Ladies and gentlemen, young and old! In the next zhang big less.
1 Crystal patterns and mathematical history
Speiser (1927) proposed in his brief Prehistory of Group Theory that the origins of higher mathematics (then thought to be Greek around 500 BC) should date back to the egyptian use of one- and two-dimensional patterns a thousand years ago (see Jones 1856 for an explanation of this design). In his view, creating these two-dimensional patterns with many complex symmetries is a major mathematical discovery.
In contrast to these Egyptians, the later Greeks, whose study of geometry in other respects was very profound, seemed less interested in this infinite pattern. However, they developed finite design theory—in the form of regular polygons, especially regular triangles, squares, regular pentagons, and regular hexagons—to a very high level. Euclid analyzed 5 regular polyhedra in detail, and later 13 Archimedes polyhedra. Both types of polyhedra can be interpreted as patterns on a spherical surface. However, the Greeks apparently did not emphasize similar (infinite) motifs in the plane: three regular mosaics and eight semi-regular mosaics.
After these Greek constructs, there have been few records of purely mathematical research for hundreds of years. But the work of the Byzantine craftsmen of Ravenna and Constantinople and their heirs in Venice, as well as the islamic motif makers throughout the Mediterranean and east to India, were carrying out what we must consider mathematical work. Although they did not call themselves mathematicians, looking back (see Muller 1944), we find that both their methods and results have important geometric content.
During the Renaissance, Italian artists and architects made extensive use of limited designs. It is believed that Leonardo da Vinci consciously studied the symmetry of the finite designs and determined all of them in order to be able to attach chapels and niches without breaking the overall symmetry. His conclusion is now known as Leonardo's theorem (Martin 1982:66), namely that the only possible (monochromatic) finite designs are those that have only rotational symmetry, such as the swastikas used by the Nazis (Figure 1.1a), and those that have both rotational symmetry and reflective symmetry, such as the Greek cross (Figure 1.1b) or square.
Figure 1.1a with rotationally symmetrical swastikas
Figure 1.1b crosses with rotation and reflection symmetry
We will use the symbol c4 to represent a design with quadruple rotational symmetry but no reflective symmetry, such as a swastika. More generally, cn represents any design that has only n-weight rotational symmetry, such as a swastika with n arms. For the corresponding design with reflective symmetry, we use the symbolic dn. Thus, the Greek cross and square have d4 type symmetry, and the regular hexagon has d6 type symmetry.
In Italy, other regular polygons in the form of fountains are everywhere. A typical example is the Maggiore Fountain in Perugia, which has two layers, with 12x (i.e. DL2) rotational symmetry on the upper level and 25x rotational symmetry on the lower level. The rose window of Assisi San Chiara has 15x and 30x symmetry (Munari 1966:63). The pulpit of St. Stephen's Cathedral in Vienna alternates triple (C3) and quadruple (C4) symmetry (Weyl 1952:67). In St. Johnny's Church in Lineburg, an unusual set of examples of symmetry of c3, c4, c5, c6 and c7 were found, carved into wood to mimic Gothic windows.
Dürer's Geometry Book (1525) brought this and other information about regular polygons to Germany for artists to use. A hundred years later, Kepler conducted a careful study of regular polyhedra and in 1611 wrote a treatise on snowflakes in which he considered circles on planes and the accumulation of balls in space. Kepler's work can be considered a precursor to crystallography, and the study of crystallography in the nineteenth century led to until recently almost all of our mathematical information about repetitive patterns.
In the early 19th century, Hessel discovered 30 major crystals (i.e., three-dimensional repetitive patterns), which are still in use today. The names of Bravais, Jordan, Sohncke, Barlow, and Schoenflies played an important role in Fedorov's complete list of all 230 three-dimensional repeating patterns published in 1891. Fortunately, these 230 figures are not directly related to the study of flat graphics, as recently listed by Brown, Bulow, Neubuser, Wondratschek and Zassenhaus (1978).
Monochrome pattern
In 1891, Fedorov also published a count of 17 two-dimensional (monochrome) patterns. Because the paper appeared only in Russian and had little interest in crystallography, it was not until the 1920s, through papers by Niggli (1924, 1926) and Po1ya (1924), that the classification of one-dimensional and two-dimensional patterns became widely known.
The second edition of the Speiser group paper (1927) first attracted the attention of mathematicians to these results. Speiser adopted the symbols used by Niggli, but unfortunately swapped the two Niggli symbols. The consequences of this error affected the mathematical literature for the next fifty years. It was eventually corrected by Schattschneider (1978). Fortuna Teliz crystallographers continue to go their own way, so this mistake does not appear in their work.
It was thanks to Spicer's student Edith Mueller that it was possible for the first time to systematically use these tools to analyze material culture. Her 1944 dissertation was a detailed study of the art of Alhambra patterns. While Mueller is often thought to have found all 17 monochromatic planar motifs there, she explicitly mentions only 11; the other two are documented elsewhere. (In the notation explained in chapter II, the 11 recorded by Mueller are p1, pmg, cm, pmm, cmm, cmm, p4, p4m, p3, p6, and p6m.) Grunbaum and Shephard [1986] recorded P31m and Pm, Jones [1856:P1]. 41,5] Record PG [ambiguous]. )。 It was not until 1987 that the combined efforts of Spanish mathematicians and topologists provided a record of all 17 monochromatic planar patterns in the Alhambra region.
Monochromatic bands and planar patterns began to be discussed more frequently after Speiser, especially in the mathematical writings of Coxeter (1961), Fejes Toth (1964), Burckhardt (1966), Guggenheimer (1967), Cadwell (1966), and O'Daffer and Clemens (1976). Lockwood and Macmillan (1978) and Martin (1982)'s new books deal extensively with these topics. Of these references, the last four are probably the easiest for non-mathematicians to understand. Schattschneider's (1978) paper is also recommended.
2 Applications of geometry in design: historical precedents
With the exception of crystallography, there is little interest in applying the principle of geometric symmetry to other fields. "Reinventing the wheel" is ubiquitous in the literature, as many individually find symmetry to be a useful analytical tool. Many authors seem unaware of other similar work, and it seems that until the last decade, these isolated, groundbreaking introductions had little follow-up.
Our brief historical discussion here is merely an introduction to the literature in this important area. We cannot guarantee an exhaustive list of these efforts. Publication in unknown journals, and the isolated, unique nature of many of these studies, make omissions almost inevitable. However, we hope to discuss here the main applications of symmetry in the analysis of decorative objects.
Social and scientific activity at the turn of the century seems to have fostered a focus on design. Since the machines of the Industrial Revolution could endlessly print and weave patterns, there was a great need to encode and sequence these new material riches. At the same time, scholars in the humanities are classifying and bringing into a new order the vast amount of artifacts accumulated by explorers who have begun to fill museums with artifacts they have traveled around the world.
Interestingly, during this time, although both crystallographers and designers were describing repetitive patterns, neither of them seemed to notice the work of others. As a mathematical exercise, crystallographers derive the geometry of crystal structures, but designers have a practical need to organize countless patterns from home and abroad in some systematic way of describing them. Although designers saw the inherent rhythm and repetition of patterns, they never found that the symmetry of patterns could describe patterns more systematically, accurately, and objectively.
Literary scholars have published a large number of compendiums illustrating patterns from all over the world and from various periods in prehistory and history. An early example is Franz Sals Meyer's Handbook of Ornaments (1894). Meyer divides "all geometric ornamentation" into "ribbon-like forms (bands), bezel-like graphics (panels), and borderless planar patterns," which usually correspond to one-dimensional, finite, and two-dimensional categories, respectively. Meyer called the auxiliary line system (necessary to form geometric patterns such as mosaics) "nets", laying the groundwork for the widespread use of the term in two-dimensional planar pattern layouts.
As another example, a textbook on woven fabric decorative design (Stephenson and Suddards, 1897) shows students how to make repetitive patterns, not by specifying geometric movements, but by illustrating basic layouts such as "water drop patterns" (cmm), "flip patterns" (pg), "cotton satin" (various two-dimensional patterns), and "borders" and "stripes" (various one-dimensional patterns). Students use these as a basic layout on which they can repeat a wide variety of themes. The term "inverse transformation" applies to patterns whose shapes are designed to leave perfectly similar and equal shapes on the ground (1897:18).
A whole set of design encyclopedias appeared, some of which were elaborate folio illustrations from a certain region or medium, such as Perleberg's Peruvian Textiles (N.D.), Clouzots Organs NOgres (N.D.), Flenv ming's Textile Encyclopedia (N.D.), Dolmetsch's Der Ornanieiiterischat (1889) and racinef Uornon eiit Polychroz (1869). Others try to cover designs around the world. Spertz's three-volume Colorful Ornaments of All Historical Styles (N.D., 1914, 1915) covers motifs from antiquity to modern times. Hamlin (1916) wrote a history of decoration from the primitive period to the Gothic period, and like Meyer, he divided decorative patterns into "linear", "comprehensive" and "radiating"—his name refers to the three types of designs we will study here: one-dimensional patterns, two-dimensional patterns, and limited designs.
There are also writers who try to have some discussion about the composition of the pattern. In this regard, four volumes are particularly noteworthy: Owen Jones's Grammar of Decoration (1856), Z W. and G. Odesley's Outline of Decoration in the Dominant Style (1882), Lewis Day's Decoration and Its Application (1904), and Archibald Christie's Pictorial Design (1929). Much of their work is very close to describing patterns through geometric symmetry.
Odesley's statement of objectives is perhaps the best summary of the design activities and interests of this period.
Over the past two decades, a large number of important ornamental works have been published in this country and in other countries, but we are sure that there is still room for this more rudimentary work that we are now before the public, because it is different in its objectives and treatment from any previous work. In this direction, large-scale, informative works such as Owen Jones's Grammar of Ornaments and Rassinette's Multicolored Ornaments may not appear again in our time. They offer an excellent collection of decorative and ornamental designs, grouped according to the respective countries or genres in which they are produced, and provide an almost inexhaustible source of inspiration for decorative artists and viewers, and if they have any shortcomings---- they do not directly and clearly present the true scope of each decorative design to the students. Clearly leave him with an impression of structural principles and point out how these principles have been modified or developed by different artists and different artistic eras. (1882: Preamble)
Odesley sorts patterns according to different layouts: stripes, striped diapers, interweaving, powders, diapers and traditional leaves.
The term "pattern" necessarily means a design consisting of one or more devices that are multiplied and arranged in an orderly order. A single device, no matter how complex or complete it is, is not a pattern, but a unit that the designer can compose according to some clear action plan. (1929：1)
Christie divides all the decoration into two broad categories: isolated units that are "dotted" on the ground, and continuous units that are "striped" on the ground. (Christie's "background" is the same as our "background," discussed and defined below.) He gives many examples of how various patterns developed from these types through processes such as "interweaving, branching, interlocking, and reversing." He realized that different designs can be built from the same unit simply by arranging them differently into spots and stripes. "Structural methods, rather than the elements used, are the sole basis for classification," Christie sees as his model classification "largely following the line laid down by zoologists, who divide all living and extinct organisms into clearly defined categories" (1929:77, 66, 59).
The work of Andreas Speiser (1927) and his student Edith Muller (1944) (discussed in section 1.2 above) may be the first attempt by outsiders to describe symmetry in a repetitive pattern. By the 1930s, both aestheticists and designers had noticed the geometric shapes inherent in repetitive patterns. For example, mathematician George David Billhofs argues that the formal structure of Western music is a mathematical problem that can be defined by the formula M=O/C, where M is the aesthetic scale of any object relative to O and G is its complexity. Later, he developed formulas for the aesthetic measurement of vases and other objects. He defines an ornament as "any figure depicted, drawn, embossed, or otherwise made on a surface ... As long as there is at least one possible movement, move the figure as a whole rather than point by point to its initial position" (1933:49). In addition, in his famous book Aesthetic Survey (1933), he defined and illustrated four symmetrical movements, the 'basic area', the 'basic part', and the finite design and one-dimensional and two-dimensional patterns.
Another early example of systematic coding of repeating patterns was H.J. Woods of the Textile Physics Laboratory at the University of Leeds, who wrote a four-part essay for the Journal of the Textile Institute (1935-36) describing the geometric principles behind finite designs and one-dimensional, two-dimensional, monochromatic and two-color patterns. Trying to get non-mathematicians to understand the "science" behind repetitive design, he points out that even in nature, patterns are crystalline, "and crystals are nothing more than three-dimensional patterns." ”
In fact, the "science" of design is only a simplified and specialized part of the branch of physics dedicated to the study of crystal shapes, "crystallography", just as the latter is for mathematicians, but a great branch of mathematics - the application of "group theory".
Woods claims to be the first to present this symmetry analysis to non-mathematical users: "To the author's knowledge, a designer who is not proficient in mathematics has no work to master his fundamentals." The purpose of these papers is to investigate... A section of crystallography that may be of interest to textile designers" (1935: 197). Woods is right that his paper is landmark because he offers non-scientists a new way to understand the formation of repetitive patterns. Of course, a large number of articles and texts by mathematicians and crystallographers have been around for some time, but the technical nature of the proof undoubtedly precludes the possibility of the average reader understanding these ideas.
Subsequently, other physicists mentioned the applicability of geometry in design. In a short essay in The Technical Review (1937), Buerger and Lukesh illustrate two-dimensional patterns with asymmetrical "comma" symbols. The authors classify patterns by crystallization terminology and symbology, which describes the shape of the lattice, the number of rotors (i.e., the centers of rotation), and the presence and location of gliding and specular lines.
The mathematician Hermann Weyl's (1952) classic book Symmetry describes symmetry in various situations. Rosen (1975) recently updated its scope, noting the myriad presence of symmetry in botany and art as well as in pure science. Another important book, summarizing the impressive work done by Russian crystallographers, is Shubnikov and Koptsik's Symmetry in Science and Art (1974).
The Symmetry Festival, held at Smith College in 1973, aims to emphasize the many manifestations of symmetry. The conference paper, The Pattern of Symmetry (Senechal and Flash, 1977), emphasized that symmetry is evident not only in plant and crystal structures, but more subtly in the structure of musical phrases and literary works.
A summary of the latest research on the different existence and applications of symmetry in pure sciences and the arts and humanities can be found in Istvan Hargittai (1986) edited Symmetry: A Unified Human Understanding.
In Godeh Escher, Buch: An Eternal Golden Belt (1980), Housta made a highly incendiary digestion of symmetry, comparing Bach's fugues with cannons, the latter altering musical phrases by repeating them in different harmonic rhythms; Escher's cyclical paintings, whose subtle forms, from insects to fish to birds, suggest potential infinite variations; and Gödel's mathematical definition of the cycle of logical paradoxes. In fact, these are just the props that Hou Shida uses to stimulate the reader to think about the isomorphic phenomena in the human system of thought and action. That is, he wants the reader to see that isomorphism leads to symbolic associations of meaning, which in turn are part of a larger system of forms.
Hou Shida uses the interpretation of linear B scripts to illustrate how the discovery of symbolic association systems depends on the ability to discover isomorphisms. He lamented, "It is unusual to say the least that one is in the position of 'decoding a formal system that emerged in the excavations of a destroyed civilization'" (1980:50). But that's exactly what we're here for. We can clearly see that the repetitive design is prescribed by the formal rules of symmetry. In addition, since symmetry limits the kinds of possible pattern arrangements, they form a syntax. But each language has different grammatical characteristics; the repetitive design of the "grammar" is universal. All patterns arise from the same rules; that is, the schema structure can be changed, but the rules that change them cannot be changed. The meaning of this formal system is inherent in the structure of the parts.
Scott King made an equally interesting artistic representation of the symmetry principle. Inverted (1981), on one level, contains a pleasant excursion to the world of graphic palindrome. On another level, the significance of understanding art as a graphic image of our world is almost as limited as the mosaic itself. King's play of words illustrates how different audiences perceive the same scene differently; how the viewer perceives only a portion of what they see; and how perception is conditioned by the viewer's particular culture, making the interpretation of partial or ambiguous images correlated with what the viewer already knew and knew in his world (see Wash burn 1983a).
Perhaps the most peculiar application of the laws of crystallography can be seen in the work of the Dutch artist Escher, who arranged people, animals and insects into two-dimensional monochromatic, bi-color, and multi-color symmetrical patterns. Escher began studying these mosaics in the 1920s. By 1942, he had written a notebook in which he illustrated almost all of the 2-, 3-, 4-, and 6-color rotating two-dimensional patterns, with and without slip reflections. In 1960, an exhibition of Escher's works was organized at the Fifth International Congress of the International Union of Crystallography in Cambridge, England. Subsequently, the crystallographer Carolyn McGirlafri collected and published many of these drawings as an aid to the teaching of the principle of symmetry (1965). Escher's work grew in popularity, MacGillavry's book was reprinted under the title Fantasy and Symmetry (1976), and a complete compendium of Escher's work was recently published by J. Escher. L. Locher, ed.; see also Coxeter et al. (1986).
The designer's interest in regular motifs, sparked at the turn of the century by various historical factors, led to the emergence of many surveys of decorative design, and continues to publish similar compendiums today. Arranged by culture, history, and theme, the early papers were compiled primarily as source material for craftsmen and designers. The current publication, by architects for architects and other designers, includes details of the mathematical basis of repetitive patterns.
For example, March and Steadyman's (1971) Geometric Environment schematically illustrates seven one-dimensional motion classes and seventeen two-dimensional motion classes with trapezoidal units. The author highlights how Frank Lloyd Wright and Le Corbusier were influenced by the many symmetrical structures in Owen Jones's Grammar of Decoration (1856). Frank Lloyd Wright is said to have "exercised his drafting skills through night training before applying for a job with March and Steadman (1971: 38), the father of modern architecture in Chicago."
3 Symmetry analysis from an anthropological perspective
In this section, we will explain why symmetry analysis is an appropriate method for studying patterns of human behavior, as well as the laws found in material culture. We first describe symmetry as a property of the regular pattern (1.4.1) and then describe the role of symmetry in the perceptual process (1.4.2). We then show that symmetry is a fundamental feature in stylistic structure (1.4.3) and argue that this property therefore applies to systematic classification (1.4.4). Finally, we explore how systematic approaches such as symmetrical classification can be the basis for reliable methodological understanding of human behavior patterns and material culture (1.4.5).
Symmetry is an attribute of regular patterns
First of all it is necessary to define the properties of symmetry simply here, because in order to understand how symmetry becomes a useful means of measuring laws in culture, it is necessary to understand it as a mathematical entity with specific measurable properties.
In its most common and popular usage, symmetry means symmetry on both sides, i.e. mirror reflection of parts in a single (limited) graphic. The first definition in Webster's third New International Dictionary reflects this popular usage: "the correspondence in size, shape, and relative position of the opposite sides of the dividing line or middle plane or the parts distributed around the center or axis." "For example, we often think of the human body as having bilateral symmetry. Many objects in nature (flowers, starfish) have other forms of finite (radial) symmetry. In reality, however, examples of this finite symmetry include only a very small subset of the known symmetrical world as defined by geometrists and crystallographers. The finite scope allows only two symmetrical movements around a single point axis: mirror reflection and rotation.
There are other kinds of symmetry that also occur on two-dimensional planes. The one-dimensional infinite design (strip, strip) and the two-dimensional infinite pattern (full cloth, wallpaper) may allow both movements as well as translation and slip reflections. (There may also be more symmetry in three-dimensional objects, but these are not considered here.) The symmetry of the plane, then, describes the repetitive geometry of all the repetitive decorative patterns studied by anthropologists, art historians, craftsmen, or designers. Various combinations of symmetry, here called the category of motion, exist in all regular repetitive designs. These sports categories have been systematically and objectively classified.
The focus of this short trip is to emphasize the fact that some of the properties of design, such as color and symmetry, can be described in systematic derivative units. Using a range of hues listed in the Munsell color system to describe faience patterns and using crystal symmetrical classification to describe repetitive patterns on material culture are two examples of classification systems built directly from another science.
Symmetry, like color, is an attribute of design. However, unlike color, symmetry in a cultural context is non-lexical domain. When individuals distinguish colors by naming (Berlin and Kay 1969) or by comparison with other entities (green is "like the color of a cooked cassava leaf"), the only name of the symmetry class is a term specified by crystallographers.
The key question is, without creating conscious naming of categories that anthropologists can study, how can anthropologists know that the arrangements described by these crystallography are culturally meaningful and thus relevant to the analytical study of cultural behavior?
There are two ways to determine whether symmetry is a culturally meaningful property. One is to study its role in perception and how it is exploited in form recognition. The other is to study its appearance in cultural contexts – have certain symmetrical categories always appeared in patterns? We will discuss these two issues (1.4.2 and 1.43) in the following sections.
Symmetry is a factor in perception
Since most of our information about the world is obtained through vision, it is important to understand which aspects of the visual process—the reception, digestion, storage, and recall of information—are related to cultural factors, and which are universal. Here, information is considered anything that is visually accepted, processed, and used in everyday behavior. While the physiology of vision will not be reviewed here, some recent findings about cognition have been linked to studies of symmetry as a factor in this process.
Perception is the process by which an individual obtains information from its environment. But since there is always more information than the individual can absorb in any particular environment, one must learn which stimuli are prominent. Perception, therefore, involves choice. Through socialization, a person learns in a particular culture to focus on traits that enable him to predict events, reduce uncertainty, and respond appropriately. Individuals store this information in memory and use it as baseline data to compare new information.
In a particular cultural setting, different people may have different "views" of the same scenario, depending on their specific knowledge and needs. Similarly, people from different cultural backgrounds may "see" a particular scenario in different ways based on their prior knowledge, experience, and immediate background. Thus, people from a particular culture may see a pattern with symmetry, but may not see the difference between one particular symmetry and another; two "different" colors may look indistinguishable for those who classify all shades within a certain range into the "same" color category. Individuals recognize redundancy, but not necessarily different variations of the same kind of redundancy. Thus, a particular structural relationship may still be potential information; it may never be picked up (perceived) and used (manifested in cultural institutions). In this way, cultural members and situations influence the selection process of information.
The physiologist's question is, how does the brain process visual input? The question for experimental psychologists is, what are the basic units of perception, and how are they organized into features and structures? The anthropologist's question may be, how are these traits and structures used in different cultures?
So far, research by experimental psychologists has not focused on the cultural component of vision. Subjects are usually classic "American sophomores" with no control over racial origin. Instead, anthropologists are not very interested in visual processes, but only in how a culture relates to the visible environment. Observations by experimental psychologists suggest that in a given environment, not everything is "visible" and therefore used, and anthropologists do not directly exploit this.
Research on the role of symmetry in perception is prominently reflected in Gestalt psychology. For Gestalt psychologists, symmetry is a principle that contributes to order and structure in patterns; that is, symmetry contributes to the "goodness" of patterns. For modern information theorists, symmetry is an invariable thing, a redundancy that perception systems use to evaluate observed information.
Of course, symmetry is a particularly good feature that can be used to judge whether an observed object is "the same" or "different" because its various states are truly unchanging. Each class of symmetry can be "measured" mathematically, that is, distinguished by replicable precision and accuracy. In this book we will examine the information carrying capacity of symmetry, not from its "good" perspective, but from its immutability.
In the process of perception, the individual abstracts two characteristics from environmental stimuli. 1) Universal, unchanging relationships, such as symmetry or directionality, and 2) unique characteristics, such as those peculiar to the aesthetic system of a particular culture. This book is limited to discussing the universal feature of symmetry, as it is used to describe shape.
We hope to use our understanding of symmetry to solve several problems. Is symmetry a feature that individuals use to identify and distinguish categories of objects? Is it involved in the individual's perception of the environment, making decisions about the objects in the environment, and developing strategies and institutions to respond to the environment? Does the perception of symmetry differ in different cultures? Does a particular worldview and cultural system cause individuals in one culture to focus on and use only certain symmetries, while individuals in another culture prefer to use other symmetries?
To answer these questions, we must consider findings from other disciplines. So the answer to one question is, why should anthropologists and art historians consider symmetry when analyzing patterns and designs produced by culture? By understanding the role of symmetry in the visual recognition process, we can better recognize the universality of symmetry in many cultural fields.
Perceptual psychologists have identified a range of features and developed models that describe how to use these features, for example, the shape of an abstract object or the distinction between squares and circles. At the other end of the spectrum, anthropologists develop models and explanations that explain how visually acquired information is organized in cultural systems. We propose to connect the two disciplines by developing a model that classifies cultural applications of symmetry observed by anthropologists based on fundamental structural characteristics identified by perceptual psychologists. All in all, if we build on the knowledge of other disciplines, we will gain a deeper understanding of the many fundamental properties and characteristics of cultural activity.
4 Conclusion
We have seen that, to date, the only attempt to discover general structural relationships in the study of material culture is the idea borrowed from the grammar of language. However, when applied to a design, the syntax can only extract some principles that apply to the invariant form of a particular dataset. These units are specific data and technical limitations – to date, no theory has been able to deal with the entire design phenomenon. To establish a theory about design, we need classifications that define specific units and specific relationships between units throughout a category of phenomena.
Despite all the above arguments about systematic classification, the user must be convinced that the abstract properties of a large class of phenomena in nature can be successfully and meaningfully applied to the classification of material culture. Gumerman and Phillips are fairly critical of archaeologists' practice of borrowing models from a number of different disciplines fairly casually.
Archaeologists cannot simply adopt models because they are useful ways to organize specific data sets; a more adequate justification of the selection process is needed to examine the fundamental validity of these models in new applications. (1978:187)
However, the use of symmetry analysis meets two criteria for effectiveness, namely. (1) The same measurable order exists in the phenomena systematized and revealed by the classification; (2) it resolves or describes the data.
We can now ask whether culture as a structured, ordered holistic model and symmetrical classification can admissible questions of national identity and interaction. Early research suggests that not every property of the design is sensitive to these issues. Friedrich's now classic Tarascan study (1970) shows that design elements are very easy to disperse across groups, but layout configurations seem to be a more specific indicator of shared work groups. Washburn's re-study of the comments made by O'Ncale's information provider (1986a) showed that basket weavers in Northern California acknowledged that a small fraction of the element configuration was correct and appropriate for the tribe. These and many other studies suggest that a structured approach seems to best illustrate patterns of behavior stemming from cultural preferences and choices.
Clearly, the symmetrical classification of pattern arrangements—which is a systematic measure of what is commonly referred to as a design structure, arrangement, or layout—produces a reproducible description. It reveals that in a cultural group, design elements are arranged in a consistent, non-random structure by craftsmen. It is a powerful tool for organizing and objectively presenting these cultural preferences. It doesn't explain these preferences, but it does organize the data in such a way that hypotheses can be proposed and tested, and ultimately the theory can be developed. Without clearly defined categories of phenomena, it is impossible to develop a general theory of design that is relevant to all cultures.
Design is a multifaceted phenomenon and there can be many different classification methods. Here we propose a classification scheme that focuses on the attributes of design structures that have been shown in previous surveys to be sensitive to issues such as group identity, communication, and interaction, which are at the forefront of today's anthropologists, archaeologists, and other humanistic theorists of behavior. If the purpose of history, art history, archaeology, and anthropology is to describe and study the products of human behavior, which are constantly reproduced, thus forming non-random patterns, and if we regard these patterns as manifestations of the ideas shared by the makers and users of crafts, then we must first pay attention to the classification of phenomena related to these non-random ideas and modes of behavior. Here we offer a more rigorous method of defining the unit of analysis of an entire class of phenomena – repeating design, which allows us to solve important problems related to group formation, maintenance, and interaction. The question of why people do something similar is universal, profound, and not trivial. It deserves the best systematic effort we can make.
The green mountains do not change, and the green water flows for a long time, and retreats under the sun.