Some patterns are easy to recognize—a checkerboard, the hexagonal units of a beehive, stripes of alternating colors on a flag, but biological patterns are often more subtle and less easy to discern. As Gould (1991) has pointed out, however, our visual perception of patterns tends not to be accurate. Discriminating the type of pattern is difficult. As we search to understand pattern, we may believe we see figures or objects in random patterns that are not there, like the mythological characters, objects, or animals in constellations. Random arrangements often look like clustered patterns (Fig. 1A) and ordered patterns may look random (Fig. 1B). Our minds, or those of most, are incapable of intuitive probability calculations while viewing a pattern. We perceive patterns where none exist, as viewers of modern art have done for years. We look for and find the familiar in the unfamiliar. Moreover, our terms for pattern have mathematical as well as common meanings, and unless we use language precisely, confusion may result. For example, we might describe the leaves of a palm tree as clumped at the top of the stem. In reality the leaves are present in a precise and regular order, albeit at the top of the stem.
Two-dimensional patterns, such as random, clustered, and ordered, are easy to grasp and to study. In part, this ease comes about because patterns can be reduced to dots on a sheet of paper, and, in part, because examination is simpler than for three-dimensional patterns. Patterns, even simple two-dimensional ones, are difficult to assess when they form over time. At an early stage, no pattern may be apparent, for example, three data points on a sheet of paper. Or, the pattern may change so its final configuration differs from its earlier form. Consider, for example, seat occupancy on an early morning bus. Passengers at the beginning of the bus route may select seats on a more or less random basis. If two neighbors get on at the same stop and take adjacent seats while they continue a conversation, the pattern takes on a clustered aspect. When the bus is at the end of its route and passengers fill all the seats, the pattern is highly ordered. Stomatal pattern on dicotyledonous leaves is the biological equivalent of bus occupancy. The kinetic development of the pattern confounds our ability to understand the patterning mechanism. Our knowledge of pattern is not yet sophisticated enough that we can detect the placement of the bus seats and thereby predict the final pattern.
Random vs. nonrandom
Distribution of developing stomata is reportedly random, although an exclusionary distance is present around each stoma (Sachs, 1974 ; Rasmussen, 1986 ). This may be an instance where random is being used in the everyday sense of the word, meaning that an ordered pattern is not visually obvious, rather than in the mathematical sense. Asserting that a pattern is mathematically random must be based on a standard statistical method.
A statistical method developed to assess plant distribution in ecological studies (Clark and Evans, 1954 ) is often used to determine whether stomatal distribution is ordered, random, or clustered. The mean neighbor distance between stomata in a population is compared to that in a randomly distributed population; the ratio of these two distances is called the R value. The two populations must have the same density, samples per unit area, but not necessarily the same size. When the sample distribution is random, the R value is 1, when distribution is clustered, the R value is near 0, and when distribution is ordered, the R value is significantly greater than 1. For example, if stomata were arranged in precise hexagonal arrays, the R value would be 2.1491. R values reported for stomatal distribution range from ~1.4 to 1.6. The R value indicates the type of pattern, but yields no information on dimensional or space-filling aspects of the pattern.
Arrested stomata and stomatal distribution
Published data indicate that stomata are ordered initially and become more ordered as leaves mature (Sachs, 1988 ; Kagan and Sachs, 1991 ; Croxdale et al., 1992 ; Boetsch, Chin, and Croxdale, 1995 ; see Tessellation section below). The increase in order results from stomata that fail to complete their development, so-called aborted, arrested, or immature stomata. Their presence on leaves is very common, if not universal. In Tradescantia, arrested stomatal initials can be identified shortly after the initials form. When developing stomatal initials have the first pair of subsidiary cells, arrested initials in the same cell file lack them (Boetsch, Chin, and Croxdale, 1995 ). The strict basipetal development of stomatal complexes in a cell file permits the early identification of arrest. In other genera, arrested stomata have been identified primarily by their structural configuration on mature leaves. Depending on when stomata arrest and their subsequent development, they might not be recognizable on mature organs. Based on illustrations of monocot and dicot stomata, 10–50% of the total number of initiated stomata arrest. In Ruscus and Sansevieria, 30–40% of stomata arrest (Kagan and Sachs, 1991 ; Sachs, Novoplansky, and Kagan, 1993). Such a high rate of failure is felt to be noncompetitive in evolutionarily successful plants and unlikely to result merely from developmental failure. Instead, it has been argued that patterning is epigenetic and Darwinian.
The change in R values reveals the relationship between stomata that arrest and those that complete their development. Since R values increase with leaf maturation, stomatal arrest is based on position. If arrest were random, R values measured at maturity would not differ from R values measured on immature leaves. Although R values change during leaf development, whether the change in pattern is statistically different rarely is tested. Standard procedures are available for testing whether values are significantly different from one another, including one by the authors of R values (Clark and Evans, 1954 ), which should be used routinely.
Does this adjustment in the stomatal array yield clues to stomatal patterning? In Tradescantia stomata arrest only at the earliest stage of stomatal development (Chin et al., 1995 ; Boetsch, Chin, and Croxdale, 1995 ), but in Pisum stipules stomatal arrest occurs at several stages of development (Kagan, Novoplansky, and Sachs, 1992 ). Variation in the time of arrest indicates there are redundant pathways of arrest. In Tradescantia (Boetsch, Chin, and Croxdale, 1995 ) the position of an arrested stoma is linked to the distance of only the nearest of the five closest stomata. Although proximity does not prove a causal link in arrest, the location of possible participants in the process is revealed. Similar positional analysis has not been done in other taxa, but a group of stomata rather than a single stoma might be necessary to suspend development of stomata in other genera. The presence of arrested stomata indicates that guard mother cells (GMC) or stomatal initials are committed, but not determined in the stomatal pathway. Nevertheless, to understand how cells are patterned, as a developmental biologist, I am interested not simply in a mathematical description of pattern. I am interested in a theory of patterning that accounts for the origin of all stomata, including those that arrest.
Scale, boundaries, and pattern
Assessment of pattern is best done on an entire organ or on large areas of an organ. Simply looking at small fields of pattern (cf. Fig. 1C, D, which are subsections of Fig. 1A, B) shows that pattern can appear ordered when it is random or it can appear random when it is ordered. These extreme examples show how erroneous conclusions might be reached using data from small-sized samples. The effects of scale on stomatal distribution have not been studied, but clearly are important in our understanding of patterns.
When the physical size of the sample is small, scale and the possibility of edge effects on pattern also must be considered. While the leaf edge is an obvious physical boundary, the areas above the midrib and the vascular bundles generally lack stomata and serve as a boundary. Pisum stipules (Kagan, Novoplansky, and Sachs, 1992 ) and Avena coleoptiles are exceptions to this general rule for stomata lie above their vascular bundles (Dollahon et al., 1988 ); exceptions may exist in other species. Since stomata are not the only differentiated cell type in the epidermis, lithocysts, trichomes, or papillae that differentiate earlier than stomata will serve as absolute barriers (Smith and Watt, 1986 ; Rasmussen, 1986 ; Larkin et al., 1996 ). Because of the labor involved in evaluating pattern on entire organs, we do not yet know whether pattern is adjusted near boundaries and is different from pattern in boundary-free areas.
A common and useful measure of stomatal distribution is frequency, although it provides no information on pattern type. Frequency measured on an area basis (stomatal number per unit area) simplifies and speeds the collection of data but restricts its usage to organs of the same developmental age and the same taxon. Even within the same taxon, one cannot make valid comparisons between stomatal frequency of young and mature leaves because of cell size changes that take place during leaf expansion. Measuring frequency on an index basis (100 cells) permits leaves of different age, but not different genera to be compared. Additionally, experimental treatments that influence stomatal frequency might also influence cell size so frequency comparisons between treatments also must examine cell size to ensure uncensored data.
The mode of organ growth may also complicate analysis of pattern. Generally speaking, entire (simple) leaves are reasonably straightforward to study, although there are exceptions such as Tropaeolum, which is a simple leaf at maturity but bears leaflets early in development (Fuchs, 1975 ). Compound leaves generate leaflets in ways that complicate analysis. For example, leaflets may be generated basipetally or acropetally or from the middle toward the two extremes of the leaf blade region. Thus, the leaflets of an immature leaf are not developmentally equivalent to one another; each leaflet will develop according to a timetable established at its initiation. This information as well as the growth mode of the leaflet itself must be known before embarking on a study of stomatal pattern. Leaves of dicotyledons have scattered clonal growth with new cells being intercalated between existing cells. Such growth may be rhythmic, and comparisons of stomatal frequency during leaf expansion may not be valid. Fern, gymnosperm, and monocotyledon leaves exhibit polarized growth, simplifying analysis of their patterns.
Frequency determinations on a cell or index basis permit comparison of leaves or leaf areas at different developmental stages (Radoglou and Jarvis, 1990 ; Croxdale et al., 1992 ). Again, this measure provides no information on two-dimensional order (pattern), but a sense of spatial coverage can be gleaned using the index basis. These measures also can be misleading, however, unless the investigator is familiar with the typical configuration of the stomatal complexes in the species. For example, Tradescantia plants grown in elevated CO2 have an increased stomatal frequency on an index basis, 27.2 vs. 23.9 in ambient CO2 (Boetsch et al., 1996 ). However, the number of subsidiary cells associated with nearly half the stomatal complexes (44%) increases from four to six or seven cells. The additional cells are recruited from the epidermal cell population. The shift in cells from the epidermal population to the stomatal complex population increases the stomatal index, although stomatal frequency on an area basis remains the same, 23.1 in elevated CO2 vs. 23.9 in ambient CO2. This example demonstrates that no single measure of pattern is adequate in all circumstances.
Regardless of whether frequency is measured on an area or an index basis, reports must state whether the stomata have subsidiary cells and whether they are counted as part of the stomatal complex or as epidermal cells. Although distinguishing subsidiary cells from neighboring cells has been a controversial issue (Baranova, 1992 ), authors need to state how they assessed cells surrounding each stoma. Information on the size and minimum number of stomata in the sample field is useful in all reports of stomatal frequency. Additionally, the number of plants and leaves sampled needs to be statistically relevant. Individual plants and individual leaves will show variance in stomatal frequency, and the sample size must be adequate to provide meaningful data. Because of plasticity in stomatal pattern, studies should report whether the plants were grown at the same time or at different times to account for possible sources of variation. Investigators always should be aware of possible genetic changes that may be carried forward in vegetatively propagated clonal material or be found in sexually produced seed. New individuals resulting from either means of propagation should be surveyed to determine whether stomatal frequency is within the confidence limits established in earlier experiments. The reporting of stomatal frequency has limited utility for understanding stomatal patterning. Investigators need to examine stomatal pattern, not as a mathematical exercise, but in ways that lead to testable hypotheses about the mechanism of stomatal patterning.
Tessellation is a powerful visual and quantitative means of assessing stomatal pattern. Tessellation consists of dividing a region into tiles using markers (stomata for example) as tile centers and drawing lines halfway between tile centers to form the tile edges. We have photographed entire blade panels, marked relevant features, and assessed relationships using tessellation programs. Figure 2 is an example of a tessellated leaf blade panel with stomata, arrested stomata, veins, and leaf margins marked. The latter two are marked because they are boundaries. If stomata and arrested cells are the markers of the tessellation, a portion of the tiled leaf area looks like that in Fig. 2A. Tiling changes visually (becomes more ordered) when tessellation uses only stomatal positions (cf. Fig. 2A to B), and order can be readily assessed statistically. The distribution curves of tile areas for the two populations differ markedly (Fig. 2C). The stomata and arrested cell population contains a large number of small areas while in the stomata population tile areas are shifted to larger values. The curve for the stomatal population is more symmetrical and has a lower maximum than that for the mixed population, which peaks at small tile areas but slowly trails off at larger areas. Tile areas and their distributions can be analyzed by common statistics—averages, standard deviations, variance, normal probability distributions, skewness, and kurtosis. Tessellation provides spatial and dimensional aspects of pattern that R values and frequencies do not. Assembling the raw data is tedious, but the visual immediacy of tessellations is memorable and the power of statistics provides a robust platform for pattern analysis.