As the modularity will be considered the use of several basic elements (modules) for constructing a large collection of different possible (modular) structures. In science, the modularity principle is represented by search for basic elements (e.g., elementary particles, prototiles for different geometric structures...). In art, different modules (e.g., bricks in architecture or in ornamental brickwork...) occur as the basis of modular structures. In various fields of (discrete) mathematics, the important problem is the recognition of some set of basic elements, construction rules and an (exhaustive) derivation of different generated structures.
In a general sense, the modularity principle is a manifestation of the universal principle of economy in nature: the possibility for diversity and variability of structures, resulting from some (finite and very restricted) set of basic elements by their recombinations. In all such cases, the most important step is the first choice (recognition or discovery) of basic elements. This could be shown by examples from ornamental art, where some elements originating from Paleolithic or Neolithic art are present till now, as a kind of "ornamental archetypes".
In many cases, the derivation of discrete modular structures is based on symmetry. Using the theory of symmetry and its generalizations (simple and multiple antisymmetry, colored symmetry...) for certain structures it is possible to define exhaustive derivation algorithms, and even to obtain some combinatorial formula for their enumeration.
As the examples of modular structures laying on the border between the art and mathematics could be considered:
"Things are not always what they seem". If we try to explain what we see as an object, we could conclude that we always make a choice between an infinite series of real three-dimensional objects having the same flat retinal projection. From this infinity, our perception selects usually only one "natural" (most probable or, sometimes, simplest as possible) interpretation. In the perception of the usual 3D world, such interpretation is strongly connected with the coherent interpretation of complete picture, but in the cases of isolated objects, when the common reference system not exists, sometimes occurs the ambiguity: impossibility to determine a unique "natural" interpretation. The situation is even worst, if such "natural" interpretation is an impossible 3D object.
The interest of the mathematics and psychology of vision to research impossible 3D objects is mostly stimulated by Penrose impossible triangle and works by M.C.Escher, but their interesting interpretations could be found in the works by Piranesi, by Albers or in some Op-art works. If we search for the history of such objects, we need to remember the mosaics from Antioch. One of them, that consists of Koffka cubes, represents at the same time a regular plane ornament and the possible 3D structure. In the both cases, it is modular.
Let us consider first its basic element: a Koffka cube. It is multi-ambiguious: it could be interpreted as three rhombuses with joint vertex, as convex or concave trihedron, or as a cube. If we accept its "natural" 3D interpretation - a cube, than for a viewer they are three possible positions in space: upper, lower left and lower right, having equal right to be a point of view. So, for the corresponding three directions, a Koffka cube represents a turning point. Having such multiple symmetry, it fully satisfies the conditions to be a suitable basic modular element. Now, we could return to the well known impossible objects: Thiery figures (proposed at the end of XIX century) consisting of two Koffka cubes, objects created by Oscar Reutesward in 1934, Penrose tribar, Vasarely constructions, alphabet by T.Taniuchi... All of them could be derived as modular structures from a Koffka cube. For each of them, it is possible even to simply calculate the volume: for example, the volume of tribar with each bar formed by five cubes is 12. According to E.Jonesco, we all know sounds applause by two hands, but how sounds the applause by only one hand? What will happen with tribar, getting smaller and smaller, when its edge consists of three, two or even only one unit. In the last case, the result is a singular Koffka cube. If the edge is 2, the result is the next impossible object, having as an envelope another interesting geometrical figure: one-sided Moebius band. Going in the opposite way, from Koffka cubes we could construct an infinite family of impossible figures. In the process of their growing, in every point we have a possibility to proceed in three directions, i.e. to choose each from six oriented ways.
Our vision of some 3D objects could be reduced on the perception of their consisting parts, regarded from different points of view. For example, for the corners of a rectangular frame, we have only three such possibilities (or four, if we distinguish enantiomorphic forms). Then, the drawings of its constitutive parts could be the basic elements (prototiles) of a modular plane tiling. Having in mind that in plane they are only three regular tilings (by regular triangles, squares and hexagons), I derived in 1995 the modular puzzle "SpaceTiles" for the construction of pictures representing possible and impossible objects in plane. For example, if we compose the corner pieces of a rectangular frame, only three combinations will result in possible objects, and the others will be impossible. One of them that represents the frame in opposite perspective could be associated with the mentioned mosaic from Antioch, with some M.C.Escher works, or with many of middle-age paintings using counter-perspective. The basic elements of "SpaceTiles" describe different possible vertex-situations, and could be used for the experimental research of visual perception, in attempts to find criteria for realizability of a 3D object reconstructed from its drawing. If we introduce in our game Archimedean (or uniform) plane tilings, we could obtain an infinite collection of (possible) and impossible figures, beginning from the elementary ones, and including more sophisticated forms, similar to that occurring in the book L'aventure des figures impossibles by B.Ernst, or to the artistic creations by T.Farkas.
"Are impossible figures possible?" This question is the title of the paper by Z.Kulpa, explaining that the property "to be an impossible figure" is not the property by the drawing alone, but the property of its spatial ("natural") interpretation by a viewer. We already noticed that the vision is a choice among infinity of objects, where some of them are possible, and others are not. From the same drawing of impossible object, we could derive also some of its objective realizations: a possible object having the same retinal projection. Anyway, in every such situation when the impossible object is simpler than the possible one, the eye and mind will accept it as the interpretation of a drawing. After that, checking the space relationships of its constitutive elements, here comes the conclusion: this is an impossible object.
The next question concerning impossible objects is their "degree of impossibility", this means, our ability to recognize them as impossible. For example, the impossible frames will be easily recognized as impossible, but for the "truncated tetrahedron", and especially for line-segment figures (where the first is not possible, and the other is), it is very difficult to make the conclusion. In the same way, for the well-known Borromean rings the visual arguments are not sufficient, so we need the mathematical proof of their impossibility (this means, that they could not be realized by three flat rings).
Such a three-component link is named "Borromean rings" after the Borromeas, an Italian family from the Renaissance that used them as their family crest. As it is proved by B.Lindstrom and H.O.Zetterstrom "Borromean circles are impossible" (this is the title of their paper), but Borromean triangles are possible. A hollow triangle is the planar region bounded by two homothetic and concentric equilateral triangles, that is, a flat triangular ring. The Australian sculptor J.Robinson assembled three such triagular rings to form a structure (entitled Intuition), topologicaly equivalent to Borromean rings. Their cardboard model under its own weight collapses to form a planar pattern. Peter Cromwel has find it as the detail from the picture-stone from Gotland. This and other symmetrical combinations of three and four hollow triangles are considered by H.S.M.Coxeter.
Borromean rings are only a simple example from the large field of knots and links. In the practice, knots and links occur as the result of knotwork, weaving or plaiting, the human activities old as a world. From the mathematical point of view, a knot is a homeomorphic image of a circle, this means, some placement of a circle in 3D space, and a link is such a placement of several circles. As the plane representations of knots and links, we use their projections. In the mathematics, the golden age of knot projections is the end of the XIX century. To the most of the knots correspond more than one projection. Anyway, in all tables of knot projections, that are some kind of mathematical tradition transferred from one author to another, you will find only one representative projection for each knot. Their first choice maybe belongs to K.Reidemeister (Knotentheorie, 1932), and it is interesting to find some consequent principle for the choice of particular projections.
In the art, the knot designs are present from the ancient times. One of their highlights are Celtic knotworks, analyzed from the mathematical point of view by P.Cromvel. Their symmetry is distinguished by P.Gerdes, discussing so-called mirror curves resulting from knot and link sand drawings from Lunda region (Eastern Angola and Northwestern Zambia) or Tamil designs. If we have a some polyomino in a regular plane tiling, with the set of (two-sided) mirrors incident to the edges or perpendicular in their midpoints, the ray of light starting from such midpoint, after series of reflections will return to it, forming a closed path: a mirror curve. If the polyomino is completely covered by a singular curve, it always represents a knot projection; otherwise, if it is exhausted by several components, it is a link projection. The drawing will be symmetrical or asymmetrical, depending on the placement of internal mirrors, so the symmetry is not a necessary property of mirror-curves. Anyway, they possess the other remarkable property: the modularity. Each such knot projection could be obtained as a plane tiling, using only five basic elements, that I have introduced in 1994 and named "KnotTiles". The possibilities for the modular design of such structures are unlimited. The variety could be obtained by using topological variations of the prototiles, but also if we use different basic polyominoes, resulting from Archimedean (uniform) plane tilings. It is interesting that the modularity of knotworks is maybe for the first time discovered by M.C.Escher, who created several prototiles for their production.
Analyzing mirror-curves, P.Gerdes discovered Lunda designs: if the successive smaller squares through which the curve passes are colored in a color-alternating manner (black-white), we obtain the black-and-white mosaic. Such designs possess the local equilibrium property: each edge-midpoint is equally surrounded by black and white small squares. Certainly, from the local equilibrium results the global equilibrium in every row and in every column. Every square Lunda design is a modular black-and-white design, formed by only three kinds of prototiles (two kinds of internal and one kind of border prototiles).
Because P.Gerdes derived Lunda designs from mirror-curves by their black-and-white coloring, the natural question is: could you find any sample of Lunda designs in ornamental art? Many of them possess amazing property: equality between figure and ground (black and white). This means that they are antisymmetrical. To find the answer to this question, we will return to the origins: to the Paleolithic and Neolithic ornamental art.
The term "key-patterns" is used to denote key-like patterns occurring in the Egiptian, Greek, Roman, Maya, Chinese, and especially Celtic art. Because they are specific in visual sense, they are distinguished in G.Bain book Celtic Art in a separate chapter. In mathematical books (even in Tilings and Patterns by B.Grünbaum & G.C.Shephard) they are considered as usual patterns. But, in the same book, in the comment of the cover-illustration of the chapter about patterns, it is noticed that this pattern is inspired by mazes. G.Bain explicitly noticed the connection between key-patterns, spirals, meanders, mazes and labyrinths.
The oldest example of key-patterns, given in my monograph "Theory of Symmetry and Ornament", belongs to the Paleolithic art, (23000 B.C., Mezin, Ukraine). If we compare this ornament with other ornaments existing in Paleolithic art, we will conclude that it strongly differs from all the other Paleolithic ornaments and represents absolutely unexpected and even unbelievable scope of the Paleolithic art. All the ornaments from Mezin represent a systematic study of possibilities to derive different ornaments from two basic prototiles: squares with a set of diagonal parallels. Similar constructions could be obtained by using only one or two such prototiles.
The next peak of key-pattern ornaments are Celtic ornaments. In the book by G.Bain there is an attempt to explain their construction, where every key-pattern is described by a series of numbers denoting the number of steps in a particular direction, so from each key-pattern it is simply possible to read the corresponding series. Anyway, it is not clear how to obtain such series for the first time. Also, we could see the very strong and specific visual impression that ornaments make to an observer, similar to the effects produced by op-art works. Such op-art works are considered by C.Barrett as "interrupted systems", where "the pattern or system is broken or interrupted". The result is an extraordinary degree of flicker and dazzle. Similar structures are well known in the theory of visual perception.
Some of key-pattern ornaments produce the same strange and somewhat frightening visual impression, so they are used as a charm against enemies or as a symbol of labyrinth. The maze-pattern from the Wall of Palace (Knossos) with the motif of double-axe (labris), from which maybe originated the name "labyrinth" also could be constructed by the symmetrical repetition of such single prototile.
Analyzing Roman mazes we can register the same elementary meanders or meander friezes occurring also in key-patterns. In searching for the common ground on which are based the both, we can conclude that this is antisymmetry. In the first example (Roman maze, Avenches, Switzerland), we can see a regular system formed by concentric circles, which is interrupted by four "dislocations": three rectangles with 4, and one with 5 diagonal lines. They are obtained from the rectangle of the dimensions 6x4 with 9 diagonal lines, using the principle of antisymmetry. But now, we must remain that antisymmetry is not only the contrast of opposites ("black-white"), but also the principle of complementarity: two opposites giving a unity. The other examples of Roman mazes (and their reconstructions) are only the variations of the same idea: the regular system of concentric squares is interrupted by several (regularly arranged) antisymmetric rectangles. The circular mazes are the simple topological equivalent of square mazes, and are derived directly from them.
If we consider again the Paleolithic key-patterns, Celtic ornaments and op-art works, now we can see their joint basis: basic (anti)symmetric prototiles obtained by a division of a rectangle with their diagonal lines into two antisymmetric (complementary) prototiles, where one of them or both are used. Also we could consider its division into two black-white prototiles. For a rectangle with the sides a and b, the number of diagonal lines is a+b-1, so we distinguish the cases a=b, a=b(mod 2) and a+b=1(mod 2). Considering the simplest case (a=b=2), and following also the alternation of the direction of diagonals ("ascending- descending" or "left-right"), we have the two-multiple antisymmetry scheme. In such ornaments we can recognize its "black" and "white" (red) component, which are equivalent if the multiple antisymmetry is consequently used. This also explains somewhat hesitating visual impression that such patterns produce: the constant effect of flickering, when the eye recognizes black or white pattern and oscillates between them. From "black-white" prototiles we can obtain the corresponding black-white patterns. The series of such tilings derived from the four prototiles is represented by "OpTiles".
Analyzing the black-white ornaments occurring in the history, there is the simplest method for their construction: alternating black-white coloring of some isohedral tiling, but for some of them it is very difficult to comprehend how they are constructed. Maybe in that cases is used the multiple antisymmetry: the division of a fundamental region into several parts, and then use of multiple antisymmetry. Let us only notice that multiple antisymmetry is not a very sophisticated rule: it is only multiple 0-1 way of thinking (or the simple use of binary numerical system, i.e. of Boolean spaces) in geometry. This idea, introduced in my paper Periodic Antisymmetry Tilings is used, e.g. in order to obtain some non-standard isohedral tilings by use of multiple antisymmetry.
The Paleolithic key-patterns mentioned represent probably the first use of antisymmetry in ornamental art, and Celtic key-patterns, Roman mazes and some op-art work are based on the same principle discovered 23000 B.C., so the question: "Do you like the Paleolithic Op-art?" is maybe not a nonsense.
After the discovery that many of antisymmetric ornaments could be derived by recombination of few basic "Op-tiles", as a modular structures (or simply, patchworks), my recent research connected with archaeology and ethnical ornamental art, concerns the basic elements (modules) and origin of such ornaments. Such modules are, for example, a square with the set of diagonal lines, two antisymmetrical squares, black-white square (abundantly used in the prehistoric or ethnical art, known also as the element of mosaics: Truchet tile), and their topological equivalents obtained by substituting straight diagonal lines by circle arcs.
After finding that key-patterns are based on antisymmetry, I proceeded my research of the antisymmetry in ornament. If you start from simplest antisymmetrical squares with only one diagonal field, you could derive a infinite series of black-and-white ornaments, including in this class many key-patterns, different neolithic ornaments, as well as Kufic writing. The same prototile is well known in the Renaissance and later European ornamental art as a basic element for the Persian scheme. Certainly, Kufic writings could be obtained from different basic elements (e.g. from a unit black and white squares), but in this case they may appear 2x2 squares, and the proposed antisymmetric prototiles guarantee that the thickness of all the lines will be exactly 1.
In many cases, the mathematical symmetry-approach differs from the construction rules used in ornamental art. For example, the mathematical concept of the basic asymmetrical figure or asymmetrical fundamental region multiplied by symmetries is not always used in ornamental art. There are used basic symmetrical or antisymmetrical figures (modules): rosettes, friezes..., and their overlapping (e.g. many of Islamic ornaments are probably obtained by superposing very simple patterns).
Also, the mathematical approach gives us the answer to the question: "which ornaments are derived", but not to the questions "how they are derived" (or why?). Their origin could be some working procedure (e.g. matting, plaiting or basketry), or the very simple rules for making different arrangements of modules. As examples there are the similar or the same ornaments derived from some modules, occurring in different cultures (e.g. neolithic black-white ornaments from Tell Halaf or Cacaudrove patterns from Fiji). After Neolithic, it is almost impossible to find the culture that have not used that kind of patterns, derived from a black-white square. The question is only how many of them are derived from the same prototile (in the sense of exhaustive derivation according to symmetry rules) by particular cultures. Researching the history of (intuitive or visual) mathematics expressed in ornamental art or its ethnomathematical aspects, it is important to follow the use of such basic elements in different cultures, their (geometrical and topological) change in time and possible intercultural relationships.
Such relationships could be followed considering the basic prototile used in Mezin (Ukraine), the similar paleolithic artefacts from Schela Cladovei culture (Romania), and its occurrence in all neolithic cultures: Cucuteni (Ukraine, Moldavia, Romania), Gumelnitsa (Romania), Tisza (Hungary), Vincha (Yugoslavia), Dimini (Greece).
In the Ph.D. dissertation L.Tchikalenko for the first time noticed the possibility that ornaments from Mezin are composed by repeating one module: a rectangle with parallel diagonal lines. Its black-and-white variant appears as the result of matting. According to this, having in mind the occurrence of the similar or the same black-and-white ornaments in the ethnical ornamental art, as well as in the neolithic ornamental art of different cultures, suggests the idea that all of them originated from textile, and after that have been transferred to the other materials (ceramics, wood-carvings, stone-carvings...). In the book Gotter aus Ton by Nandor Kalicz, such antisymmetrical ornaments are qualified as textile ornaments. The same conclusion follows from Vincha, Tisza or Vadastra figures, where they occur as the dresses. For the most of them it is strictly respected the equality (congruence) between figure and ground (black and white part). From some completely preserved neolithic ornaments, this follows immediately. On the other hand, this rule could be the very important basis for the reconstruction of many neolithic ceramic ornaments, that are preserved only in parts, or even of some uncolored ornaments originating from the colored ones. Because it is not so probable that some part of a certain ornament, after its multiplication will completely cover the plane, respecting at the same time such very strong counterchange condition, I believe that the most of my proposed reconstructions are also correct.
The most interesting problem is to explain the way of planing of such ornaments. How it is difficult to construct that kind of ornaments, we could see from Escher notebooks, or even if we try to construct them independently. We already mentioned some of possible modular solutions: different kinds of op-tiles, multiple-antisymmetry tiles or Lunda designs. Let us notice that this such black-and-white monohedral patterns are more general than Lunda designs: they satisfy the global equilibrium condition (the congruence of black and white part), but usually not satisfy the local conditions for Lunda designs. Anyway, it is interesting that all monohedral Lunda designs with some plane antisymmetry group from P.Gerdes book Lunda Geometry are discovered in the Neolithic. Therefore, as an open field for research appears the connection between technology of work (matting, plaiting, knotwork, textiles, fabrics...) and the complete paleolithic and neolithic ornamental art preserved on stones, bones or in ceramics and the implicite mathematical knowledge as their underlying basis. In all such ornaments we could notice the domination of binary systems (black-white, left-right, above-below...), i.e., of simple and multiple antisymmetry.
As a style in art we will consider the specific properties in art expression, the mode and the manner of execution, and construction rules characterizing a particular epoch. By time, in the concept of a pattern, inseparable from the idea of symmetry, prevailed the descriptive theory of ornamental styles. By classifying ornaments according to their concept: the underlying symmetry, it is possible to follow the constancy - symmetry rules discovered and the change - their ornamental variations. From that point of view, every epoch or ornamental style will be characterized as well by the geometrical and constructional problems solved in ornamental art.
The style is a term mainly used in the art analysis of modern epochs: you will not find the terms as "paleolithic style" or "neolithic style". On the other hand, in a collection of ornaments it is not difficult to recognize paleolithic or neolithic ornaments from the others: they posses their "style". We will try to explain exact geometrical-symmetrical criteria, closely connected with the theory of visual perception, appearing as defining properties for every ornamental style.
The approach to the ornamental art from the theory of symmetry point of view is almost completely taken from the mathematical crystallography, giving us the answer to the question: "which ornaments are derived", but not to the questions "how they are derived" (or why?). Trying to explain the origins of ornamental art, and to understand anthropological, social, cognitive and communicational sense of ornaments, together with the symmetry recognition and evidence, for every epoch we need to reconstruct the complete process of the construction of ornaments. This especially concerns the oldest periods, from which we have preserved only archaeological artifacts. In construction of ornaments we could distinguish two possible ways: the symmetry extensions from local to global symmetry, and desymmetrizations ("symmetry breaking") going from high-symmetrical structures to their symmetry subgroups. By a comparative analysis, we will follow the use of some basic ornamental (and geometrical) elements and patterns in different cultures, their (geometrical and topological) change in time, and possible intercultural relationships.
This work was supported by the Research Support Scheme of the OSI/HESP, grant No. 85/1997.