Faculty of Science, Palacký University, Olomouc
Structures can be seen, examined and created, but they can also be ignored, changed and destroyed. Every structuralism that studies structures always emphasises the whole over individual sections (the whole is greater than the sum of its parts), with a crucial role ascribed to the organisation of structures and the functional relationships between their elements (constituent parts). The same principle forms the basis of Hermann Haken’s (1927) synergetics1 and my fractal analysis2 of structures in quantitative linguistics – which, like the majority of structuralist movements, was preceded by Swiss linguist and semiotician Ferdinand de Saussure (1857–1913). However, this approach deliberately highlights the inadequacy and limited applicability of Descartes’s analytic method (Discourse on the Method, 1637).
The French today understand structuralism or post-structuralism primarily as a monumental philosophic movement represented by Michel Foucault (1926–1984), Roland Barthes (1915–1980), Jean Baudrillard (1929–2007), Gilles Deleuze (1925–1995), Jacques Derrida (1930–2004) and others. In Czech circles, structuralism is justifiably often associated with the Prague Linguistic Circle, whose core members were Roman Jakobson (1896–1982), Jan Mukařovský (1891–1975) and Vilém Mathesius (1882–1945). The Prague structuralists’ aesthetics evaluated a work semiotically as a sign whose parts and whole are bearers of meaning.3
In no way is my aim here to evaluate. However, I would like to proceed from a mere description of structures to at least an introduction to possible perceptions of structures. It is not important to understand the technical principles in this regard. My approach could be called radically “anti-Bernsteinian”, as the final goal is everything and the means are nothing – a concept that was once calculatedly attributed to the Jesuits. Recognising this risk, I will use an analytical approach in an attempt at describing the structures of François Morellet (1926–2016) and Zdeněk Sýkora (1920–2011) in as “structural” a manner as possible.
For me, the French tradition of visual, combinatorially-variable structure does not start with philosophers or linguists, but perhaps surprisingly with the Carmelite Order, which gave rise to two remarkable figures: Sébastien Truchet (1657–1729) and Dominique Douat (1681– date of death unknown). In addition to fulfilling their monastic obligations, they both worked with different, primarily practical disciplines. In 1704 Truchet published a study6 on creating patterns out of tiles split by a diagonal line into two differently-coloured triangles. Instead of combinatorial patterns, the arrangement options are demonstrated on lovely copper-plate engravings with the use of symmetry. Since 1987 Truchet’s studies have been available in English translation by Pauline Boucher with additional modern commentary provided by Cyril Stanley Smith.7
Truchet’s monastic brother Dominique Douat built on his studies in 1722 with a book8 in which Douat works with the simple concept of successively rotating each tile by 90°. In so doing he codified the four basic “Truchet tiles” for a gridded field. Naturally, the patterns that arose were again consciously chosen based on symmetry. Douat’s book can be considered a period manual for the purely deterministic production of predominantly (but not necessarily) symmetrical structures.
Fig. 1: Truchet’s graphic breakdown of reductions from a total of 64 possible pairs
Fig. 2: Four “Truchet tiles” and 16 possible variations with repeating pairs from Douat’s book
François Morellet did not develop his compatriots’ work any further; in fact, he did not even know about it. 9 Instead, he relied more on the opposite principle of stochasticity, which I will hereinafter refer to as randomness.10 Into his best-known structures, Morellet encoded numbers from the phone book11 and the decimal expansion of the Ludolphine number π = 3.14159…,12 expressing the ratio of a circle’s circumference to its diameter. As coding means creating a bijective (one-to-one) assignation (in this case, digits to elements), the specific method of coding depends on the number of selected elements. Morellet then plotted the resulting coded sequence lexicographically on a gridded field (chequerboard).
For his early 1960s binary (digital) structures made using two elements – which had long, yet perfectly fitting titles: Répartition aléatoire de 40 000 carrés suivant les chiffres pairs et impairs d’un annuaire de téléphone, 50 % bleu, 50 % rouge (resp. 50 % noir, 50 % blanc, etc.) – it was sufficient to assign colour to even numbers and another to odd numbers. Different coding for two differently-coloured square elements could be executed, e.g., in a binary system (1 = black, 0 = white), which to a certain extent would today correspond to the increasingly popular use of QR (Quick Response) codes. In the same period, Karl Otto Götz (1914) completed his related binary two-colour structures titled Statistisch-metrische Modulation, Statistische Verteilung and Density. In the 1970s Ryszard Winiarski (1936–2006) created his extensive numbered Area (Polish: Obszar) series based on a similar principle. 13 In 1971 Galerie Denise René in Paris published an album containing eight corresponding Morellet screenprints with the shortened title Morellet, 40 000 carrés.
In 1958 François Morellet created his famous black-and-white structure, titled Répartition aléatoire de triangles suivant les chiffres pairs et impairs d’un annuaire de téléphone, again by encoding numbers from the phone book – but this time he used four Truchet tiles plus white (empty) and black (full) squares. However, I do not know how he assigned the six elements to individual digits. He might have used only the first six digits (1, 2, 3, 4, 5 and 6) and ignored the rest, or perhaps he used dice, but the title of the structure rules out this option.
Fig. 3: A page from the telephone book prepared by Morellet for encoding into the structure
Fig. 4: The first 875 digitally-differentiated (even=white, odd=grey) digits of the decimal expansion of the Ludolphine number π
On several occasions Zdeněk Sýkora provided a detailed explanation of the constructional principle (“My System”) behind his structures himself14 or in conjunction with mathematician Jaroslav Blažek “Computer-aided Multi-element Geometrical Abstract Paintings”, Leonardo 3, 4 (1970), 409–413. These sources clearly indicate that a computer was always used (likely for the first time15 in Europe) as an aid to facilitate technically complex operations. Once again it should be emphasised that Zdeněk Sýkora made sure (perhaps too solicitously) to preserve the “painterly nature” of his works.
It is known that Sýkora learned more about combinatorics and the potential it offered from Hlaváček’s translation of chapter 6 of Werner Haftmann’s The Mind and Work of Paul Klee (Faber & Faber, London, 1954). In the chapter titled “Pedagogical Sketchbook”, Haftmann describes and interprets Klee’s teaching methods from his eponymous brochure,16 which was an official Bauhaus textbook. Sýkora especially connected with a passage about the repetition of units forming a structure: “Since these figure arrangements rest on the principle of repetition, any number of parts can be added or taken away without changing their rhythmic character. Therefore the structural character is divisional.”
Less well known is the fact that Sýkora’s system is essentially deterministic. After making the first well-considered insertions of tiles into a grid (establishing “nests”), subsequent additions are made based on a set of fixed rules. Randomness enters the game only if the instructions are ambiguous – but it is more a question of choosing from several options than randomness per se.
For example, in the 1966 painting Black and White Structure (oil on canvas, 220 × 110 cm) described by Sýkora and Blažek in Leonardo, the restrictive rules of a pre-set scenario would only have led to possible modifications to the original, which my former student Martina Losová was able to computer simulate (see fig. 5–9) using permissible variations (there are four different relational rules).17 The same principle can be applied to practically all of Sýkora’s Structures, including his prints, with the exception of his early attempts (Grey Structure and all of his paintings from 1963).
Fig. 5: Computer simulation of Sýkora’s Black and White Structure
Fig. 6: Another variation based on the same rule #2 (differing red elements)
Fig. 7: Variation based on rule 0
Fig. 8: Variation based on rule 1
Fig. 9: Variation based on rule 3
The rules of Sýkora’s system show a certain formal similarity with the best-known cellular automaton – Conway’s Game of Life.18 This interesting question was already being studied in the late 1940s as part of the exploration of discrete dynamic systems by such brilliant mathematicians as John von Neumann (1903–1957) and Stanislaw Ulam (1909–1984). John Horton Conway’s (1937) Game of Life was first presented in a 1970 article in Scientific American by popular science writer Martin Gardner (1914–2010). Stephen Wolfram brought research into cellular automatons to the next level with his aptly titled A New Kind of Science (Wolfram Media, Champaign, Il., 2002), a 1,200-page book that went on to become a bestseller.
Other interesting comparisons are the prints of Anni Albers (1899–1994),19 who studied at the Bauhaus, or the artist’s books of contemporary young American artist Tauba Auerbach (1981),20 whose work can be found in the collections of the Centre Pompidou in Paris, among other places – but this would go beyond the scope of this study. In closing, I will limit myself to noting that two of Zdeněk Sýkora’s (combinatorial) Structures and one of his Macrostructures were used for the covers of Czech university mathematics textbooks:
- G. Kuroš, Kapitoly z obecné algebry (Alexander Gennadyevich Kurosh, Lectures on General Algebra), 1st edition, Academia, Prague, 1968.
- G. Kuroš, Kapitoly z obecné algebry (Alexander Gennadyevich Kurosh, Lectures on General Algebra), 2nd edition, Academia, Prague, 1977.
- Ralston, Základy numerické matematiky (A First Course in Numerical Analysis), Academia, Prague, 1973.
More detailed information about the two artists’ structures within the scope presented here, as well as a comparison of their approaches, can be found in my articles:
- Andres, Tvorba Zdeňka Sýkory očima matematika (The Work of Zdeněk Sýkora through the Eyes of a Mathematician), unpublished, final version from 2008, 15 pages.
- Andres, Zdeněk Sýkora and François Morellet: Parallels and complementarity. Leonardo, 47, 1 (2014), 27–31, 34.
- See H. Haken, An Introduction. Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry and Biology, Third Revised and Enlarged Edition, Springer, Berlin, 1983.
- Andres, “On de Saussure’s principle of linearity and visualization of language structures”, Glottotheory 2, 2 (2009), 1–14.
- An engaging book that focuses on the semiotics of images is Peter Michalovič and Vlastimil Zuska’s book Znaky, obrazy a stíny slov, published by AMU Press, Prague, in 2009.
- Measuring the aesthetic value (quality) in quantitative terms can be tricky and misleading. However, a whole range of evaluation criteria is available based primarily on the concept of Shannon entropy. This type of aesthetics is associated with names such as George David Birkhoff (1884–1944), Abraham Moles (1920–1992), Fred Attneave (1919–1991), Herbert Werner Franke (1927) and others. In the Czech Republic, Jaroslav Nešetřil (1946) has systematically explored the topic of mathematical aesthetics.
- The opposite of Eduard Bernstein’s (1850–1932) motto: “The movement is everything, the final goal is nothing.”
- Sébastien Truchet, Mémoire sur les combinaisons, Mémoires de l’Academie Royale des Sciences, 1704, 363–372.
- S. Smith, “The tiling patterns of Sébastien Truchet and the topology of structural hierarchy”, Leonardo 20, 4 (1987), 373–385.
- Dominique Douat, Methode pour faire une infinité de desseins differens, avec des carreaux mi-partis de deux couleurs par une ligne diagonale: ou observations du pere Dominique Douat, religieux Carme de la province de Toulouse, sur un memoire inseré dans l’Histoire de l’Academie royale des sciences de Paris l’annee 1704, presente par le reverend pere Sébastastien Truchet, religieux du même ordre, academicien honoraire, Florentin de Laulne, Claude Jombert, André Caillau, Paris, 1722.
- Confirmed in a private discussion with the artist in February 2010.
- In mathematics these two concepts differ; in the 1950s, the Prague school of probabilists led by Antonín Špaček made a fundamental contribution to the study of random processes in connection with probabilistic generalisation of deterministic operator theory. Neither of these concepts can be mistaken for another important mathematical concept, deterministic chaos, which in a certain regard can be understood as a higher (dynamic) form of order.
- The first four digits, which represented the city code, were always omitted.
- A 40,000 decimal expansion of π to 40,000 digits is available online. The world record for calculated digits exceeds this limit many times over and is continuing to rise. See, e.g., R. P. Agarwal, H. Agarwal, S. K. Sen, “Birth, growth and computation of pi to ten trillion digits”, Advances in Differential Equations 2013, 2013:100, 1–59. Morellet also used the coding of the decimal expansion if π for his lines. See François Morellet, Konstruktionen mit der Zahl π, Chorus-Verlag, Mainz and Munich, 2001.
- See J. Grabski (ed.), Ryszard Winiarski. Prace z lat 1973–1974, IRSA, Krakow, 2002.
- See Kappel, L. Sýkorová: Zdeněk Sýkora – 90. Verzone, Prague, 2010, pp. 64–71. It was first formulated in 1967 at the request of Umbro Apollonio from Archivio Storico delle Arti Contemporanee in Venice; see Lenka and Zdeněk Sýkora Archive, Louny.
- Sýkora’s first Structure created with the support of a computer in the sense described here is Black and White Structure (oil on canvas, 100 × 100 cm) from 1964, originally titled Variation I VI E. In addition, similar structures were created (but not in the manner described here) in Czechoslovakia in the early 1960s – such as an illustration by Miloš Noll (1926–1998) on the last page of J. R. Pick’s book Monoléčky (Mladá fronta, Prague, 1961) Jiří Krejčí’s illustration on the cover of J. Oliverius and R. Veselý’s conversational Arabic book Egyptská hovorová arabština (Státní pedagogické nakladatelství, Prague, 1965). Both structures were composed of Truchet tiles.
- A Czech translation was published in 1999: Paul Klee, Pedagogický náčrtník, Triáda, Prague, 1999 (translated by Anita Pelánová from the German, which had been published in 1925 as the second in a series of Bauhaus books).
- Losová, Matematické aspekty výtvarného díla Zdeňka Sýkory (Mathematical aspects of the work of Zdeněk Sýkora), thesis, Palacký University, Faculty of Science, Olomouc, 2010.
- The Game of Life is described in a broader context in the book by Z. Neubauer and J. Fiala Střetnutí paradigmat a řád živé skutečnosti, Malvern, Prague, 2011, pp. 39–49.
- Fox Weber, B. Danilowitz, The Prints of Anni Albers. A Catalogue Raisonné, 1963–1984, The Josef and Anni Albers Foundation / Editorial RM, Everbest Printing, Co., Panyu, Guangdong, 2009.
Prof. RNDr. Dr. Hab. Jan Andres, DrSc. studied Numerical Mathematics at the Faculty of Science, Palacky University in Olomouc, where he also serves as head of the Department of Mathematical Analysis and Mathematical Applications. He has been a visiting professor at American and European universities, particularly in Rome and in recent years at the Sorbonne in Paris. He gained the highest scientific rank not only in the Czech Republic but also in neighboring Poland.
Professor Jan Andres is also a member of many professional scientific journals around the world. He specializes in mathematical methods of exploring nonlinear dynamics and fractals and their applications in quantitative linguistics.
Many thanks to Professor Andres for allowing Kulturebite to post this text.