INTRODUCTION

Is it possible to reason by means of images? If it is, then with what kind of images can we organize thoughts? How can the rules governing the relations between images be established? Could these relations be as complex and productive as those defined within the grammar of the verbal language?

It is now more than twelve years since I began to be interested in various problems related to a rather unclear and polysemic concept of the visual language. These or similar questions were the roots of my interest which lead me to work on formal aspects of visual signs developing some manner of visual grammar.

I take language to be primarily a vehicle for thought of self-communication. In my opinion, it is also a window to the world. One can say that there is no world at all. Initially, the window was small and sight was limited and poor. Throughout the ages the window became wider, more elements were added, many colors appeared, and many details were altered and changed. Hence today, through the window of verbal language we learn to perceive a very complex and sophisticated landscape of the world. However, since the window is merely a frame and the landscape is simply a picture, our vision of the world is naturally limited and distorted.

Any new language, based on premises other than verbal language, can open another window in our room, allowing us to see a different picture of the same world. These different pictures will give us a much wider and more complex vision of the universe. I believe that a highly organized and developed visual language provides such a new window.

The basic construction of any language, especially a developed one, is a structure of formal rules which regulate the relations between its signs or elements. For verbal language it is a syntax which regulates all relationships between elements of a certain language: alphabet, words and sentences.

This work is an attempt to explore and establish a set of formal rules between a large and complex group of standardized visual signs which I call discrete visual structures. A fundamental characteristic of a discrete visual structure is its possibility to be visually represented. The relations between these structures depend primarily on their graphic organization and structural characteristics. Elements of each structure can be presented as finite parts of the plane surface. There are four basic types of discrete visual structures: spatial structure, qualitative structure, state of space and visual process.

Spatial structure is defined as a structure of position. Each element of this structure is presented as a definite part of the plane with defined neighborhood relations with other elements of the same structure, presented in standard form. A position is a basic characteristic of each element within a spatial structure. Therefore, any particular spatial structure represents a specific universe of positions with its own topological integrity.

Qualitative structure is defined as a structure of content. Each element of this structure is presented as a definite part of the plane with defined neighborhood relations with other elements of the same structure. Content is a basic characteristic of each element presented visually by various degrees of shading, from white to black. There are two different presentations of a qualitative structure: natural and conventional. A natural presentation keeps the natural neighborhood relations between qualities. This means that in a white-gray-black structure, white can be a neighbor of gray but not black. In a conventional presentation the neighborhood relations can be presented arbitrarily.

State of space is defined as a union of both spatial and qualitative structures. Here we have a complex visual structure with defined both positions and content for each element within a structure. With a state of space it is possible to analyze in more detail topological characteristics of a spatial structure and to distinguish various kinds of figures within a structure. State of space can also be very helpful in analyzing the relations between different qualities. It is shown that some qualities, like gray for example, are not independent and can be generated by uniform distribution of two basic qualities: black and white. Therefore, within a state of space we can get much more information about both spatial and qualitative structures. A matrix state is a distinct kind of state which represents a visual image of neighborhood relations of a certain spatial structure.

A visual process is defined as appearance of equal or different states of space presented in sequences. The order of appearance of the same or different states of space is a rhythm. With a given number of different states we can generate a random process. In this kind of process it is known what states can appear but the order of their appearance is not predictable. Another group of processes, generated by unary operators, are predictable, since we know the exact order of appearance of same or different states throughout the entire process, by knowing which particular operator is employed.

I have based this presentation of discrete visual structures on two different types of signs: visual and verbal, but the visual presentation of images is the essential subject of this analysis. Verbal signs (written text) are used here as a necessary meta-language in order to communicate the basic ideas on discrete visual structures to the readers.

Chapter I

SPATIAL STRUCTURE

1.1 Let some finite set be A = (a, b, c, d, e) and let the neighborhood relations between elements of the set be defined in the following way: a*b, a*d, b*c, b*d, c*d, d*e.

Neighborhood relation is symmetric, which means that a*b = b*a. If we presume that the elements of A-set can be presented as definite parts of a plane, with a defined form and size in Fig. 1.1a,

for example, then the observed structure can also be presented in the following way (Fig.1.1b): We can see that the spatial neighborhood relations of elements in the structure are equal to those given by the relation above. A structure whose elements are definite parts of a plane with defined spatial neighborhood relations we will name the spatial structure. A set of elements which can define a spatial structure is the generating set. However, with the set of elements shown in Fig.1.1a it is possible to present a given structure in a number of ways as shown in Fig. 1.2. In order to avoid such polysemic

presentation of a spatial structure, we will adopt the form of the structure presented in Fig 1.1b as the standard form for any spatial structure. Therefore, only the structure presented through such a standard form can be named a spatial structure. In the following text it will be presented as shown in Fig. 1. 3. At the same time the one-element (monoelement) spatial structure will be presented in this way as well.

1.2. According to the previous explication we have some idea about neighborhood relations between elements of the spatial structure. For example, in the structure shown in Fig 1.1b elements (a) and (b) are neighbors, but not elements (a) and (c).

Two elements are in a neighborhood relation if they are presented in the spatial structure in the following ways (Fig.1.4): Two elements are not in a neighborhood relation if they are presented in the spatial structure in the following ways (Fig.1.5):

Therefore, in spatial structure Sa (Fig.1. 6), elements (a) and (b) are neighbors but not elements (a) and (c). In the structure Sb, elements (d) and (e), or (f) and (g) are neighbors. In the same structure elements (d) and (f) are not neighbors, nor are (e) and (g). In structure Sc element (k) is a neighbor of (h) and (i), but elements (h), (i)

and (j) are not mutual neighbors. In structure Sd, for example, elements (l) and (m), and (l) and (n) are neighbors, but elements (m) and (n) are not. In the last structure (Se), element (p) is a neighbor of (o) and (q), but these elements are not mutual neighbors.

1.3 If we have two different generating sets: A = (a, b, c, d, e) and B = (f, g, h, i, j) with an equal number of elements but with a different form and size, as shown in Fig.1.7a, we can define two different spatial structures (Fig. 1.7b).

The neighborhood relations between elements of these two spatial structures are:

a*b, a*d, a*c, b*d, c*d, d*e;

f*g, f*i, g*h, g*i, h*i, i*j.

Comparing these two relations we can see that they are equal, allowing us to come to the conclusion that two different spatial structures defined by two different generating sets can have equal relations of neighborhoods between elements. These kinds of structures are isomorphic spatial structures. For example, all structures presented in Fig. 1. 8 are isomorphic

spatial structures. It is obvious that structures with a different number of elements by definition, could not be isomorphic. Therefore, the spatial structures shown in Fig. 1. 9, for example, are not isomorphic. According to this we can say that all two-element spatial structures are isomorphic (Fig.1.10). However, this statement is not correct for structures with a number

of elements, n>2. For example, the three-element structures in Fig. 1. 11 are not isomorphic. Each element in structure Sa has two neighbors, but in structure Sb there are two elements with one neighbor and one with two neighbors.

1.4 The five-element spatial structures shown in Fig 1. 12 are not isomorphic, since in structure Sa there are two elements with two neighbors; in structure Sb there is none with two neighbors. Observing carefully we can come to the conclusion that

they are defined by the same generating set shown in Fig. 1.1a. All non-isomorphic structures defined by one generating set are homogeneous spatial structures. The number of homogeneous spatial structures define the potentiality of the corresponding generating set. For the generating set from the Fig 1.1a, for example, the potentiality is two (p=2). A potentiality of any two-element generating set is one (p=1), since there is only one possible non-isomorphic configuration with two elements. With a three-element generating set it is possible to define two non-isomorphic configurations (Fig. 1. 11). However, these two structures are not homogeneous since they are defined by two different generating sets. Therefore, it would be useful to define a three-element generating set whose potentiality is two (p=2). With the three-element generating set shown in Fig. 1. 13a, it is possible to define both non-isomorphic configurations (Fig. 1. 13b). It is not possible to define

a three-element generating set whose potentiality is greater than two (p>2). With a four-element generating set it is possible to define six non-isomorphic spatial structures as shown in Fig. 1. 14. However, these structures are not homogeneous since

they are defined by six different generating sets. Therefore, it would be interesting to define a four-element generating set whose potentiality is six. With the four-element generating set shown in Fig. 1. 15a it is possible to define only four

homogeneous spatial structures (Fig.1.15b). The potentiality of this generating set is four (p=4). With another generating set (Fig. 1. 16a), it is possible to define five homogeneous spatial structures presented

in Fig.1.16b; therefore its potentiality is five (p=5)*. Until now the four-element generating set whose potentiality is six (p=6) has not been found.

With a five-element generating set it is generally possible to define 20 non-isomorphic spatial structure. However, until it was possible to define a five-element generating set whose potentiality is only ten (p=10). Two such examples are presented in Fig. 1. 17. It would be interesting to find a five-element generating set which can define all 20 non-isomorphic configurations, if such a generating set exists at all.

However, we can easily find some five-element generating sets whose potentiality is 1, 2, 3 or more. The five-element generating set shown in Fig. 1. 18, for example, can define only one non-isomorphic spatial structure; therefore its potentiality is one (p=1). Another five-element generating set whose

potentiality is 2 (p=2) is shown in Fig. 1. 19a, and finally an example of five-element generating set whose potentiality is 3 (p=3) is presented in Fig. 1. 19b.

1.5 In previous explications it was shown that a certain three-element generating set can define two non-isomorphic spatial structures (Fig.1.13). However, it would not be difficult to demonstrate that isomorphic spatial structures can also be defined by the very same set (Fig. 1.20). These structures are

at the same time isomorphic and homogeneous. Spatial structures that are both isomorphic and homogeneous we will name homomorphic structures. Two homomorphic structures are equal if all corresponding elements occupy the very same position in the structure (Fig. 1. 21). However, if we

observe element b in different structures, as shown in Fig. 1.22, we could note that its position remains unchanged in relation to the structure as a whole. We may conclude that the position of the element

within a structure is not conditioned by its neighborhood relations with other elements of the structure. This means, that besides a given size and form and corresponding neighborhood relations, only when the spatial structure has been defined does the element acquire one additional characteristic and this is its position within the structure. The position of the element is typical of the spatial structure.

1.6 The configuration presented in Fig. 1. 23 is a four-element spatial structure. Element (a) has one neighbor: (b), element (b) has three neighbors: (a, c, d), element (c) has two: (b, d), and element (d) has two: (b, d) neighbors. If we are in position (a) there is only one possibility of changing position: (b).

From position (c) and (d) there are two possibilities: (b, d and b, c respectively), and finally from (b) there are three possibilities of changing position: (a, c, d). A spatial structure has the dimension n if it contains at least one element with its n neighbors. Examples of some structures with different dimensions are presented in Fig. 1. 24. When a structure whose dimension is n contains elements with

n-1, n-2, n-3, … neighbors, then this is a limited spatial structure. Elements with n-1 neighbors are limits of the first order, elements with n-2 neighbors are limits of the second order, … elements with one neighbor are limits of the n-1 order. The three-element spatial structure shown in Fig. 1.24a (Sa) is a limited two-dimension structure with two limits of the first order. The five-element spatial structure Sb is a limited four-dimension structure with one limit of the first order, two limits of the second order and one limit of the third order. The limited structure Sc has 13 elements and its dimension is 12, since there is one element in this structure with 12 neighbors. The other 12 elements are limits of the ninth order. The limited ten-element structure Sd is of dimension nine with nine limits of the eighth order. A one-element (monoelement) spatial structure (Fig. 1. 3) is dimension zero (d=0), since there is no neighborhood relation defined within this structure. All two-element spatial structures are of dimension one since they contain two elements, each with one neighbor. Three-element spatial structures shown in

Fig. 1. 25 are homogeneous and both two-dimensions. Spatial structure Sa is limited with two limits of the first order. However, in spatial structure Sb all three elements have two neighbors. A structure whose elements all have the same number of neighbors is an unlimited spatial structure.

1.6.1 A two-element spatial structure is of dimension one. As both elements of this structure have the same number of neighbors (one each), the structure of such a type is unlimited and one-dimension. It can

be said that all two element spatial structures are unlimited (Fig. 1.26). The minimal number of elements for such a structure is two, and, at the same time, this is the maximum number, since an unlimited one-dimension spatial structure with the number of elements n 2 does not exist.

1.6.2 A three-element spatial structure has only two non-isomorphic configurations (Fig. 1. 27). One is a limited two-dimension spatial structure with two limits of the first order, as shown in Fig. 1. 27a, the

other is an unlimited two-dimension spatial structure, as shown in Fig. 1. 27b.

1.6.3 A four-element spatial structure has six non-isomorphic configurations as was shown before. Two are unlimited (dimensions 2 and 3) and four are limited (dimensions 2 and 3). Some four-element unlimited two-dimension spatial structures are shown in Fig. 1.28a. Another unlimited configuration is a three-dimension spatial structure (Fig. 1. 28b). There is only one configuration for a limited four-element spatial structure whose dimension is two. This kind of structure has two limits of the first order and some

examples are shown in Fig. 1. 28c. The spatial structures in Fig. 1. 28d are limited three-dimension structures with three limits of the second order. The next four-element configurations are limited three-dimension spatial structures with two limits of the first order and one of the second order (Fig. 1. 28e). The last four-element configurations are also limited three-dimension structures with two limits of the first order (Fig. 28f).

1.7 A minimal unlimited two-dimensional spatial structure contains three elements and can be realized in several ways as shown in Fig. 1. 29. In order to consider such cases it would be interesting to determine

the procedures of generating some general examples of an unlimited two-dimension spatial structure, starting from its corresponding minimal form (Fig.1.30).

1.7.1. An unlimited three-dimension spatial structure can be realized with a minimum of four elements. Starting from the minimal three-dimension spatial structures presented in Fig.1.31, it is possible to give several examples of generating unlimited spatial structures of the same dimension but with the number of elements greater than the minimal (n>4) as shown in Fig.1.32. Upon the basis of the minimal

structures shown above (Fig.1.31) it is possible to define some isomorphic configurations with changed form and position of elements while maintaining relations of neighborhood and the way in which they are realized. A number of examples for such structures is shown in Fig. 1. 33. While using the given procedures for generating unlimited structures shown in Fig. 1. 32, and the minimal structures shown in Fig. 1. 33, we can easily determine analogous procedures for generating unlimited three-dimension

spatial structures whose number of elements is greater than minimal (Fig. 1. 34). The given examples demonstrate that all unlimited three-dimension spatial structures contain an even number of elements. With a four-element generating set it is possible to define only one non-isomorphic unlimited three-dimension spatial structure (Fig. 1. 35a). With a six element generating set it is possible to define only one unlimited tree-dimension spatial structure (Fig. 1. 35b). With eight-element generating set it is possible to define three non-isomorphic three-dimension spatial structures (Fig. 1. 35c).

With ten-element generating sets there are nine (Fig. 1. 35d), with 12-element sets there are 32 (Fig.1. 35e), and with 14-element sets there are 132 unlimited non-isomorphic three-dimension spatial structures. Examples of procedures for generating some of these non-isomorphic configurations are presented in Fig.1. 36.

1.7.3 A minimal unlimited four-dimension spatial structure can be realized with a six-element generating set. Fig. 1. 37 shows two isomorphic configurations of such a structure. Some examples of unlimited

four-dimension configuration with a different number of elements (n=8, 9, 10, 11, 12) are shown in Fig.1.38. These are all non-isomorphic unlimited four-dimension spatial structures with 8,9,10,11 and 12

elements. Starting from minimal structures (Fig. 1. 37), the examples in Fig 1. 39 show the procedures generating corresponding unlimited four-dimension spatial structures with a number of elements greater than the minimal.

1.7.4 In the case of the unlimited five-dimension spatial structures, the minimal structure requires a 12-element generating set (Fig. 1. 40). Starting from the structures shown below it is possible to define

two different procedures for generating unlimited five-dimension spatial structures (Fig. 1. 41). Some non-isomorphic unlimited five-dimension structures with 20 and 22 elements are shown in Fig. 1. 42.

1.8 If we compare the examples shown in Fig. 1. 30, 32, 39 and 41, we can see a rather distinct analogy with the ways in which these procedures are represented. In order to have a better comparison, these procedures are shown, in part, in Fig. 1. 43.

As is shown, we can define unlimited spatial structures of 1, 2, 3, 4 and 5 dimensions, but there is no one generating set which can define an unlimited spatial structure with dimension greater than 5. In other words, there is no spatial structure possible in which all elements have six neighbors each, regardless of the number of elements. This is one of the interesting characteristics of spatial structures in general.

*This generated set is suggested by Ranko Bon, Assistant Professor at M.I.T, who also gave a solution for a second example in Fig. 1. 17.