis-
tances one behind another. The introduction of space into the
drawing produces three-dimensional illusory forms of pyra-
mids-corridors, receding far back into the background.
Though monothematic from the theoretical and practical
point of view, this exhibition is extremely varied. Yet as
regards its message, it turns out not no be so varied as the
materials and forms used by the artist. In fact, the K-dron is
the final solution to the philosophical and aesthetic problem
which fascinated Kapusta when he first began working on
his drawings, Plus Minus Infinity, an attempt to represent
infinity by visual means: as seemingly material space reced-
ing from us into the perspective depth, and at the same time
- paradoxically - approaching us. In contrast to all the other
solid figures, the K-dron has an inner and an outer form. In
addition, one of the surfaces is partly inside and partly on
the outside of it. The artist made a two-colored model in
blue and black to show some of the paradox and mystery
reminiscent of those which baffle us about the Moebius
strip. Like the Moebius strip, the K-dron has its own philo-
sophical implications, provoking us to reflect on the
absolute and the relative, and to ask questions on whether
we should draw borderlines between entities, dividing them
into same and different, as in the European cultural tradition
deriving from Aristotelian philosophy, or rather discover
their unity in the processes of transformation, as taught by
The discovery of the K-dron was the most important event in
Apostasy work up to now and the third step of his imaginary
ladder. It is also still the object of the artist’s reflection.
Kapusta argues that the appearance of the K-dron has
enriched reality; the shape is very expressive and never
monotonous. However, he has already begun work on a dif-
ferent project: the superimposed striped circles which pro-
duce unexpected effects when moving, thanks to the phe-
nomenon of interference. He is trying to find a key there to
phenomena close to his interests: fractions, perspective, and
Kapusta’s outlook on art is briefly summed up in the follow-
ing fragment: “Art is for me the domain of human activity in
which we humans have permitted ourselves to conduct a dif-
ferent, sometimes more subtle, but independent exploration
of the world. For me, art does not express what is known, but
is a tool for gaining this knowledge. It is a kind of medium
allowing for the most personal experiments. It is in art that
we recognize the natural or artificial limitations on our
humanity, beauty and imagination.”(9)
These statements are another manifestation of the artist’s
unique way of thinking, his rich imagination, engaged in
unusual pursuits. Who else reflects on infinity while bicycling
to work? Who tries to draw it, knowing that this is an impos-
sible task? Who exercises his imagination for six months,
thinking out hundreds of definitions of the point? Who imag-
ines the world filled with cubes? And who creates a scale -
a ladder whose steps he is to climb, but not before building
them up from scratch?
Someone who can do all this may still have a lot to offer us.
1. J. Kapusta, K-dron. Opatentowana nieskonczonosc
[K-dron. Patented Infinity]. Warszawa 1995, p. 29.
2. ibid, p. 29.
3. ibid, p. 30.
4. ibid, p. 34.
5. ibid, p. 42.
6. ibid, p. 16.
7. ibid, p. 20.
8. ibid, p. 52.
9. Janusz Kapusta, from the speech on the occasion of being
awarded the Alfred Jurzykowski Prize in Arts in 1998.