Environment and Planning B: Planning and Design, volume 27 (2000), pages 537-547. © Pion Limited.  An earlier version of this paper was entitled "Complexity and Visual Images".


A Pattern Measure


Allen Klinger
School of Engineering and Applied Science, University of California at Los Angeles, Los Angeles, California 90095, USA; e-mail: klinger@cs.ucla.edu

Nikos A. Salingaros
Division of Mathematics, University of Texas at San Antonio, San Antonio, Texas 78249, USA; e-mail: salingar@gmail.com


Abstract. In this paper we propose numerical measures for evaluating the aesthetic interest of simple patterns. The patterns consist of elements (symbols, pixels, etc.) in regular square arrays. The measures depend on two characteristics of the patterns: the number of different types of element, and the number of symmetries in their arrangement. We define two complementary composite measures L and C for the degree of pattern in a design, and compute them here for 2x2 and 6x6 arrays. The results distinguish simple from high-variation cases. We suspect that the measure L corresponds to the degree that human beings intuitively feel a design to be "interesting", so this model would aid in quantifying the visual connection of two-dimensional designs with viewers. The other composite measure C based on these numerical properties characterizes the extent of randomness of an array. Combining symbol variety with symmetry calculations allows us to employ hierarchical scaling to count the relative impact of different levels of scale. By identifying substructures we can distinguish between organized patterns and disorganized complexity. The measures described here are related to verbal descriptors derived from work by psychologists on responses to visual environments.

  1. Introduction
  2. Information measures
  3. Complexity and patterns
  4. Two-symbol 2x2 example
  5. Hierarchical generalization
  6. Four-symbol 6x6 example
  7. The effect of imperfections
  8. Psychological responses
  9. Conclusion

1 Introduction

Visual images are immediately understandable or not, based on the ease by which their message can be processed by our minds. This depends on both content (information) and relationships (organization). Among the operations leading to cognition, our perceptive mechanism identifies coherent units, notices a number of repeated occurrences, and measures the extent of a field property. Symmetry relationships come from comparing pictorial entities or elements. All of this occurs instantaneously to give us a unified impression of an image, and contributes to what we perceive to be "reality". A visual pattern is easily recognizable if it is mathematically simple, and not so if it is random. Although simplicity correlates with ease of understanding, that misses the point of interest in a design whose success depends on substructure.

The greatest creations of humanity -- be they buildings, cities, artworks, or artifacts -- are neither simple, nor random, but show a high degree of organized complexity. A satisfactory complexity measure that can distinguish between organized and disorganized complexity has not been available (Maddox, 1990). Work has been done motivated by interest in human-computer interfaces in engineering, as well as in trying to understand complexity from a thermodynamic basis in physics. These questions are relevant to architecture, in studying how buildings and structures impact on human beings by virtue of their geometry and shape. From this starting point, researchers have developed "shape grammars" (Stiny & Gips, 1978), "information aesthetics" (Krampen, 1979), and "space syntax" (Hillier & Hanson, 1984). Atkin (1974) introduced a theory of relations that he calls "Q-analysis" and which he uses to analyze building facades, street plans, and Mondrian paintings. Our own work is in keeping with these endeavors.

The human visual system is especially receptive to patterns (Salingaros, 1999). We apparently enjoy the input from patterns, and enjoyment often increases with the complexity of a pattern; however, this is true only for complex patterns that have some sort of ordering. The precise nature of this effect remains imprecise and largely intuitive. We propose to combine these and other ideas into a measure for the degree of interest of a design. We call this measure L and claim that it distinguishes between organized patterns and disorganized complexity. The information in a structure is increased by having more subelements and is decreased by correlations between subelements, which reduce the coding length. (By contrast, uncorrelated subelements add raw information and make a design more random and thus difficult to input). Apparently, a viewer's interest does not correlate with the total amount of information, but rather with the degree of pattern, which represents a net measure of usable information.

The model developed here is applied to two simple examples: first to 2x2 two-symbol arrays; then to 6x6 four-symbol arrays. The quantitative results obtained clearly distinguish simple from complex arrays; they further differentiate the more complex cases according to their degree of internal organization.


2 Information measures

Some authors measure complexity by the total number of different subunits. Although useful, this does not account for connectivity and internal organization. Others consider the code that generates a pattern, and measure complexity by the length of such a code. Thus self-similar fractal structures such as a fern leaf, which requires only a brief code (see Barnsley & Hurd, 1993) are simple, whereas random structures requiring a code as large as themselves are complex. This latter approach identifies complexity with randomness, but neglects complex systems that are highly ordered. We believe this to be a crucial point: legibility depends on organization independently of internal structure, and influences the perceptual qualities of images.

A vast literature and a historic description (Spence, 1984) show how events can be organized by hierarchy in order to assist human memory. The psychological limit on the number of distinct items people can easily recall (Miller, 1956) underlines the need for chunking or grouping of data to aid both memory and perception. Computer technology has developed compression schemes that take advantage of identical data in portions of an array. Distinct methods for condensing uniform data in visuals, such as the JPEG and GIF encoding formats, make possible the transmission of images on the Internet. Distinct encoding schemes, however, will likely treat the same visual image differently. For instance, a fern leaf is simple according to the Fractal Image Compression (FIC) algorithm (Barnsley & Hurd, 1993; Fisher, 1995), but is complex according to the Graphics Interchange Format (GIF). Such approaches are clearly program specific and cannot be made universal.

The information theory model of symbol transmission has us learning the most from receiving a specific sequence of data when element values are random and equally likely (Cover & Thomas, 1991; Hartley, 1928; Shannon & Weaver, 1949). The figure 1 sequence shown below (exactly fifty ones followed by that number of zeroes) labeled by (1) conveys little additional information at the thirtieth element after observing the first twenty-nine. But if sequence (1) is replaced by the random sequence (2), the thirtieth symbol conveys a great deal of information.


11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000

Orderly sequence


00100 01111 01010 11000 01011 10010 11111 10101 11111 11010 01110 00110 10001 00101 10111 01100 10111 11001 11101 01100

Random sequence



Mathematica generation rule


Figure 1. Two sequences of fifty binary digits.


When information is being received one digit at a time, then the probability that a 1 appears as the thirtieth digit is 1/2 for sequences (1) and (2), but that does not concern us here: we want to know how information is organized in the ensemble. Shannon and Weaver (1949) extended a logarithmic measure introduced by Hartley (1928) that proposed a numerical way to describe information based on the probability of each individual symbol in a transmission. Weaver (1948) termed the symbol randomness to be disorganized complexity. While it is impossible to recognize disorganized complexity (such as sequence (2)) since it does not match something we remember, and cannot fit into a perceptual framework, there are other situations where patterns enable recognition.

An early application of information theory to aesthetics was undertaken by Moles (1966), who did not develop any measure for visual patterns. A substantial literature on information aesthetics exists only in German; it is reviewed in Krampen (1979). Stiny and Gips (1978) proposed algorithmically-based aesthetic measures that basically address the complexity of sequences and geometrical figures. One of the few authors to distinguish between organized and disorganized complexity (Papentin 1980) introduced a quantitative model that is distinct from ours. Later work by Landsberg (1984) uses physical entropy (which describes the lack of organization, but without any geometrical reference) to measure the complexity of systems; that model was not applied to visual arrays. We discuss the details of Landsberg's model below.

In Salingaros (1997) one of us introduced two pragmatic measures, termed temperature (T) and harmony (H). Temperature describes symbol variation, and harmony measures the correlations of subunits via symmetries. Originally an index ranging from zero to ten, T sums five statistical measures, each ranging from zero to two. The temperature components for complex structures were: (1) intensity and size of details; (2) differentiation density; (3) line curvature; (4) color intensity; (5) color contrast. Harmony is a similar five-part sum composed of the following symmetry values: (1) vertical and horizontal reflections; (2) translations and rotations; (3) shape similarity; (4) form connectedness; and (5) color matching. These statistical estimates apply to any object. Although the two models are entirely distinct, Atkin's (1974) Q-analysis is based on relationships, which is basically what H measures.

Another approach looks at hierarchy and substructure. Figure 3 given later shows the four-way and nine-way decomposition of 6x6 arrays into nonoverlapping blocks. Measurements could compute complexity within perceptual regions such as lines/stripes, blocks, and rings. Uniformity is easy to see in such areas. A set of six thirty-six-element four-level arrays in figure 4 below shows these perceptual regions. A method for aggregating such measures was introduced by Klinger (1980). Suppose a perceptual region with n elements has some index 0.7, and another with m points scores 0.9. An overall value could be the size-weighted average:

(0.7n + 0.9m)/(n + m)


This paper combines hierarchy with information measures to estimate the degree of pattern directly. Although the examples given here are limited to square arrays, the ideas behind this model are general, and we eventually hope to extend them to measuring the same qualities of coherence in more complex situations.


3 Complexity and patterns

We propose two linearly independent descriptors: T , a simplification of traditional measures of information; and H , a representation of symmetry. T measures the number of different subunits. Symmetry as a factor in the perception of visuals is analyzed by many authors (for example, see Attneave, 1954; Berlyne, 1971) and we give here a practical means of quantifying it. The main comparisons will involve two composite measures, L and C, derived from T and H :

L = T H


C = T (Hmax - H)


The combination of T with H follows an analogy with thermodynamics (Salingaros, 1997) where the potentials that describe all properties of a system arise as products such as TS (in which T is the physical temperature, and S is the entropy). H corresponds to the negative entropy (disorder), since the presence of symmetries corresponds to the absence of visual disorder. Their relationship may be written as S = Hmax - H , so that C = TS from expression (6). In practice, it is much easier to measure H (by counting symmetries) rather than S (where one has to count the degree of disorder). With the maximum harmony Hmax being constant for each specific system, our two composite measures L and C differ by a constant times T, and equation (6) implies that C + L = Hmax T.

Other authors have proposed similar measures. Birkhoff (1933) was clearly searching for something like this when he tried to quantify beauty mathematically. Birkhoff's model fails, however, when compared to experimental observations (Krampen, 1979). Eysenk (1941) followed that work with a better model and derived statistical correlations. (This early work is reviewed in Berlyne (1971) and in Stiny and Gips (1978)). Landsberg (1984) clarified the role of entropy as distinct from order. Labeling a system's maximum entropy as Smax = b implies that Hmax = b. Note, however, that whereas Landsberg's "complexity" SH/b2 (which is closest to our L and C measures) uses only S, our model combines the two complementary measures of disorder, T and S , where S = Hmax - H. We believe that our generalization offers an advantage in describing different aspects of visual complexity.


4 Two-symbol 2x2 example

We begin with four-element, two-by-two arrays with two different symbols shown in Figure 2 below. Rotation, reflection, or taking complements (replacing 0 by + and vice-versa) can transform each array into others not shown here. Up to those permutations, Figure 2 shows all such possible arrays.






0 0
0 0

+ 0
0 0

+ +
0 0

+ 0
0 +

Figure 2. Four four-element two-symbol arrays.


In the general case of arrays with different entries, the temperature T is just the number of different element-symbols (smallest units) minus one. The basic idea is that a uniform surface represented by a single symbol has T = 0. Since there are only two symbols in the Figure 2 arrays, T takes binary values 0 or 1. In terms of the Figure 2 labels the uniform-symbol array I has T = 0. Altogether, we have:

T(I) = 0, T(II) = T(III) = T(IV) = 1


Now we look at the internal subsymmetries in each array. The harmony H measures the presence or absence of symmetry, and assigns a binary value (present corresponds to one). Six possibilities are described as follows:

  1. h1 = reflectional symmetry about the x-axis
  2. h2 = reflectional symmetry about the y-axis
  3. h3 = reflectional symmetry about the diagonal y = x
  4. h4 = reflectional symmetry about the diagonal y = -x
  5. h5 = 90° rotational symmetry (either +90° or -90°)
  6. h6 = 180° rotational symmetry

Summing the six hi causes H to range from 0 to 6. The Figure 2 arrays have H values:

H(I) = 6, H(II) = H(III) = 1, H (IV) = 3


With Hmax = 6, equations (5), (6), (7), and (8) give the Table 1 summary:


















Table 1. Pattern measures for four-element two-symbol arrays.


This example shows how the L and C measures differentiate between uniform and varied arrays. We are going to propose that L corresponds to the level of interest of each array (based on a more complex example given below), whereas C corresponds to its internal complexity. Here, array IV has L three times those of II and III. Uniformity in I leads to zero values for L and C, but all the other cases have a comparable number for measure C. Having established the basis for the model we now generalize it to non-trivial cases.


5 Hierarchical generalization

Using thirty-six-element square arrays allows us to incorporate hierarchy, and to combine measures on small regions into the overall global measure. (Though this could be via expression (4), in the example below the averaging calculations are weighted equally). Either four 3x3 arrays or nine 2x2 arrays could be adjoined, as shown in figure 3 below, to obtain a 6x6 array. We can now measure symbol diversity and the presence of symmetry within subblocks; the decomposition of arrays into subblocks relates our model to tiling, or tessellation.












Figure 3. Subdividing a 6x6 array into 3x3 and 2x2 subblocks.


Subblock calculations introduce recursion into large arrays. As before, T counts the number of different symbols minus one, and for thirty-six elements it can be evaluated on three different scales. The first is all thirty-six elements at once. The other two are based on either nine-element or four-element subblocks. A square thirty-six-element array is four nine-element or nine four-element subblocks. To simplify matters, we ignore all other regions that overlap, such as 4x4, and other 3x3 and 2x2 different from the ones shown in Figure 3. Using the subblock labeling of letters and numbers in Figure 3, we can compute the T (or H) values as averages of the different size subblocks:

T(3x3) = [Ta(3x3) + ... + Td(3x3)]/4


T(2x2) = [T1(2x2) + ... + T9(2x2)]/9


Each of the measures T(6x6), T(3x3), and T(2x2) expresses something about the overall array. As there is no built-in preference for one over the other, the three combine as:

T = [T(6x6) + T(3x3) + T(2x2)]/3


H was previously defined for 2x2 arrays in terms of six different symmetry operations. Now, similarity-at-a-distance measures are required to account for interactions between subblocks. This adds three measures of translational symmetry:

  1. h7 = similarity to another element (yes or no gives a 1 or 0)
  2. h8 = relation to another element by translation plus a reflection about either the x-axis or the y-axis (glide reflection).
  3. h9 = relation to another element by translation plus a rotation by either +90°, -90°, or 180°.

In cases of high symmetry, h8 and h9 sometimes double-count h7 . Two different subblocks may be similar as oriented and also so after a reflection or a rotation. If a subblock is related to another via glide-reflection, and is similarly associated to one more by glide rotation, it counts as 2. (We do not consider glide rotation by multiples of 90°, as that would lead to a more complicated model. Empirical experiments show that glide reflections about the two diagonal axes do not provide a strong visual connection, so they are not counted here).

H sums the nine hi , i = 1, ... , 9, each a binary value, so it 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 (the last two by equations (9, 10)). The results are then combined via a form of equation (11). The role of the symmetry measures hi is to determine the degree of visual pattern in the arrays. The hierarchical subdivisions (equations (9), (10), and (11) for H) combine the pattern measures on each individual level of scale into an overall symmetry measure. When elements on one scale are related through symmetry, they create an element on a higher scale. This process is taken into account here by the hierarchical decomposition of H. Our model is therefore consistent with the Gestalt process of grouping individual elements so that a single percept emerges. On the other hand, even though a recognizable form (Gestalt) that jumps out of a pattern has a major effect on aesthetic interest, the model at present cannot identify this except in the simplest, most symmetric cases.


6 Four-symbol 6x6 example

A game proposed by Sackson (1969) led one of us (Klinger, 1980) to design six thirty-six-element square arrays shown in Figure 4. Each array element is one of four symbols, visually indicated here by 0, #, +, and x . These may be easily shown as colors or gray-values on a computer screen. (With a numerical equivalent for these symbols, for example 0 = 0, x = 1/3, + = 2/3, # = 1, one can identify such arrays with zones of pixels, and their actual intensities in a digitized image). The obvious generalization to arrays large enough to represent digitized images makes it possible to compute L and C for photographs. The algorithm is essentially what is already developed in this paper. It is also of interest to apply this model in image analysis to distinguish between different visual textures (Haralick et al., 1973; Julesz, 1981).



+ + + + + +
+ + + + + +
+ + + + + +
+ + + + + +
+ + + + + +
+ + + + + +

+ + 0 0 x x
+ + 0 0 x x
# # # # # #
# # # # # #
x x 0 # + +
x x + x + +

+ # x # x #
# 0 # x # x
x # + + x #
# x 0 0 # x
x # x # + #
# x # x # 0


+ 0 # x 0 #
x # 0 + x x
# 0 x # 0 +
0 # + 0 + #
+ x + # x x
# 0 + x 0 +

+ + + + + +
+ 0 0 0 0 +
+ 0 x # 0 +
+ 0 # x 0 +
+ 0 0 0 0 +
+ + + + + +

+ # x x # +
# 0 # # 0 #
x # + + # x
x # + + # x
# 0 # # 0 #
+ # x x # +

Figure 4. Six thirty-six-element four-symbol arrays.


As in Figure 2, array I is uniform: it presents no perceptual information beyond the 6x6 frame and the single symbol + (equivalently, any other single symbol). Array II has information strongly organized by 2x2 subblocks. Array IV was constructed by an algorithm based on outer and middle ring structures. Array V, which vividly displays those rings, is visually simpler than IV. Array VI varies III by taking its 3x3 subblock a (see Figure 3) and reflecting it four ways, so it has high internal symmetry.

Computations for T and H are straightforward and they readily lead to the L and C measures. Results appear below as Table 2. Unlike the exhaustive 2x2 two-symbol arrays studied earlier, the tabulated cases here represent only a few examples of all possible 6x6 arrays; they are given solely for comparison among differently-structured designs. Their more complex structure leads to some high entries in Table 2. These sample computations use the definition L = TH, and require writing equation (6) as:

C = T (9 - H)







































Table 2. Pattern measures for thirty-six-element four-symbol arrays.


L and C differ in essential ways: L corresponds to organized nontrivial structure of visual patterns, whereas C measures disorganized internal structure. L is the "pattern measure" of this paper's title. C is a "randomness measure", which estimates a separate visual quality. A key element of the model is that C is not just the opposite of L. How does each array get its values? For example, even though T is high for VI , the number of internal symmetries lowers C and raises L. By contrast, IV is high T and low H; it possesses little internal organization, which raises C and lowers L. These calculations point out differences between similar arrays such as III, IV and VI, which are superficially alike due to their multiple substructures. Table 2 provides the basis of rank orderings such as the following alternatives:

Decreasing L: VI -> V -> II -> III -> IV -> I


Decreasing C: IV -> III -> V -> II -> VI -> I


Inspection of the H values in Table 2 explains why we did not choose H alone as the pattern measure: it is highest for the pattern-less uniform array I, and, consequently, a poor indicator of structure. Multiplication by T , however, gives a measure that corresponds more closely to our intuitive assessment of degree of pattern. To measure the randomness, or absence of pattern, we need C . One might ask the question why T alone cannot be used as the measure of disorganized complexity. To see the reason, note that the ranking by decreasing T is:

Decreasing T: IV -> VI -> III -> V -> II -> I


The two rankings (14) and (15) would coincide if VI were not included. Intuitively, VI has low disorganized complexity, so we should expect it nearer I than IV, which validates (14) rather than (15). It is essential to measure symmetries (and not just visual temperature or symbol variation expressed by T) as can be seen from the position of VI in the ranking (13) of organized complexity. The combination of measures in C is therefore useful as a way to detect visual disorder or disorganized complexity; better than the use of T alone. The following imperfection example suggests use of the L and C measures to study image degradation.


7 The effect of imperfections

Suppose the design II in figure 4 were made more symmetric by removing the imperfection in subblock 8 (see figure 3). How does that change become reflected in the measures we have proposed? Figure 5 shows the two alternatives: II, and the remedied version VII.



+ + 0 0 x x
+ + 0 0 x x
# # # # # #
# # # # # #
x x 0 # + +
x x + x + +

+ + 0 0 x x
+ + 0 0 x x
# # # # # #
# # # # # #
x x 0 0 + +
x x 0 0 + +

Figure 5. Locally imperfect and symmetric versions of a design.


Although VII changes just three elements from II (an amount that is only about 8% of the data) it raises L about 28% and lowers C approximately 26%. (The indices for VII are: T = 1.67; H = 3.67; L = 6.11; C = 8.89). This shows a significant impact on the L and C measures from added symmetry introduced by changing a small amount of data. Other questions, not addressed here, concern the enhancement of images and why objects with very high L are still recognizable even with imperfections.


8 Psychological responses

Using two separate measures such as L and C for the perception of information represents a departure from previous one-variable approaches to complexity and organization. The model helps to establish that observed states are two-dimensional combinations of theoretical states. This is common in physics, where mathematical quantities that are computable from some model (such as T and H here) are often not the ones to be measured directly. Instead, it is combinations of these quantities that correspond to actual measurements. We conjecture that T and H are not perceived directly by an observer, and that it is L and C that create an impression. This is the reason we defined these particular combinations. Although tentative we mention a possible realization.

A group of researchers classify emotional reactions to physical environments in terms of opposite psychological relationships (Nasar, 1989; Russell, 1988; Ward & Russell, 1981). Emotional states are associated with points on a circle with reference to two orthogonal axes defined by attention versus rejection and pleasantness versus unpleasantness (Schlosberg, 1952). We suspect that these characteristics parallel human responses to different degrees of complexity and organization. Easily-absorbed visual information uses a structural organization of complexity. Structures generate a psychological response on a viewer because human beings judge the content and organization of environmental information. An environment will have a low degree of usable information if either: (a) there is little variety or novelty among elements; or (b) the elements are unrelated and, consequently, overload the human perceptual system (Rapoport, 1990).

A circular diagram documenting opposite psychological responses to physical environments labels them as: unpleasant versus pleasant; gloomy versus exciting; sleepy versus arousing; relaxing versus distressing (Nasar, 1989; Russell, 1988). As outlined in Russell (1988), Schlosberg (1952), Ward and Russell (1981), this diagram establishes a two-dimensional field of responses which is a mixture of these variables depending on their relative proportion (see figure 6). The L and C measures for the patterns in visual images fit within those psychological descriptors. The most consistent choice is that going from a low to high C visual image or environment would tend to alter one's response from sleepy or relaxing to arousing or distressing. On the other hand, going from low to high L would change one's response from unpleasant or gloomy to pleasant or exciting. The arrays in figure 4 and figure 5 illustrate this enough to encourage a serious program of properly designed psychological experiments.



Figure 6. Two-dimensional space of psychological responses.


A statistical extension of the present model was used to evaluate L and C for the facades of twenty-five famous buildings (Salingaros, 1997). While only a first, tentative step towards a more comprehensive model of aesthetic content and other perceptual qualities of images, the results were encouraging. Those results agree with rigorous experiments reported in Krampen (1979) that test observers' responses to eighteen facades of buildings, built either before 1900, or after 1945. The former have much higher L measure than the latter. The older set was judged as more "pleasant", "exciting", and "alive", compared to the newer set, which was judged more "unpleasant", "calming", and "dead" (Krampen, 1979). This was done at a time when modernist aesthetics were dominant, that is, before the various reactions to simple forms and surfaces that evolved into the postmodern architectural aesthetic, and before the present time when architectural variety and small-scale detail are again becoming fashionable. For this reason, the preferences are all the more significant.


9 Conclusion

In this paper we developed two complementary measures for the structure or complexity in visual arrays. By combining symbol and symmetry measures based on internal substructures, we distinguished between complexity with organization, on the one hand, and disordered complexity, or randomness, on the other. The model is generalizable in a straightforward manner to all visual images. A preliminary, statistical version of this pattern measure has been used to evaluate the visual qualities in the facades of twenty-five well-known buildings. The measures described here might shed some light on how two-dimensional information is perceived by the human mind, and how it establishes an emotional link between object and observer. Although we do not claim that these measures in any way encapsulate the entire aesthetic experience, the measures do seem to distinguish among the different arrays analyzed. By raising more questions than we can possibly answer, we hope to spur interest in pursuing these topics.

Acknowledgements. Computations were checked by Trask Stalnaker, Department of Mathematics, UCLA. Nikos A. Salingaros is supported in part by a grant from the Alfred P. Sloan Foundation.



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