Complexity and Visual Images

Allen Klinger ^{1}and Nikos A. Salingaros^{2}

1. Department of Computer Science, University of California at Los Angeles, Los Angeles CA 90095.2. Division of Mathematics, University of Texas at San Antonio, San Antonio TX 78249.

AbstractThis paper is concerned with quantitative modeling of image complexity. A less theoretic viewpoint restates this as

measuring design complexity. Here the simplest design is a visual covering a planar region, i.e., an image. We discuss the relationship of design substructures to order and disorder in measurable terms. A hierarchical weighting of estimates on different scales leads to judgements about vividness, clarity and even beauty of two-dimensional visual designs. We develop a method of measuring visual complexity in arrays that is supported by previous experimental work on architecture. The key is to introduce two practical and distinct measures of complexity: C representsdisorganizedcomplexity, whereas L representsorganizedcomplexity.

INTRODUCTION

Computer users encounter visual complexity through multiple icons representing applications, directories or folders, and files. Visually-presented information on a computer screen or designed into a page posted to the world-wide-web, like its antecedents in publishing, advertising, art, and architecture, can be either immediately understandable or intimidating. What makes the distinction? How can intermediate levels of viewing difficulty be quantified? In other words, what is the meaning in numeric terms of verbal descriptors such as cluttered, clean and spare?

If artificial intelligence concerns the construction of inductive algorithms, pattern recognition adds to this feature extraction, abstraction and learning. Since even an approximate means for measuring the direct impact of a visual could have many practical uses (e.g., in computer interface design, or in aircraft cockpits) we consider this problem and its roots in entropy in physics, information theory in the branch of electrical engineering that concerns communication, and in the several more generally accessible fields that are aspects of the media of artistic expression. This paper examines the applicability of these formal concepts of pattern recognition towards measuring complexity.

Each of us independently arrived at pragmatic indices for describing two-dimensional information. We view that as concerning the content - the entities in an image or other array; and the relationships between elements. One way to measure such aspects is to examine statistical correlations. That process is one of several used to describe textures [1, 2]. Briefly, statistical correlation computations compare the amount of match between subordinate fields within a general image. However, there is no existing theory in that domain indicating how to delimit the size or shape of such sub-fields. With no basis for defining the central concept of sub-domain, there remains room for a structural definition of coherence.

The following material describes measures that differentiate levels of complexity and coherence. We do this by means of array examples. We begin with simple cases of visual information and build to more complex displays. We show that information-theory entropy relates to two quantitative descriptors. The first is called

temperature; the second,harmony. Combined use of both allows portraying the degree to which the visually-displayed information is organized. Our invention of those and other numeric devices enables us to measure visual complexity directly.

BACKGROUND

Both authors have quantitative-discipline training, degrees, and university appointments;

andas painters both have created numerous art works. We share interest in two kinds of references in the published literature: some concerning mathematical measures of beauty [3], information and complexity [4, 5], and others about human capabilities in psychology [6] and memory [7]. One of the authors (N.A.S) has worked for many years with the architect Christopher Alexander, the originator of "patterns" in object-oriented programming. The present work is heavily influenced by Alexander's attempts at understanding order in patterns and architecture, both in computer science and in buildings.The issue of memory relates to the ability to perceive harmony: people are comfortable when a seen sequence is repeated, possibly at a different scale. A seminal paper [6] discusses the number of distinct items that people recall with ease, and argues that seven is an ideal number while five to nine is acceptable. Yet an extensive literature on memory bases recall of many more items - even hundreds of distinct things - on hierarchy. The method seems to antedate paper [6] and it is associated with ancient Egypt and then ancient Greece: Mercury serving as a god of wind is related to that use of four-way hierarchy. For an interesting review of that method in action in China by a priest who came there from Italy, see [8].

Symbol transmissions present a modern version of the memory issue. We learn the most from receiving a sequence when the prior values come in a random or disordered way. For instance, consider the sequence

fifty plus-signs followed by the same number of minuses. Very little is learned when the twenty-ninth value is known and then the thirtieth is viewed. By comparison, if plus and minus elements are introduced at random, a great deal is learned. In [4] Shannon extended a logarithmic measure introduced by Hartley [9] to the probability basis of such symbol transmissions. Weaver [5] described the general notion presented here as randomness or probabilistic generation asdisorganized complexity.We believe that it is not possible to recognize (

i.e., match to something we remember)disorganized complexity,hence we cannot fit those things into a perceptual framework. On the other hand there is complexity that is organized. Several measures discussed here describe highly-organized complex architectural structures; they first appear in [10], where the termstemperatureandharmonywere coined. In the original formulation, the temperature is an index ranging from zero to ten that is the sum of five components: each partial measure is able to take on values from zero to two. The components are: 1) intensity and size of details; 2) differentiation density; 3) line curvature; 4) color intensity; 5) color contrast. The harmony is a similar five-part sum with components: 1) vertical and horizontal reflectional symmetries; 2) translational and rotational symmetries; 3) similarity of shapes; 4) connectedness of forms; and 5) harmony of colors. Another approach computes degree of uniformity within perceptual regions and produces an average over those considered.The organization of complex systems in many apparently unrelated disciplines seems to follow the same universal rule. Biological forms exhibit a definite pattern of combining complex subelements into coherent wholes. The role of information and entropy in the development of biological forms is now beginning to be recognized. Initially chaotic processes and components are stabilized through the creation of new internal links, with both higher and lower levels of scale interacting. In [10] we called the organized complexity the

architectural lifeL, because of its correlation with the degree of biological life.

THE BASIC MODEL

The temperature T and harmony H can either be measured directly, or estimated via some algorithm. They are not the quantities that directly influence our perception of complexity, however. A basic equation relates the two measured quantities to two quantities that are actually perceived. We define the

lifeL andcomplexityC of a design as two distinct products of temperature and harmony.L = T H , C = T ( 10 - H ) Equation (1)

This definition leads to experimentally-verifiable conclusions for architectural forms [10]. In the present context, we state this rule as an assumption, then present various arguments throughout this paper to support its validity. We have sufficient data to believe that this is the way it should be done, at least as an initial approximation.

The number 10 serves only to convert the negative of H into a positive measure; 10 is replaced by the maximum of H in higher-dimensional models. H corresponds to a negative entropy (degree of disorder). As H measures the number of internal connections and correlations, their absence is a direct measure of the entropy of a design. We measure H instead of the more theoretically appealing entropy S, because it is far easier to measure the

presenceof symmetries and connections than it is to measure their absence. While the entropy is useful for discussing this model theoretically [10], it is more practical to use H as the main measure instead.Over the years, many people have looked at this issue. Among the related approaches to aesthetic measure, one can draw a direct comparison between our results and the work of Michael Benedikt [11], an architectural theorist, who uses

valueas another quantitative measure of organized complexity. His paper [11] written primarily for economists, but having wide applicability to architecture and design [11], defines value in terms that show a close relation to our L measure.The mathematician George Birkhoff's pioneering book, entitled "Aesthetic Measure" [3], is driven by similar concerns. Birkhoff defines the

valueoraesthetic measureM, which has some correspondence to our L. He defines M in terms of two other variables, thecomplexityC, and theorderO. The latter measures symmetry content, so O may be compared to our harmony measure H. The defining relation is M = O/C . Taking logarithms gives lnM = lnO - lnC, which compares to our formula L = 10T - C from Equation (1).There exist both similarities and crucial differences between our results and Birkhoff's. In a general way, lnM is related to our L. The complexity, however, is not uniquely defined. His complexity represents both our C and our T. Birkhoff's C measures the number of differentiations, and so is more directly related to our T. On the other hand, Birkhoff uses C to represent the instinctive feeling for complexity in a design, though it is another thing altogether to measure it. The design temperature T measures the gross information content, and is a key part of our theory. This concept is confused with complexity in Birkhoff's formulation, which is not surprising, as it is pre-Shannon.

Therefore, while Birkhoff clearly was on the right path, he did not arrive at the same result. Two further differences are important in practice. First, he did not give simple rules for obtaining numerical values like we have done. Second, and perhaps more important, Birkhoff wanted to derive aesthetic differences between two shapes like a rectangle and a triangle. He ultimately applied his theory to rank-order different shapes of vases. This is not a fruitful comparison in our scheme. Rather, we would like to compare similar objects with different internal structure, which can be measured quantitatively.

The next sections compute several indices that measure complexity, and determine to what degree it is organized or disorganized.

THE SIMPLEST EXAMPLE: 2X2 MATRICES WITH BINARY ENTRIES

We apply the model for complexity to the simplest example: the set of all 2X2 matrices with entries either 0 or + . Consider the four distinct matrices that represent all possible matrices of this type. We count arrays that can be transformed into each other by rotations or reflections, and by changing +'s into 0's, as similar. The four distinct examples are labeled (1) through (4) in Figure 1.

Figure 1. Four distinct 2X2 arrays with 0 and + entries.

1

2

3

4

0 0

0 0

+ 0

0 0

+ +

0 0

0 +

+ 0The design temperature T measures the degree of internal contrast; the density of differentiations; the smallness of subdivisions. This is the raw amount of information contained in each design. The least complex pattern is a uniform placement of the same symbol over the array, so a measure of temperature is based on the number of symbols in a region. Following a thermodynamic analogy introduced in [10], the basic criterion is that a uniform surface, which has no differentiations, should have T = 0. We define T as the number of different elements (smallest units) minus one. The smallest units in this example are the 0 or + symbols. T for the four 2X2 arrays takes values 0 or 1 .

T(1) = 0, T(2) = T(3) = T(4) = 1 Equation (2)

The design harmony H is the symmetry content of each square, measured according to the following criteria. If any individual symmetry is present, it rates a 1, otherwise a 0.

- h
_{1}= reflectional symmetry about the x-axis- h
_{2}= reflectional symmetry about the y-axis- h
_{3}= reflectional symmetry about the diagonal y = x- h
_{4}= reflectional symmetry about the diagonal y = -x- h
_{5}= 90° rotational symmetry (either +90° or -90°)- h
_{6}= 180° rotational symmetryH is the sum total of all the individual contributions h

_{i}. Each h_{i}takes values of 0 or 1, so H in this example can take values from 0 to 6. For the four arrays 1 through 4 in Figure 1, we have:

H(1) = 6, H(2) = 1, H(3) =1, H(4) = 3 Equation (3)

The life and complexity are computed from equation (1) using T from Equation (2) and H from Equation (3).

Table 1. Values of L and C for the four 2X2 arrays in Figure 1.

array

(1)

(2)

(3)

(4)

Life L

0

1

1

3

Complexity C

0

9

9

7

The relative numbers have meaning for comparing between different arrays. Square (4) has the greatest degree of life L - three times that of array (2) or (3). The reader will agree that this is our intuitive feeling for the relative degree of interest among all four arrays. Array (4) is more interesting to most people than either (2) or (3). On the other hand, array (1) generates no interest at all because of its uniformity, and that is reflected in its null L value.

The complexity C distinguishes only between array (1), which has no complexity, and the other three, which have about the same degree of disorganized complexity.

These calculations reveal L to be an important quantity in visualization: it measures the difference between organized and disorganized complexity. To the best of the author's knowledge, this is the first time that this is done. Our measurements show that the model clearly links what we perceive as the visual interest or "life" of a design with a numerical value that can be computed. This is the key to our model, and underlines why we consider it so important. The complexity C does not distinguish between what is interesting and what merely has substructure. These points become strikingly evident in more complex examples, which we consider next.

USING HIERARCHY TO GENERALIZE TO HIGHER DIMENSIONS

The rest of this paper considers 6X6 arrays. One basic component of problem reduction is geometric: in addition to the entire field of 36 cells we consider smaller arrays which are similar to the original array. Thus we look at the nine 2X2 arrays and four 3X3 arrays. We call these arrays subblocks. Values for the complexity of these geometric arrays will be computed in the next section, where we define the complexity of a complete pattern design. A second type of problem reduction is symbolic: we count how many symbols the area being searched has. The area may be the entire pattern of 36 cells, a subblock, or a series of cells associated by an algorithm.

The two types of problem reduction must interact in any computation of complexity, since some subblocks may have the full alphabet of symbols while near groupings of cells with a single symbol might be present. These subblocks permit a recursive formulation based on the symbols: the entire pattern is a 3x3 array whose elements are 2x2 subblocks. A measure of the total pattern complexity is based on this subblock reduction.

The design temperature T equals the number of different symbols minus one. These measurements have to be done hierarchically on three different scales. First on the 6X6 level, then on each of four 3X3 subblocks, then on each of nine 2X2 subblocks. It will be useful to label these subblocks in terms of letters and numbers. Let the index

ntake values a, b, c, d, andmrun from 1 to 9. The regular 3X3 and 2X2 subdivisions of a 6X6 matrix will be denoted as follows:

Figure 2. Subdivisions of a 6X6 array into 3X3 and 2X2 subblocks.

a b c d

1

2

3

4

5

6

7

8

9

For simplicity, we will ignore all other subblocks. There exist 4X4 subblocks also, and other 3X3 and 2X2 subblocks that do not coincide with the above subdivisions. (A more accurate count should include all possible subarrays, which are indeed picked up by the human eye when the mind computes L and C).

For each different size, the T or H value is the sum of all the T or H values for the different subblocks.

T(3X3) = SIGMA _{n = 1}^{4}T_{n}(3X3), T(2X2) = SIGMA_{m = 1}^{9}T_{m}(2X2)Equation (4)

The same goes for H. The three contributions are then combined as a weighted sum of each different scale.

T = T(6X6) + T(3X3)/2 + T(2X2)/3 Equation (5)

The weights are 1 for the 6X6 array, 1/2 for the 3X3 subblocks, and 1/3 for the 2X2 subblocks. These give the proper equipartition when we count the subdivisions in terms of their width. (The significance of this weighting for 1/

fscaling will be discussed in a separate paper). Note that we are not finding the average over the number of matrices. Thus, even though there are four 3X3 matrices, their width is 1/2 of the original 6X6 matrix, so we divide by 2 instead of 4. Similarly, the width of the 2X2 matrices is 1/3 of the original 6X6 matrix. We will use the above combination for computing both T and H totals.One could ignore weighting altogether, and simply add all contributions from all sizes of submatrices. However, that would skew the numbers so that the smaller elements contribute much more than the larger elements. Elements of different size contribute simultaneously to our perception of the whole, so it is necessary to count them in the proper balance.

The harmony H is generalized from the previous example by including measures of similarity at a distance. Different subblocks may interact with each other. This makes it necessary to count translational symmetry, which did not apply when dealing with isolated 2X2 arrays. In addition to the six symmetry measures h

_{i}, i = 1,...,6 given in the previous section, we introduce three measures of translational symmetry:

- h
_{7}= similarity to another element (yes or no gives a 1 or 0)- h
_{8}= relation to another element by a translation, plus a reflection about either the x-axis or the y-axis (a "glide reflection").- h
_{9}= relation to another element by a translation, plus a rotation by either +90°, -90°, or 180°.Note that h

_{8}and h_{9}will sometimes double-count h_{7}in cases of high symmetry. That is justified mathematically. Two different subblocks may be similar as they are oriented, and also be similar after a reflection or a rotation. A subblock may be related by glide reflection to another subblock, and by glide rotation to yet another, which counts as 2. (We do not consider each glide rotation by different multiples of 90° separately, because that would lead to more complication than we want in this model. Also, empirical experiments show that glide reflections about the two diagonal axes do not provide a strong visual connection, and for that reason they are not counted here).The design harmony is defined as the sum of the h

_{i}, i = 1,..., 9. Each h_{i}takes values 0 or 1, so H for a given array (of any size) ranges from 0 to 9. As in the case of T, these computations have to be done on three different levels, 6X6, 3X3, and 2X2 (Equation (4)), then combined with the appropriate weights in Equation (5).

SIX 6X6 ARRAYS WITH FOUR DIFFERENT ENTRIES

The complexity model is applied to the six 6X6 arrays listed in Figure 3. Each location (cell) contains one of four symbols. The goal is to guess the complexity accurately after as short a visual inspection as possible: the result obtained is a rank-ordering of the arrays in terms of decreasing complexity. The extreme scores attainable, maximum and minimum, become more widely separated as more information is put into the model.

Figure 3. Six 6X6 arrays with four different entries.

1

2

3

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + 0 0 * *

+ + 0 0 * *

# # # # # #

# # # # # #

* * 0 # + +

* * + * + +

+ # * # * #

# 0 # * # *

* # + + * #

# * 0 0 # *

* # * # + #

# * # * # 04

5

6

+ 0 # * 0 #

* # 0 + * *

# 0 * # 0 +

0 # + 0 + #

+ * + # * *

# 0 + * 0 +

+ + + + + +

+ 0 0 0 0 +

+ 0 * # 0 +

+ 0 # * 0 +

+ 0 0 0 0 +

+ + + + + +

+ # * * # +

# 0 # # 0 #

* # + + # *

* # + + # *

# 0 # # 0 #

+ # * * # +

*******????? Example 4 is of high complexity, constructed by an algorithm based on the rings: 12 elements, B: 20 elements, Y. Each ring is a cycle of all four symbols, followed by a first-to-last shift to begin a new cycle; the middle ring B starts with a different cycle, than: the outer ring Y. Example 5 is low in complexity: two types of structure present are the rings of + and 0 and is???? Example (6) is prepared from array number (3) by taking its (3X3) subblock (a) and reflecting it four ways. Array (6) consequently has a very high degree of internal symmetry, though still lower than the totally symmetric array (1).

The computations for T and H are straightforward, and all details are given in the Appendix. Unlike the simple 2X2 case treated earlier, this is not an exhaustive classification of all possible 6X6 arrays with four different entries. We just pick a sample of arrays to show how the method works in practice. The matrices chosen have very different internal structure that illustrates various possibilities.

Before reading further, we suggest that the reader study the above matrices and rank-order them in terms of decreasing C and L. Remember that C measures the intensity of design complications. In art, C measures the level of visual excitement, which often arises from chaotic aspects of a design. The life L measures the degree of organized complexity in a design; the visual interest comes from the degree to which elements interact coherently. The name "life" is chosen because continuing to increase L mathematically brings one closer to the structure of living organisms. This comparison is useful for the test that we propose. The reader can decide which of the above six arrays most resembles something that could be organic, then rank-order them in L based on this impression.

This example requires the following amended definition of C and L, instead of Equation (1):

L = T H , C = T ( 50 - H ) Equation (6)

Table 2. Computation of T, H, L, and C for the six arrays in Figure 3.

array

(1)

(2)

(3)

(4)

(5)

(6)

Temperature T

0

8.5

11.7

16.3

10

13

Harmony H

51

23

15.7

3.3

16

28

Life L

0

196

184

54

160

364

Complexity C

0

230

401

761

340

286

These numbers are larger than the earlier 2X2 case, and that reflects the existence of substructure. The amended definition of C = T(50 - H) instead of Equation (1) is necessary so that our numbers are all positive. The number 50 is arbitrary, and is needed only as an upper scale. (To be more precise, we should have chosen 51, the maximum H from array (1) in Table 2, but that makes little difference for the comparison). The six arrays can be rank-ordered in terms of decreasing L and C.

Ranking in decreasing L: (6), (2), (3), (5), (4), (1).

Ranking in decreasing C: (4), (3), (5), (6), (2), (1).

These measures are in accord with our perception of pattern complexity. Most readers will agree that these rankings correspond to what they have already concluded from direct observation. The six examples are decreasing in complexity in the same order as our feelings. We demonstrate here a strong correlation between the subconscious process of perception and a simple quantitative model. Our model can be refined by incorporating more and more input, but even at this stage, it is remarkably accurate in predicting our emotional response to a design.

Even though the design temperature of array (6) is high, the number of internal symmetries organizes the complexity so that C is lowered and L is raised. Contrast this to the high-T, low-H array (4) - it has very little internal organization, which raises C and lowers L. Array (6) shows how C essentially differs from T. Could one not skip the additional complications of measuring symmetries in this model and simply compute T as the complexity of a design? The answer is no, because the ranking in decreasing T is 4, 6, 3, 5, 2, 1, which is entirely distinct from the ranking in decreasing C. Without example (6), however, the two rankings would coincide.

Superficially, arrays (3), (4) and (6) look pretty similar. Their complex substructure emerges after looking at them analytically for some time. The more we look, the more hidden substructure is discovered as the mind moves the symmetry links from its subconscious to its conscious cognitive state. The point is that we perceive all of this information in an instant. By asking ourselves which array most resembles organic forms we can make an immediate assessment without having to analyze whether the substructure is organized or disorganized. The human mind is evidently programmed and equipped with tools to make this judgment.

VISUAL COMFORT

The notion of visual comfort concerns how an entity connects to the viewer. We believe that easily-absorbed visual information uses a structural means to organize complexity. More important, we can devise metrics that quantify that organization.

What is happening to make the previously computed variables relevant for visual perception? Why exactly do these numbers have such a close correspondence to what we actually feel? This question lies beyond the strictly mathematical treatment of this paper, yet it is after all what makes the topic relevant. Human beings connect in different degrees to various visual patterns. Any pattern will correlate to the viewer's internal understanding of order and disorder. Underlying this idea is the expectation that viewers will be more receptive to information that is presented in a pattern that connects strongly to them. The converse is also true: One is not perceptive to information that is presented via a visual pattern that fails to establish a strong connection to the viewer.

We have proposed the design life L - the relative degree of organized complexity - as a measure of this connection. As discussed in a series of forthcoming papers, human beings connect to objects that have a high degree of organized complexity. This observation explains in large part why mankind has produced artifacts having an incredible richness of pattern. Such artifacts have been central to all civilizations; in all periods of time, and in all geographic regions. Despite the amazing variety of those patterns, they are all characterized by having a high degree of organized complexity, or high value of our index L.

We have profited from a study of biological and inanimate natural forms to derive rules for achieving coherence. Organized complexity is readily accessible to the human perceptive process. Disorganized complexity is not; it escapes our perceptive abilities, and so we cannot cope with it. In a visual context, similarity of patterns, symmetries, and arrangements connect and relate one set of data to another. Establishing connections between sets of data links them in one's mind. Using this connective technique, one can handle vast quantities of information.

A human being can connect by different degrees to a design on the basis of the form's subdivisions. Obviously, this problem is entirely analogous to the visualization problem in human-computer interfaces. There, we wish to maximize the amount of information, while at the same time facilitating the transmission of that information. The solution is to organize the complexity of information in such a way that it is perceived at once, and does not confuse the user. The mathematical problem is how to maximize the information content, and minimize the complexity of the overall picture.

Every design has to control two opposite forces: the desire to simplify so as to grasp the whole at once; and the process of adding more elements, which is inevitable if one wants to include different functions and make the structure interesting. The former process reduces complexity, whereas the latter increases it. The best designs have a high degree of complexity, yet connect immediately on the visual and emotional levels. Plain designs are dull and uninteresting, and consequently fail to connect. Human beings seek a high degree of organized complexity, and are bored and sometimes repelled by structures with very low information content.

A pioneering study of this phenomenon in the context of oriental carpets is given by Christopher Alexander [12], who is developing a comprehensive theory of structural order [13]. Following this way of understanding order in both natural and man-made situations in the same terms, the growth of cities is a reflection of the organization of concepts and thoughts in the human mind [14]. The organized connections between human activity nodes that make up a city correspond fairly closely to connections between ideas and concepts that are the basis for intelligent thought.

AN EXPERIMENT IN IMPERFECTIONS.

A visual experiment uncovers some questions of fundamental relevance to both art and computer science. A reader looking at example (2) in Figure 3 might feel an urge to "clean it up" by fixing the eighth 2X2 subblock. The obvious choice is to replace it by all 0's.

Figure 4. Example (2) "cleaned up" and relabeled as (2').

(2')

(2)

+ + 0 0 * *

+ + 0 0 * *

# # # # # #

# # # # # #

* * 0 0 + +

* * 0 0 + +

+ + 0 0 * *

+ + 0 0 * *

# # # # # #

# # # # # #

* * 0 # + +

* * + * + +L = 210, C = 140

L = 196, C = 230

Changing example (2) also changes the values of the complexity parameters from those given in Table 2. The new values are: T = 7, H = 30, L = 210, C = 140. This small change (3 out of 36 cells) does not affect L to any significant degree; raising it slightly, as should be expected. On the other hand, the change drastically lowers C. This experiment prompts some observations that are more in the nature of suggestions for further discussion than finished conclusions. They are based on the reverse process of starting with example (2') and ending up with (2) by introducing an

imperfection.

- Apparently, an imperfection does not necessarily lower the degree of organized complexity L. Most great works of art are imperfect in some local region. In fact, anything that is hand-made is imperfect, and that usually enhances rather than detracts from its value.
- Objects with perfect symmetry throughout often appear dead. One way of livening them up is to introduce local asymmetries. This is particularly relevant to art and architecture. Biological forms approach structural perfection, yet they are rarely totally symmetric.
- It is well-known that the human mind can overcome an incredible degree of image degradation and still recognize a visual as a face. Classical Greek statues and Seljuk carpets usually have pieces missing, but we are still moved by their power and intensity.

In the past, such questions were thought to be outside the domain of a quantitative treatment. The above calculation belies that conclusion and opens up new possibilities for seriously addressing these issues. Our result, which is based on only one example, does not attempt to answer these fundamental questions; yet it is clear that our model of complexity is directly relevant.

CONCLUSION

A model developed in this paper computes the design complexity in terms of simple measurements on arrays. Two examples were worked out: first, a simple 2X2 case established the method, and then a more intricate 6X6 case showed the considerable power of the model. We are in fact measuring the organizational entropy (degree of disorder), which is the negative of the degree of connections established via visual symmetries. The entropy is not sufficient by itself, however. It was also necessary to measure the degree of internal differentiation of an array. Both measures were then combined to give an estimate of the perceived complexity.

This model introduced a useful distinction for discussing complexity in theoretical terms. We drew a sharp distinction between disorganized complexity, which we called simply

complexity, and organized complexity, which we calledlife. The choice of words is justified in part because of the essential structural and evolutionary connection between biological forms and organized complexity. These results should be useful in resolving issues in visual complexity and preventing human information overload, and hence find applications in maximizing rates of visual information transfer and communication.

APPENDIX. COMPLEXITY PARAMETERS FOR THE 6X6 ARRAYS.

For the reader's convenience in following our calculations, we have listed all the components that contribute to the complexity of the 6X6 arrays.

Table 3. Computation of the design temperature T.

array

(1)

(2)

(3)

(4)

(5)

(6)

(2')

T(6X6)

0

3

3

3

3

3

3

T(3X3)

0

9

10

12

8

12

8

T(2X2)

0

3

11

22

9

12

0

Summing as prescribed in Equation (5) gives the numbers we have listed for T in Table 2.

The harmony H is rather more involved. The individual figures for the different subblocks contribute to the total value, which is computed from Equations (4) and (5).

Table 4. Computation of the design harmony H(6X6).

array

(1)

(2)

(3)

(4)

(5)

(6)

(2')

H(6X6)

6

0

0

0

3

6

1

Table 5. Harmony measures H(3X3) of the four (3X3) subblocks a through d.

array

(1)

(2)

(3)

(4)

(5)

(6)

(2')

H(a)

9

0

3

0

2

6

1

H(b)

9

0

1

0

2

6

1

H(c)

9

0

1

0

2

6

1

H(d)

9

0

3

0

2

6

1

Table 6. Harmony measures H(2X2) of the nine (2X2) subblocks 1 through 9.

array

(1)

(2)

(3)

(4)

(5)

(6)

(2')

H(1)

9

9

2

2

3

3

9

H(2)

9

6

5

2

3

3

9

H(3)

9

9

5

0

3

3

9

H(4)

9

9

5

3

3

3

9

H(5)

9

9

1

2

3

6

9

H(6)

9

9

5

1

3

3

9

H(7)

9

9

5

0

3

3

9

H(8)

9

0

5

0

3

3

9

H(9)

9

9

2

0

3

3

9

REFERENCES

[1] Brodatz, P.,

Textures for Artists and Designers, New York: Dover Publications, 1966.[2] Haralick, R. M., Shanmugan, K., Dinstein, I., "Textural Features for Image-Classification,"

IEEE Trans. on Systems Man and CyberneticsSMC-3, pp. 141-152, 1973.[3] Birkhoff, G. D.,

Aesthetic Measure, Cambridge, Massachusetts: Harvard University Press, 1933.[4] Shannon, C. E., Weaver, W.,

The Mathematical Theory of Communication, Urbana, Illinois: University of Illinois Press, 1949.[5] Weaver, W., "Science and Complexity,"

American Scientist36, pp. 536-544, 1948.[6] Miller, G. A., "The Magical Number Seven Plus or Minus Two: Some Limits on Our Capacity for Processing Information",

The Psychological Review63, pp. 81-97, 1956.[7] Klinger, A., "Mnemonic Assistance," In: Klinger, A. (ed.),

Human Machine Interactive Systems,New York: Plenum Press, 1991, pp.89-102.[8] Spence, J.,

The Memory Palace of Matteo Ricci,New York:Viking Penguin, 1984.[9] Hartley, R. V., "Transmission of Information,"

Bell Systems Technical Journal,7/28, p.535. *******date???[10] Salingaros, N., "Life and Complexity in Architecture From a Thermodynamic Analogy,"

Physics Essays10,pp. 165-173, 1997.[11] Benedikt, M., "Complexity, Value, and the Psychological Postulates of Economics",

Critical Review10, pp 551-594, 1996.[12] Alexander, C.,

A Foreshadowing of 21st Century Art, New York: Oxford University Press, 1993.[13] Alexander, C.,

The Nature of Order, New York: Oxford University Press, 1997 (to appear).[14] Salingaros, N. A.,

"Mathematical Theory of the Urban Web", to be published electronically byResource for Urban Design Information(http://rudi.herts.ac.uk/), 1997.