mathematical design concepts relative to aesthetic...

MATHEMATICAL DESIGN CONCEPTS RELATIVE TO AESTHETIC CONCERNS

J. Michaels-Paque

Introduction.

When we realize that the ability to make original statements is unique to humans, we then recognize our responsibility to use this potential wisely. Ideally, our artistic creations spring from both an instinctive and an intellectual base.

Intellectually we need to explore different fields in order to under stand the correlations that exist among all disciplines. In other words, we must study a variety of subjects such as science, nature and mathe matics to make the most of our creative capacity.

Granted, this interdisciplinary approach may seem more complicated. However, it is not necessary to completely understand a concept or principle to use it effectively. For instance, the act of observing and correlating ideas frequently results in insights and intuitions not fully comprehensible. Yet, this approach works.

If you think in terms of how logically something might be done (rather than how someone else might have done it), you are already in a cre ative, problem solving state of mind. Let the problem dictate the most logical solution to a given set of circumstances. Then proceed on the assumption that whatever works is valid. Let the idea dictate the medium and techniques that are chosen to express it.

The groundwork is laid by feeding our mind. We then proceed in stinctively, allowing the subconscious to do most of the work. It is less pragmatic and restrictive to combine an instinctive and rational ap proach. (An element of intellectual skepticism is also necessary.) This allows answers to suggest themselves. Then too, innovation often comes from outside a field rather than from within. This is because a degree of detachment allows a greater measure of objectivity.

ARS TEXTRINA 8 (1987), pp. 155-191

We must not assume that any one philosophy or person is an absolute authority. Each individual and era makes contributions to the whole. We arrive at truths through a consensus of knowledge and experiences gleaned over the centuries.

This approach serves to simplify problems and define goals. It pro vides a workable system for learning, teaching and creating. In addi tion it facilitates communication and expands our potential. Recognizing this enables us to form a frame of reference to which ev erything encountered may then be related. It provides a source for analysis and a resource for comprehension. Knowledge thus acquired becomes versatile, easily applied and transferable. It provides its own proof and justification.

Ultimately, it is a process of refinement in which we each play a con tributing role. There is no specific method for getting at these truths. However, from this perspective we can relate concepts to a variety of techniques, media and disciplines. This is done by making compar isons, drawing analogies, and experimentation.

As Albert Einstein stated:

The supreme task is to arrive at those universal elemen tary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws, only intuition resting on sympathetic understanding of experience. 1

This paper outlines the importance of interdisciplinary studies in rela tion to nature and art generally and textiles specifically. My purpose is to promote the use of simple concepts and universal principles to en courage the development of style and originality.

Fiber and its universality.

Through research and comparative analysis we have become increas ingly aware of the historic importance of fiber techniques. Their in fluence on the course of history and the development of art and design has permeated all aspects of our culture.2

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There is a general consensus that fiber art antedates the art of pottery, and that clay preceded the metal era in mankind's growth. From crib to coffin man relies on cellulose and protein for clothing, shelter and containers. Man first fashioned huts, baskets, reed boats, headgear, weapons, ropes, mats, fences, rugs and fabrics to serve both functional and aesthetic needs. The flexible, linear elements used were derived from animal, vegetable and mineral sources. These were employed to fabricate buildings, bridges, furniture, tools and clothing. Thus, tech niques such as weaving, wattling, wickerwork, basketry, thatching, matting and thonging became the forerunners of technical procedures still in use today.

Antiquarians and archaeologists have been able to collectively establish the chronological relationship of the fiber arts. They documented the methods of spinners, weavers, net makers and the materials used. They categorized the kinds of weaving, twining, coiling and all other techniques at our disposal. Our predecessors laid the ground work for all present art forms, designs and processes.

Historians have been able to document and catalog this information, bringing us to the realization that there are no new techniques but only new combinations, relationships, and applications. This is not meant to deter or discourage current efforts. It is meant to encourage study, analysis, and exploration so that new forms, variations and styles are developed. My personal enthusiasm is directed to this end.

Most of the earlier designs evolved from logical and practical tech niques and processes. It is evident that these logical manipulations of fiber resulted in concepts and principles that are apparent in all art forms. These drawings illustrate that point (Illustration A). Several of these drawings are loom-woven textiles. Note that the crossing of warps and wefts, in the combination of simple, plain, satin, and bas- ketweaves are used to create a variety of designs such as plaids, checks, stripes, diagonals, and chevrons.

Other techniques that have exerted influence and contributed patterns are embroidery, braiding, lace techniques, quilting and basketry. All textile techniques are analogous in that they utilize the same proce-

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dunes. They differ only in the term or tool used or in the choice of medium and application.

Pattern principles and geometric tessellations used by all artists were preceded by the alternate or mesh arrangement repeat patterns found in textiles (Illustration B). These are evident in knotting, netting, caning, quilting and similar techniques. These motifs result from the alternation of the active and passive functions assumed by the fibers during the process of construction. These synonymous terms are based on the logical, technical process and mirror nature's packing system.

These examples illustrate the similarities and differences among classic designs. The basic pattern principles and geometric tessellations that ah1 artists use are the result of the mechanics of the technical processes. This is an interesting study of cause and its effects. These manipula tions are responsible for repeat motifs that are universally used in all art forms. History is replete with them. In fact, it is not possible to design and compose without them. Because of their ubiquity we tend to underestimate their importance and overlook their unending potential.

To recognize the limitations imposed by the technical processes is to liberate the individual artistic spirit. New variations emerge when translating from one medium and technique to another. An element of originality automatically occurs. Any new variable that is introduced requires a new strategy and solution. This results in unique variations of design themes, and it is an important distinction in defining origi nality. The farther afield one goes for inspiration the more creative one becomes.

The intent, in observing textiles in their historical context, is aimed toward an overview which then enables us to relate concepts to specific and general purposes. It provides a means of summarizing and clari fying complex subject matter.

Coriolis force or factor and the S and Z twist.

In order to understand these principles and concepts it is necessary to trace and study the history and relationships of our world.3 We derive

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our raw materials for textiles from nature, either directly or indi rectly. They emanate from animal, vegetable and mineral matter. The animal kingdom contributes wool, hair, sinew, fur, feathers, quills, leather, silk and similar fibers. Plants produce seed and fruit hairs for cotton, kapok and the like. Leaf fibers are sources of sisal and yucca. Bast fibers provide elements for linen, jute, hemp and ramie. Some tree barks can be shredded for fabric filaments and roots are made into brooms. Palm leaf is processed into raffia and stem fibers come from moss, reed, and grass. Man-made fibers are derived, indirectly, from nature in the form of protein, cellulose, minerals and chemicals. Both rayon and acetate are manufactured from cellulose. Rayon is physi cally changed by the industrial spinning process while acetate is the re sult of a chemical process. Nylons, polyesters and olefins are melt- spun fibers that are extruded. Acrylic, modacrylic and vinyl are man ufactured from synthetic polymers. Minerals contribute glass, as bestos and metals. Gold, silver and other alloys are annealed to make them temporarily flexible and workable. All these materials emanate from nature, where everything that grows or moves has a tendency to spiral. This distinctive characteristic carries over into textiles in the form of the S and Z principle. Nature's influence on textiles then gives way to universal applications.

There are three distinct applications for the S and Z principle in re spect to textiles. The first occurs in the spinning process. The twisting of the materials into the S, Z or a combination of the two produces lin ear filaments (Illustration C). This twisting and plying (by hand or in dustrially) provides both strength and stability. After the fiber is pre pared it is manipulated into an endless variety of textile techniques. This is the second application of the S and Z concept. All textile tech niques evolved out of the basic requirement that they physically hold together to serve their intended purpose. The acts of twisting and fab ricating them together achieves the necessary structural integrity. Twining serves as a good example. The material is first spun into sin gle elements. Next, the paired or grouped cords are twisted with the S, Z or combination of the two to create the fabric structure. Lastly, in the third application of this principle, the overall surface pattern is created using these same spiral characteristics.

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Fiber is fluid, flexible and manipulable, and inherits these characteris tics directly from growth patterns in nature. This effect is exerted by a little known factor in nature. It is called the Coriolis force or factor.* The Coriolis force or factor is the result of the earth spinning on its axis. All matter is subject not only to the pull of gravity but to this force. If the Earth were to stop spinning, the Coriolis force would no longer exist. All objects attempting to move through space in a straight line begin to create a spiral path. This spiral movement affects everything capable of growth or movement. Generally speaking, things in the southern hemisphere spiral clockwise in an S pattern or twist. In the northern hemisphere it spirals counter clockwise, in a Z twist. It does not spiral at either pole, where the Coriolis force is zero. At the equator, the force is at its maximum, which results in a straight vertical direction. This extreme eliminates the observable spiral completely.

The force is responsible for wind patterns, water currents, cyclones, tornadoes, and growth patterns in nature. It is quite evident in vines, the horns of ruminants, molluscan shells, or the florets of sunflowers. It is less obvious but still discemable in a lock of hair, a staple of wool, a coil of an elephant's trunk, the circling spires of a snake, or a mon key's or chameleon's tail. The influence of this factor is also consid ered when planning flight patterns and shipping routes. One of the easiest ways to observe this principle is to watch water going down a drain. At the north pole, the vortex will spiral down counter clock wise. The south pole is just the opposite.

It is surprising that little is written about this subtle but profound in fluence that is exerted on all flexible matter. Nowhere, in my re search, could analogies be found for this influence on the growth of our raw materials, the spinning of them and their fabrication into tex tile processes and products. This fact has been virtually ignored by scholars. Observing this becomes a rare moment of conceptual com prehension. This is one of the few generalized principles that underlie the natural growth processes on which we are all dependent.

There are many references that allude to the spiral's importance in art and design, and this is not to be denied. What I'm suggesting is that spirals be viewed in a broader perspective. That is, particularly, in

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relation to the contribution of textiles and their historical connection. Studying these relationships makes one cognizant of the importance of investigating and observing the whole in order to understand the parts. This force or factor is named for Gasparo Gustave de Coriolis (1792 - 1843), a French scientist who first defined it.

To observe the enormous number of spiral influences in the universe is to begin to comprehend the incredible beauty of systems. Spirals are evident in cells, seeds, stems, flowers and fruit. In man and animals it follows the entire course of development from sperm, the umbilical cord, the ear cochlea to the bone and muscular framework of the body. These are just a few examples.

It is interesting to note that the very essence of life, the DNA molecule, is a double helix composed of two intertwining spirals. The basic building block and determinator of genetic endowments consists of one S spiral and one Z twist.

The Coriolis force, along with its many spiral examples, provides us with a vehicle for understanding principles that govern the cosmos. This awareness inspires the creative use of spiral concepts in textiles and art.

The mathematical aspects of spirals are also evident in nature's use of Fibonacci number systems (Illustration D). These Fibonacci numbers are used to describe incremental growth patterns in nature.^ These spirals are based on the summation series of numbers 0,1,1, 2, 3,5, 8, 13, 21, 34, etc. Each term is the sum of the two preceding terms. While these numbers serve as a generalized principle they are not ab solute. A spiral is defined as a curve that is the path of a point moving about a fixed point and is constantly moving away from or approach ing the fixed point. In order to classify it as an S or Z twist it is neces sary to establish a point of reference and the direction of movement. In textile textbooks this aspect is not taken into consideration nor ex plained and causes endless confusion.

Many design variations are possible by combining, countering and changing the Fibonacci growth rate, and the number of radiating lines. My concern is to analyze spirals so that we can understand their incli nations and limitations. This facilitates their adaptation to textiles and

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art. The lefthanded or Levogyre spiral is the most prevalent in nature. It seems to follow the Maxwell Law of Electromagnetism. In addition to the spiral's influence on everything that grows and moves, it rules the motion of the planets and all celestial bodies.

The ancient Greeks recognized the link between natural phenomena and the proportional harmonies of our physical world. For example, the spiral growth formulae of plankton, seashells, sunflowers, cabbage leaves, pine cones, butterfly wings, and even the galaxy of stars are identical. Most of nature's horns, claws, fangs and teeth exhibit the spiral, especially the ram's horn, parrot's beak, and the lion's claw.

Growth spirals are incremental, logarithmic and equiangular. They retain their proportions at all stages of development. This reciprocity is the same formula found in the famed "golden section". The Greeks called it the "golden mean". It is defined as a point that divides a line into two parts in such a manner that the smaller part is in proportion to the larger part as the larger part is to the whole. This magic extreme and mean ratio manifests itself in all that is beautiful.

A number of cultures, notably the Egyptians, used the divine propor tion intentionally. Others applied this natural law intuitively. Leonardo da Vinci represents one of many masters who used this sys tem in painting and sculpture. A rectangular approximation of this formula is applied in the shape of Aubusson tapestries, Navajo rugs and painters' canvasses. These Fibonacci numbers provide propor tion, harmony and order in many fields including art, architecture, oceanography, botany, biology, astronomy and music.

Pattern principles, geometric themes relating to textiles.

The Coriolis force exerts its effects on all aspects of our universe. It influences the growth of our raw materials, the spinning of them, in addition to the fabrication of textile techniques. This continues in all areas of the visual arts. This classic pattern illustrates an application of spirals in a graphic sense to design and composition (Illustration E). It epitomizes the influence of textiles on art and design throughout his tory. It also exemplifies the interrelationship of all disciplines and media. This design is frequently translated into a variety of techniques

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and processes, and is found in all media including clay, metal, fiber, wood, glass, plastic, the graphic arts, painting, has relief and sculpture. It is based on one of the most beautiful and prevalent occurrences in nature: the spiral.

This design is also an excellent example ofnotan, which is essentially the negative/positive aspect of design. In this instance, it is shown in the exact, alternate repeats of light and dark patterns. Philosophically speaking, opposites complement. Hence, the visual images of these polarities naturally work well together, relating in harmony.

Notan is a Japanese word meaning dark/light. It is further defined as the interaction between (light) positive and (dark) negative space. This idea of interaction is embodied in the ancient eastern symbol of yin and yang (Illustration F). It consists of mirror images, one white, one black, revolving around a point of equilibrium. In both notan and yin/yang, opposites complement; neither seeks to negate or dominate the other. Pattern, by definition, is the result of the repetition of basic elements. These may be simple or complex. The most common pat tern networks or grid systems are shown here (Illustration G). These systems form guidelines for simple repeat units. Shapes that tessellate are the common factor in these patterns. They interlock or connect perfectly, one into the other, and are the result of repeating and com bining basic geometric shapes such as the square, triangle, circle, diamond, hexagon and rhomboid.

More complicated patterns are combinations of these and other shapes. The Roman word tessella is the root of our English term tessellation. It stems from tessara or tile (mosaic shapes). In mathematics, plane tessellations are repeat patterns that cover a designated plane, without spaces or overlapping. In textiles these design principles are very evi dent. They result from the logical manipulation of these techniques and processes. These natural tendencies are innate and follow the laws of cause and effect. Sensitivity to these laws results in both beauty and function.

The important point is that all these principles, and the underlying laws connecting them can be arrived at by searching for the simplest con cepts and the link between them.

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Principle of minima.

The minima principle is one used to some extent by all. To deliber ately analyze this law provides a means of further simplifying and re fining all aspects of our work. This minimal usage principle is gener ally defined as the judicious use of time, materials, tools, energies, space, movement and finances.

Not surprisingly, to implement these logically and conservatively re sults in good design and function. It is a natural way of achieving quality without misuse of quantity.

This principle guided eighteenth-century physics and according to D'Arcy Thompson, inspired Hamilton and Maxwell. It is still some times referred to as the Hamilton Maxwell principle. It is, however, better known as the law of least effort. It rules form in physics, art, architecture, and human behavior, among other disciplines.

Other authorities refer to this as the "economy of means" law, lines of least resistance, or the principle of least action. This study of minimal surfaces is best exemplified by soap bubble films, which cover the greatest volume with the least material. A German architect, Frei Otto, is renowned for his flexible, stretched skin architecture. These curved constructions are derived from soap bubble studies. His build ings epitomize the application of this principle at its best.

To illustrate the law of least effort, let us consider a design problem frequently encountered by landscape architects. A group of buildings have been constructed and the architect must plan for connecting walkways. If the people who are to use these buildings are allowed to freely, unconsciously, create pathways between them they would in stinctively round corners and make curved paths around obstacles. They would naturally form lines of least resistance. In design, the shortest distance between two points is not a straight line but a slalom, or curve. Here, the people are creating a two-dimensional slalom. This results in an economical, logical pattern flow not possible if right angle, straight sidewalks are used.6

Artists, architects and designers frequently ignore this law of circula tion in design. Understanding this axiom enables us to create in con-

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junction with natural forces and inclinations, rather than to contradict them.

The tangram, the ancient Chinese game of shapes, is another example of economy of means - making the most of the least. It is a simple square divided into seven pieces: five triangles, one square, and one rhomboid. Cutting apart and reassembling this configuration helps us develop skills for posing and solving problems. This develops an abil ity to abstract and refine. It also sharpens powers of observation through comparison of natural and geometric forms. Yet, it uses a minimum of material and simplified shapes to equate ideas. The na ture of the materials and techniques in textiles lend themselves to the use of this minimal principle.

Textiles and the five dimensions.

The practical use of our resources logically leads us into the problems of dealing with dimensions in space. Traditionally, we have thought of space in terms of having only three dimensions: length, width and depth. Then, the advent of non-Euclidean mathematics changed our initial perceptions. Now we need to think in terms of utilizing all five dimensions. The fourth is gravity and the fifth is time. The effects of time and gravity on textiles are particularly obvious,

C. H. Hinton comments on living in a dimensional universe:The space sense, or the intuition of space, is the most fundamental power of the mind. But I do not find any where a systematic and thorough-going education of the space sense. It is left to be organized by accident. Yet, the special development of the space sense makes us ac quainted with a whole series of new conceptions.7

These effects of gravity are evident in growth patterns in nature. This is especially true in relation to the scale of animal, vegetable and min eral matter. Gravity is a natural restrictive factor that helps to deter mine the size and shape of things. This same force continues its influ ence in technical textile applications. This extends further in its use in design and composition.

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The catenary curve serves as a good example. It is easily illustrated in this manner. Take a length of flexible cord and suspend it between your hands (which are held apart at the same height). Next, allow the cord to slaken so that it drapes into a curve. This is a catenary curve.

It is a result of gravitational forces on a flexible material. Strictly de fined, a catenary curve is formed by a cable of uniform size and weight suspended between two horizontally separated points. This same force is evident in the draping and shaping of two and three dimensional fabrics and forms. The fluid, flexible and manipulable characteristics of textiles make them strongly dependent on this effect.

This same catenary curve is translated into architecture as a basic building principle. The arch is a prime example of an inverted cate nary curve. In early history, ropes and cables were used to hoist drawbridges and suspend bridges. These were originally made with animal and vegetable fibers. Today's twisted metal cables serve the same purpose.

The simple but profound God's eye.

The simplest of textile structures, the God's eye, lends itself to a syn thesizing of these concepts and ideas. It embodies the interdisciplinary principles of minima, pattern principles, and dimensions in space in cluding the effects of gravity and the Coriolis force. It also creates ge ometric shapes that mirror nature's packing systems and growth structures.

In order to learn we must first associate with known facts and experi ence. For Helen Keller it was necessary to learn the concept of "water" before she could grow intellectually and creatively. So, too, we must devise and use systems that allow and encourage creative growth.

As children we crossed two sticks to create a frame for a God's eye. In spiralling the yarn around the spokes we mimicked the spider weaving its web in space. Thus our innate curiosity instinctively and logically led us from the simple to the profound. This evolution is essential to our intellectual development and intuitive perception.

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Conclusion.

Out of this research and experimentation has emerged a universalist attitude toward life, and an appreciation for the inspiration provided by the study of textiles.

The Coriolis force serves as a point of departure. Its effect on growth processes and raw materials is then easily understood. This logically leads into spinning of these materials into linear elements. Manipulating these into varied textile techniques naturally follows. These influences further inspire endless applications in all art forms and other disciplines. This allows some understanding of the com plexities of the universe and how they interrelate.

There is one limitation in this pursuit of creativity. One realizes the endless possibilities in the universe and that one lifetime and space can never begin to accommodate, not only its wonders and challenges, but also its beauty.

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Endnotes

1. Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance (New York, Bantam Books, 1975), pp. 106 - 107.

2. Shirley E. Held, Weaving, 2nd Ed. (New York, Holt, Rinehart and Winston, 1978), Chapters I and n.

3. J. Michaels-Paque, A Creative/Conceptual Analysis of Textiles, 2nd Ed. (Milwaukee, Wisconsin, J.M.P. Publications, 1985), pp. 7 - 9.

4. Martin Gardner, The Ambidextrous Universe, 2nd Ed. (New York, Charles Scribner's Sons, 1979), pp. 46 - 49.

5. William Hoffer, "A Magic Ratio", Smithsonian Magazine, 6 (9) (December, 1975),pp. Ill - 124.

6. Paul Jacques Grille, Form, Function and Design (New York, Dover Publications, 1975), paraphrase of pp. 180 - 182.

7. Richard K. Thomas, Three-Dimensional Design (New York, Van Nostrand Reinhold Company, 1969), C.H. Hinton Quotation on Frontispiece.

Illustrations: J. Michaels-Paque, A Creative/Conceptual Analysis of Textiles, 2nd Ed. (J.M.P. Publications, 1985).

4455 North Frederick Avenue Milwaukee, Wisconsin, 53211 U. S. A.

Verification of the concepts purported in this paper are substantiated in the form of the original artworks shown in the following photographs.

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Illustration A

plaiting plaid oblique twining

twill-herringbone interlaced stitch counter change

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Illustration A (continued)

broken twill woven knot diagonal stripes

1-beam looping flower knot

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chevron knit cablestitch strips

arrowhead mesh netting Celtic knotwork

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plain weave mat knot satin weave

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shepherds check florentine embroidery interlaced bands

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block pattern tongue & groove basket weave

overlapping circle chevron braid caning

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Illustration C

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Illustration D

Fibonacci Spirals

MICHAELS PAQUC \*

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Illustration E

Concepts that relate to textiles

Illustration F

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Illustration G

Pattern principles - geometric tessellations

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Illustration G (continued)

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Title: "Overall S pattern done in knotting". Artist: J. Michaels-Paque Photo: Henry Paque

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Title: Left: "Formulated felt" (44"h by 31 "w by 9 1/2"d) Right: "Fabricated felt" (25"h by 19"w by 7"d).

Artist: J. Michaels-Paque

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Title: "Legato" Flexible sculpture.Inspired by harmonic progression, in algebra, the mathematical

interpretation of sounds. (29"h by 36"w by 3 l/2"d). Artist: J. Michaels-Paque Photo: Henry Paque

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Title: "Ratios" Flexible sculpture.(5'6"h by 5'w by 3"d).

Artist: J. Michaels-Paque Photo: Henry Paque

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Title: "Collapsing rectangles" (1985) From the Clastic series. Fiberboard, gesso, cloth; experimental technique.

(12"hby 132"lby41"w). Artist: J. Michaels-Paque

Photo: Photographic Services, University of Wisconsin, Milwaukee.

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Title: "Reiterations" (1985) From the Anti-Clastic series.Experimental penetrating planes technique.

Knotted/woven natural cotton. (70"h by 57"w by 3"d).Artist: J. Michaels-Paque

Photo: Photographic Services, University of Wisconsin, Milwaukee.

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