exhibits information spidron page

Is the Spidron a polygon? As I often say, a Spidron arm is a spiralling formation constructed from a sequence of two kinds of triangles (usually isosceles ones). It is not possible to specify the number of its vertices or sides. Its area and circumference can be determined in the limit, but no matter how large or small a piece of it we take, it can always be extended by additional triangles. As professor of quantum logic Gyula Fáy put it, the Spidron is a process. It is a procedure in which, like the tower of Babel, the building of Spidrons can be continued as long as we want. The starting point can be set arbitrarily. The ratios of angles and lengths are the crucial aspects. Those ratios are constant and ever-present in the Spidron formations. The Spidron-arm cannot be finished, but it can be given a starting point with a clever trick: I can pick one triangle and declare that it will be the first and largest one. In order to prevent the addition of an even larger one, I can reflect the entire spiral formation across the base of that triangle. If it is a central reflection, I get the form we initially called the Spidron. If it is a mirror reflection, we get a figure like a pair of horns, which we have called the Hornflake. Various versions of those two figures fill the world of Spidrons. Those Spidron arms can be used to construct extraordinary shapes in the plane and in space. Our research ranges from plane tilings and regular and semi-regular solids through saddle surfaces and to the investigation of special, aperiodic tilings and quasi-crystals.

But the process induced by the subject of our shared thinking – which happens to be the Spidron, this time – in various human communities is at least as interesting. It generates action groups, provokes arguments and often results in striking scientific and aesthetic qualities. It was particularly astonishing for many people that it can be used to create structures with an entirely novel kind of movement. The Spidron has defined a position. I hope that sooner or later, through processes of their own, others will also be able to occupy that position. At last, others may also achieve their rightful positions.

Dániel Erdély

Péter Niedermüller

About the Spidron, in Parentheses

I think I will never understand now what the Spidron actually is (what's even worse, I don't quite know what it is that I am supposed to understand, either). Of course, I can see and I know that it is a sort of object (object? or plane? or planes? or does that not matter? perhaps a planar object or possibly an objectish plane? or is it a perspective? or...) living/existing/moving in space. And it is also beautiful - at least I like the shape, the movement, the intertwining planes, this whole thing without beginning or end. I also like the way it connects people, people who are in continuous communication through Internet chats, tinkering away with the form, drawing, making calculations, and when addressed, they just stare back at us as if they were outside this world (anyway, is the Spidron a part of this world, or is it another dimension? who is the other?). Viewed "from a distance", it seems that the Spidron is an addiction (so is it not an object, a plane or a geometrical figure... - but is it possible for a geometrical figure to become someone's obsession?)[...]

The relationship between geometry and art
The works of art that contain geometric constructions are the artistic depictions of more or less well known geometric objects. Some of those geometric objects have been known for several thousand years and they usually do not represent anything novel for the scientific perspective. The Spidron system discovered by Dániel Erdély is a peculiar exception: it is an interesting new development mathematically as well as artistically. The construction is great precisely because it is very simple. The original "recipe" seemed to be totally basic: take a regular hexagon, draw its shorter diagonals. Repeat this with the smaller hexagon created inside the original one. Repeat this ad infinitum. The artist's intuition was needed to notice in this construction the spiral forms called "Spidron arms", along whose external boundary the regular hexagon can be folded, resulting in a construction that can move in space. This is where the mathematician's task began: it had to be proven that the movement is not only permitted by the degree of distortion that the paper is capable of but that the artist has discovered a mathematically exact, well-defined movement. It was a great honour for me that I could be the one to undertake that study for the first time. Those were the first steps of the creation of the Spidron system. Perhaps at that time, neither the artist, nor the mathematician suspected that this simple idea would be the foundation of many other constructions with both artistic and mathematical significance.

Lajos Szilassi

> The function generated on the basis of the algorithm demonstrates that it is possible to deform a Spidron nest.


Prints by Rinus Roelofs

In the Spirit of Professionalism

The original form of the Spidron figure, consisting of a repeating, diminishing sequence of an isosceles triangle with angles of 30-30-120 degrees, then a regular triangle raised on one of the two equal sides, etc., forms a spiral-like fundamental domain for the plane “crystal group” p6. This is no. 16 among the 17 plane groups. Therefore the Euclidian plane can be tiled in a regular fashion with the Spidron figure. Interesting congested holes, “vortices” form at the order six centres of rotation. The tiling is also very beautiful visually, as can be seen in the drawings and other works of Dániel Erdély. Patterns with similar structures can be used to construct other aesthetic tilings on a spherical surface and in the hyperbolic geometry of János Bolyai. The possibilities of the symmetry groups featured were only exhausted definitively around 1970. Surprisingly, the Spidrons of Dániel Erdély also allow us to venture out into space as they produce wonderful curved faced regular polyhedra that we can admire in sculptural form. This also allows highly imaginative and inventive modelling of spatial crystal groups and the creation of sculptures. The computer can also be utilised in those tasks. It provides a perspective on the possible relationships between applied and applicable mathematics as well as that between science and art – which we may welcome with great pleasure.

Emil Molnár, professor
at Budapest University of Technology and Economics (BME), Department of Geometry

Emil Molnár observing the Szilassi polyhedron

The fundamental domain of a half-cube constructed from classic Spidrons. Three of these forms and their mirror images can be assembled into a complete cube. The figures below show the exposition of Dr. Emil Molnár.


A new area of research is opened by the possibility of replacing saddle surfaces by Spidron constructs.

Prints by Rinus Roelofs
Rinus opened a new chapter in the history of the Spidron system. We have exchanged several thousand e-mails which have resulted in innumerable new ideas and works.

The Dutch artist Rinus Roelofs, who was the first one to produce a scientifically sound computer representation of the rich forms and the mobility of the Spidron system. His contribution has enabled us to present our work at various international conferences and exhibitions. With his help, we built the “breathing polyhedra” from Spidrons.
Marc Pelletier’s brilliant ideas connect knowledge from diverse fields of geometry from Platonic polyhedra to Kepler’s rhombohedra and from Penrose tilings to quasi-crystals with the Spidron figures.

Amina Buhler-Allen helped a great deal by contributing several ideas and also creating some prototypes

  Amina and her son, Devon had the notion that the famous Yoshimoto Cube could also be spidronised

< Yoshimoto cube

The highly popular Intuition, Innovation, Invention exhibition was put on at the Budapest Palace of Art at the initiative of László Beke in 2000. The Spidron was shown there as a space filling system. Later, in 2004 and 2006, he was the curator of the ORNAMENTS exhibition and conference at the Renée Erdős House. Starting in 2005, Emil Molnár undertook to be my supervisor at the Cultural Studies Graduate School of the Pécs University of Science. The working title of my dissertation is Space and Innovation.

I used the two kinds of triangles of the Spidron to represent the prime factors of natural numbers and their powers.

The image on the right is a study of the distribution of primes within the system of natural numbers.

  Regina Márkus helped with the completion of many seemingly impossible paper models. A 3D model of the dodeca-Spidron ball constructed using a computer by Marc Pelletier was published in November 2006 by the American children's science magazine MUSE.


Pelletier, Ballegooijen, Erdély
In commemoration of the 30 year anniversary of the Scientific American cover Theory of Tiles (January 1977), a question was posed: Can the cover art be modified such that the tiles become Spidrons. Each individual tile in the resulting pattern has the same area and obeys the same rules as the original tile created by Penrose & Conway.

I met Craig S. Kaplan at the Bridges Conference. The young scientist is an assistant professor at the School of Computer Science of the University of Waterloo, Ontario and the author of several important scientific papers. At Marc Pelletier's request, he used one of his applications to create beautiful hyperbolic Spidron tilings.

Craig's site: http://www.cgl.uwaterloo.ca/~csk/projects/spidron/

Spidronized Platonic solids

Walt van Ballegooijen faithful friend, inventor and co-creator of numerous complex Spidron constructions. He will work on hopeless things. He wrestles with challenges with singular tenacity. His knowledge of geometry and software is so extensive we usually find him hard to follow.

Kirstóf Fenyvesi

“What else is it good for?”; What is the Spidron good for?

“We draw upon the iconography Whose mystery is able to contain The boundlessness, the storm of all existence, Give chaos form, and hold our lives in rein.

The pattern sings like crystal constellations, And when we tell our beads, we serve the whole, And cannot be dislodged or misdirected, Held in the orbit of the Cosmic Soul.”

Hermann Hesse: The Glass Bead Game (translated by Richard and Clara Winston)

The remote Province where the priests of the human spirit and culture play the glass bead game, that most exalted expression of the archaic unity of science, art and, in a sense, religion, is called Castalia, and it is by no means certain that it only ever existed in the imagination of Hermann Hesse. At any rate, Dániel Erdély seems quite real and by no means fictional, yet he is a glass bead game player like Hesse's hero, Josef Knecht: a one-time magister ludi who has surely been to Castalia. Everything he has brought back from There suggests the pervasive good cheer that is the maximum of humanity, the serious and deep Castalian cheer of the ever-renewing spirit. [...]

< A monumental Spidron relief was installed in the new building of the National Rehabilitation Institute at Budakeszi >

The title of the picture: “What else is it good for?“; What is the Spidron good for?
Dániel Erdély and Kristóf Fenyvesi
I created the first piece of the Spidron system, a swirling relief as a piece of homework for the theory of form class of Ernő Rubik at the Budapest University of Art and Design in 1979, though I had already created a plane lattice in 1975 or 1976 for a class by István Varga-Hajdú at the Print Industry Vocational School. At that time, I had not become aware of the spatial mobility and adaptibility of the system. That very first design showed 9-pointed Spidrons in the plane. The family of geometric shapes has undergone a number of developments and received international recognition thanks primarily to the mathematical description produced by Dr. Lajos Szilassi. In international research team is at work on the Spidron project. Its members include Walt van Ballegooijen, Marc Pelletier, Amina Bühler-Allen, Craig S. Kaplan, Paul Gailiunas, Simon Erdély and Rinus Roelofs.

In addition to those already mentioned above, I would like to thank the following people for their steadfast and committed work associated with the development of the Spidron system: the teachers and students of the Cultural Studies PhD programme of the Literary Studies Graduate School of the Pécs University of Science, Cristiana Grigorescu, Louise McCagg, Emil Molnár, Péter Niedermüller, László Beke, Miklós Hadas, Gergő Kiss, Péter Kőszegi, István Sági, Veroni Szalai, Attila Fábri, Iván Halász, Péter Szûcs, Zsófia Végvári, György Falk, Kate Jones, György Janovkszky, Zoltán Nagy, Gábor Bachman, Yasar Meral, Gusztáv Bodnár, Ervin Miklós, Miklós Laczkovich, Lukács Pálfalvi, Andrea Szekeres, Adéle Eisenstein, Balázs Földvári, Regina Márkus, Ildikó Szigeti, János Saxon-Szász, Zsuzsa Dárdai, John Hiigli, István Tenke, Peter Hanák, Jakab László Orsós, László László Révész, Rita Horváth, Gábor Roskó, Wenninger Magnus, György Darvas, Sándor Kabai, Szaniszló Bérczi and my family, Ági, Jakab, Janka and Matyi and Mara for their patience and understanding.

In addition, I would also like to thank the people who helped disseminate our work to a wide audience: Reza Sharangi, Tom Rodges, Ivars Peterson, John Sharp, Jaques Ferroul, Koji Miyazaki, Cor Wetting, Jerry Gray, Miklós Sulyok, Tamás Török, Attila Zsigmond, the C3 and Miklós Peternák, Csaba Sárvári, István Szenes, Gábor Kenesei, András Böröcz and László Böröcz, Márta Kádár, Erzsébet Bogácsi, György Szőnyei, Dóra Szepesi, Tibor Borzák, Miklós Bohanek, Tamás Tarcsay, György Szegő, Eszter Göcz, István Hajdu, Tamás Bodoki, Péter Mújdricza, Tamás Vajna, Csaba Madarász, Borisz Szántó, Irina Pintér, Gergő Péter Havas, Tamás Milhofer, Zsuzsa Tószegi, Enikő Huszár, Gusztáv Szöllősi, András Weichinger and Zoltán Vas.

Special thanks go to the organisers of the present exhibition and series of events: the Pécs Cultural Centre, the Pécs – European Capital of Culture Programme and, in particular, Andrea Brückler, Jolán Somlyay, Márton Méhes, Kristóf Fenyvesi, Attila Doboviczki and Rita Varga.

You can find more information about the Spidrons in the "Spidron catalogue".

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