5. Regularity through the ages

Regular patterns

The reconnaissance journey of the fourfold division leads into the world of regular patterns. STEVENS (1980) wrote a comprehensive book on the phenomenon of symmetry, dealing with the basic ingredients of repetition in two-dimensional patterns. He distinguished four types of movements to obtain multiplication of patterns:

 —————–    1. translation         from     b    to     b

—————–     2. rotation                            b    to     q

—————–     3. reflection                         b    to     d

—————–     4. glide reflection                b    to     p

These movements can be recognized in the basic fourfold division of a communication. The translation is the Prime Originator in the first quadrant. The prime shift generates space and the opportunity for comparison. The rotation in the second quadrant has a cyclic overtone and adds a different dynamic element to the exchange of information. The reflection in the third quadrant is a confrontation with boundaries, a view in the mirror. The glide reflection in the fourth quadrant is an elaborated combination of translation and reflection.

The essential feature in the recognition of patterns – including the mind patterns – is the so-called fundamental region. STEVENS (1980; p. 57) gave the following definition: ‘a fundamental region is the region of minimum area that can be repeated without gaps of overlaps to make a complete pattern.’ LAUWERIER (1988) called this unit a ‘primitive cell’, to indicate the repeating element. It is instructive to look at the anatomy of a fundamental region and the way it is created. Because in the end the human memory is just a set of (mind) patterns, constructed from various fundamental regions.

Four steps are necessary to outline the smallest unit, or to ‘create our own fundamental region’ (STEVENS (1980, p. 177)(fig. 10):

 1. An irregularity on a horizontal line between A and B; any shape is permitted. ESCHER (1958/1986, p. 118) recommends a simple contour line without too many deep incisions.

 2. The shape of the horizontal irregularity is repeated on a parallel line in C and D; it is essential for the creation of a pattern that the shape is the same as between A and B;

 3. An irregularity on the vertical line, starting at A and ending at C; again any shape is permitted.

 4. The shape of the vertical irregularity is repeated on a parallel line between B and D; now a geometric form is created, which is called a fundamental region: the region of minimum area that can be repeated to obtain a whole pattern.

These are the four distinct (mind)actions required to build-up a fundamental region, which forms a pattern by repetition.


Fig. 10 – The four phases in the development of a fundamental region, the minimum area that can be repeated to obtain the whole pattern.

STEVENS (1980) and LAUWERIER (1988) referred to the Dutch mathematician and graphic artist Maurits C. Escher (1898 – 1972), who plays a major role in the (re)discovery of visible regularity and symmetry in the early decades of the twenties century. Escher’s book ‘Regelmatige Vlakverdeling‘ is an excellent introduction to his work (ESCHER, 1958/1986).

ESCHER (1959/1970, p. 11) described the regular cover of a plane as ‘the richest source of inspiration I ever discovered. And it is still not exhausted. The symmetry drawings show how a plane can be divided in or filled up with regular shapes, who border each other without overlap. The Mores were masters in this art. They have, particular in the Alhambra in Spain decorated the walls and the floors by  congruent, multicolored pieces of majolica, which fits perfectly. What a great pity that they were not allowed to make portrayals! They only used figures with an abstract-geometric design. Not a single Moorish artist has ever, as far as I know, ventured (or maybe he never imagined) to use natural or recognisable figures like fishes, birds, reptiles or human beings as elements in a division of a plane. This restriction seems to me unbelievable, because it are just the recognisable elements in the patterns which is the reason of my continuous interest in this domain.’

Escher finds a response in the modern computer-generated graphics. The fascination to create regular patterns within certain boundaries is the common interest. William KOLOMYJEC (1976) rendered homage to Escher in an article on the appeal of computer graphics. He uses a computer program to draw ever-smaller fundamental regions in a square or circle (fig. 11). The dynamics are of a rotational nature and focuses on the size-reduction towards the edges. Escher has also ventured this terrain in several pictures, like ‘Vierkantslimiet’ and ‘Cirkellimiet I – IV’.


Fig. 11 – Escher in the round. Computer-graphic by William Kolomyjec, 1975. The use of computers to generate regular patterns has contributed to the popularity of the Escher’s graphic work, which was born in an artistic craftsmanship.

The modern attention to the generation of patterns can be seen in a long history of division (in any imaginable sense) and the visible aspects of it (again in the whole specter of visibility from mathematics to art). The most notable preoccupation came to the surface in the Renaissance (fig. 12).


Fig. 12 – Piero della Francesca’s painting ‘The Flagellation of Christ’ in the Galleria Nazionale delle Marche in Urbino, Italy.

Marlyn ARONBERG LAVIN (1972) said of the floor of Piero della Francesca’s painting ‘The Flagellation of Christ’: ‘The geometric floor pavements in the front and back bays of the Praetorium, disguised by their perspective foreshortening, are similar in design to certain kinds of Renaissance horoscope drawings. Recent studies have shown these pavement designs, in their reconstructed state, to be special purveyors of mathematical symbolism. The eight-pointed stars that form their central motifs, are the astrological sign for the planets’ (in: WITTKOWER & CARTER, 1953) (fig. 13).


Fig. 13 – The reconstruction of the eight-pointed star in the floor of the Praetorium in Piero della Francesca’s painting ‘The Flagellation of Christ’. Ducal Palace, Urbino (Italy).

A Dominican priest, Dominique Douat, again raised the interest for patterns in the early eighteenth century. He published a treatise ‘Methode pour faire une infinite de desseins differents avec des carreaux mi-parts de deux couleurs par une ligne diagonale’ (Paris, 1722). It is an elaboration of an idea put forward in 1704 by another Dominican clergyman, Sebastian Truchet. He published in the ‘Memoires de l’Academie Royale des Sciences’ a number of possibilities to arrange square tiles, which were divided by a diagonal line into two coloured parts (GOMBRICH, 1979; LAUWERIER, 1988).

Truchet took a pair of bi-colored tiles, and studied their position and orientation. He found ’64 combinations de deux Carreaux mipartis de deux couleurs’ (Table I in Truchet’s publication; fig. 14). The four different stages in the position of a two-fold division (ABCD) has similarities with the example of the binary stars in a cyclic (rotational) environment.


Fig. 14 – Sebastian Truchet gave sixty-four possibilities to arrange a pair of bi-colored tiles in his ‘Memoires sur les Combinations’ (‘Treatise on Combinations’), 1704.

The significance of Truchet’s arrangements (for the present investigation) lies in the graphical treatment of combinatories. The handling of a two-fold division (or bi-colored tiles) in a topological manner is ‘a kind of metaphor for the hierarchy of separations and connection in all things.’ (SMITH & BOUCHER, 1987; p. 378)

Quadralectic thinking is closely related to this figure of speech: it is a topological approach to communication and an effort to valuate the relation between the ‘invisibilia’ and ‘visibilia’ in a mathematical way. In this case ‘hierarchy’ must not be regarded as a scale from high to low, but as a measure of distance between the partners in a communication. In the end this measure is a simple number, which acts as an expression of visibility.

ARONBERG LAVIN, Marilyn (1972). Piero della Francesca: the Flagellation. University of Chicago Press. ISBN 0-226-46958-1.

ESCHER, Maurits C. (1958/1986). Regelmatige Vlakverdeling. De Roos, Utrecht (De Roos Foundation), Museum Meermanno-Westreenianum, ‘s-Gravenhage.

– (1959/1970). Grafiek en tekeningen. Koninklijke Uitgeverij J.J. Tijl N.N., Zwolle.

KOLOMYJEC, William (1976). The Appeal of Computer Graphics. Pp. 45 – 51 in: LEAVITT, Ruth (Ed.) (1976). Artist and Computer. Creative Computing Press, New Jersey/Harmony Books.

LAUWERIER, Hans A. (1988). Symmetrie. Regelmatige structuren in de kunst. Aramith Uitgevers, Amsterdam. ISBN 90 6834 032 8

SMITH, Cyril S. & BOUCHER, Pauline (1987). The Tiling Pattern of Sebastien Truchet and the Topology of Structural Hierachy. Pp. 373 – 385 in: Leonard (Journal of the International Society for the Arts Sciences and Technology), 20th Anniversary Special Issue. Volume 20, Number 4, 1987. Pergamon Press, Oxford/New York.

STEVENS, Peter S. (1980). Handbook of Regular Patterns. An Introduction to Symmetry in Two Dimensions. The MIT Press, Cambridge, Mass./ London. ISBN 0-262-19188-1

WITTKOWER, Rudolf & CARTER, B.A.R. (1953). The Perspective of Piero della Francesca’s ‘Flagellation’. Pp. 294 – 302 in : Journal of the Warburg and Courtauld Institutes. Vol. 16 (1953).

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