mathematics and beauty: a sampling of spirals and...

Mathematics and Beauty: A Sampling of Spirals and 'Strange' Spirals in Science, Nature andArtAuthor(s): Clifford A. PickoverSource: Leonardo, Vol. 21, No. 2 (1988), pp. 173-181Published by: The MIT PressStable URL: http://www.jstor.org/stable/1578555Accessed: 07/01/2010 21:49

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Mathematics and Beauty: A Sampling of Spirals and 'Strange' Spirals in

Science, Nature and Art

Clifford A. Pickover

Abstract-Computers with graphics can be used to produce visual representations of

complicated physical and mathematical structures and phenomena. This paper presents an overview of spiral geometry in mathematics, nature, physics, civilization and art, with an

emphasis on novel spiral forms created by the author. 'Recipes' for both traditional and nontraditional spirals are given. The computer-generated patterns are of interest artistically and

mathematically, and they reveal a visually striking and intricate class of spirals whose

generating algorithms range from electronic circuit equations to random segment methods.

Fig. 1. Examples of spirals in nature: (a, left) Several shells of various ammonites (extinct shell-fish). (b, top right) Spiral horns on a bighorn sheep and an African kudu antelope. (c, bottom right) Spiral representations ('spirons') of sub-atomic particles. Extremely small particles, such as electrons,

have been postulated to be composed of these spiral structures [17].

Clifford A. Pickover (computer scientist), IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A.

Received 19 September 1986.

I. INTRODUCTION

If the cosmos were suddenly frozen, and all movement ceased, a survey of its structure would not reveal a random distribution of parts. Simple geo- metrical patterns, for example, would be found in profusion-from the spirals of galaxies to the hexagonal shapes of

snow crystals. Set the clockwork going, and its parts move rhythmically to laws that often can be expressed by equations of surprising simplicity. And there is no logical or a priori reason why these things should be so.

-Martin Gardner, "Order and Surprise" (1985)

? 1988 ISAST Pergamon Press plc Printed in Great Britain. 0024-094X/87 $3.00+0.00

LEONARDO, Vol. 21, No. 2, pp. 173-181, 1988

w

- v

rr

v I

ways of characterizing the complexity of such behavior.

II. 'TRADITIONAL' MATHEMATICAL SPIRALS

Generally, plane curve spirals are of the form

r =J(O) (1)

in polar coordinates (where f is monotonic), and they possess a simple beauty which humans have copied in their arts and tools, and nature has used in the creation of many structures of life. All the mathematical forms presented in this section were first discovered in the seventeenth and eighteenth centuries, except for the simplest form, the Archimedes spiral, which is a curve having the equation

Fig. 2. Ornamental alphabets often contain spirals, such as seen here in several letters from graphic designer Carol Grafton [19].

"Whatever can be done once can always be repeated", begins Louise B. Young in The Mystery of Matter [1] when describing the shapes and structures of nature. From the tiny twisted DNA molecules in all living cells to the gargantuan curling arms of many galaxies, the physical world contains a

startling repetition of spiral patterns. The growing awareness of fundamental rules underlying the apparent general repetition of nature has led to the search to identify, measure and define these patterns in precise scientific terms [2, 3].

This paper presents an overview of

spiral geometry in mathematics and art, with an emphasis on new spiral forms

GI'EIU. bA R.m n 3aJl.

o ?

Fig. 3. Various simple spirals. Written languages often contain several spiral forms. (a) Thai printing [21]. (b) This ornate design represents the word for spiral in Farsi (pronounced 'mar-peach', which literally means 'snake curl'). (c) Sample of Tamil script (from southern India), which is famous for spiral forms.

created by the author [4]. The term 'spiral' is used generically to describe any geometric smooth curve that winds about a central point or axis while also receding from it. When thinking of examples of spirals, both the mundane and exotic easily come to mind: for example, the

gentle curl of a fern tendril, the shape of an octopus's retracted arm, the death- form assumed by a centipede, the spiral intestine of a giraffe, the shape of a butterfly's tongue, the spiral cross section of a scroll, the shape of the Yellow Brick Road in Munchkinland in the film classic The Wizard of Oz and even the characters in several written languages.

This paper is divided into four parts, which deal with traditional mathematical spirals, spirals in nature, spirals made by humans and 'strange' spirals. I have coined the term 'strange' spiral to denote entities that have many of the visual properties one normally associates with spirals as defined above, yet also have other interesting features such as being infinitely convoluted, such as those described later in the text.

I seek in this paper to emphasize the important and conspicuous role that spirals play in nature and civilization, to show the reader how to create such spirals using a computer and to demonstrate how research in simple mathematical formulas can reveal an inexhaustible new reservoir of magnificent shapes and images. Indeed, structures produced by these equations include shapes of startling intricacy. The graphics experi- ments presented here, with the variety of accompanying parameters, are good

This was first discussed by Archimedes around 225 B.c. The most commonly observed,spirals are of the Archimedean type: tightly wound springs, edges of rolled-up rugs and sheets of paper, and decorative spirals on jewelry. Practical uses of the Archimedes spiral include the transformation of rotary to linear motion in sewing machines [5].

The logarithmic spiral (also known as the equiangular spiral or Bernoulli spiral) can be expressed as

r = keal (3)

This spiral was first discussed by Descartes in 1638. The angle between the straight line, 0 = constant, and the tangent to the curve is constant. Examples of the logarithmic spiral found in nature are mentioned in section III of this paper. Other more exotic spirals include the hyperbolic spiral (or reciprocal spiral), which is of the form

A littus has the form A littus has the form

r20 = a

(4)

(5)

A Cornu spiral (or clothoid or Euler's spiral) has a parametric representation:

t I7rt2

x=aVrf o cos ( ) dt

y= ax/ t

sin ( t2

) dt 2

(6)

(7)

Pickover, Spirals

r =aO (2)

174

XKXV Tubo Cochleato

Fig. 4. Artists' spirals. Spirals often appea. n drawings of the creatures of mythology, legend or religion [31]. Clockwise from top left: (a) Maori drawing of spiral facial tattoos, New Zealand, 19th c. (b) Gargoyle from the Milan Cathedral. (c) Spiral nose on a cittern by Girolamo de Virdis, 1574. (d) German helmet with spiral horns, 1511-14. (e) Lion with spiral dragon's tail and tongue, from a Victorian border design. (f) Water dragon, from a print. (g) This 'voice amplifier' appears in Gabinetto Armonico, first printed in 1716, by Jesuit Filippo Bonanni. Bonanni wrote on art and folk instruments of all levels of

European society, and here it is suggested that a large spiral tube would considerably magnify the sound of the voice.

Pickover, Spirals 175

This curve was discovered by Euler in 1744, and M.A. Cornu later used this curve in the representation of optical diffraction.

Some of these families of spiral curves can perhaps be more simply defined by

r = amO (8)

which includes the Archimedes spiral (m = 1), Fermat's spiral (m = 2) (first discussed by Fermat in 1636), the hyperbolic spiral (m = -1) (first discussed by Pierre Varignon in 1704) and the littus (m = -2) (originated by Cotes in 1722).

The involute of a circle with parametric equations

x = a(cos + 4) sin k) (9)

y = a(sin k - b cos >) (10)

was first taken into account by Huygens when he was formulating his ideas for clocks without pendulums which might be of service on seagoing vessels. This is the curve described by the endpoint of a

string as it unwinds from a circle of radius a while held taut. The curves traced by all

points along the plank of a seesaw or the

path of a goat tied to a cylindrical post as it winds tightly around it [6] are both involutes of a circle.

Finally, the cochleoid (or snailform) is

given by sin 0

r=a( -n (11) 0

Apart from their mathematic differences, and also the varied natural forms these spirals help to describe, many of these spirals are quite different visually. For example, perhaps the most exotic looking of the group is Euler's spiral, which consists of two spirals connected together, giving it the appearance of a mustache with two curled ends. The only other spiral of the group with more than one 'center' is the cochleoid, which contains two directly adjacent spirals. Fermat's spiral is the only member that consists of two concentric lines, and it resembles the paths of two tracks of a stereo record groove. Finally, the littus is the only spiral of the group with a long, almost linear section; it looks like a fern tendril with a very long stem.

III. SPIRALS IN NATURE

Spirals in nature are ubiquitous and have a range of botanical and zoological manifestations [7]. Probably the most common examples are the logarithmic spirals of nautilus shells and other sea shells and the horns of a variety of mammals including the African kudu antelope [8] (Fig. la, b), the arrangement of seeds of many plants (sunflower and daisy) and the scales of a pinecone. Other clear examples of spirals in plants (also known as 'green spirals') occur in the purple cornflower [9], pineapples, cacti and the arrangement of buds in a pussy willow stem. It appears that some general regulatory agent is at work producing common spiral forms in plants

as disparate as sunflowers and pines, though relatively little biological experi- mentation has been done in search of the source of these patterns. Dixon [10] gives several equations for computing these spirals based on the logarithmic and Fibonacci spirals. Other algorithms for recreating shells, horns, tusks and claws are described by Kawaguchi [11]. Body parts with spiral structures include the spiralling fibers at the apex of the heart.

Martin Gardner has noted that Eperia, a common variety of spider, spins a web in which a strand coils around the center in a logarithmic spiral [12]. Aquatic life often manifest spiral bodies or append- ages. The branchial filaments of spiro- graphics (tube-worms) play spirally in the water, and when stimulated they jerk spontaneously back into a tube along a screw-like form. The tube-worms' gill filaments form a perfect spiral [13]. In the intestine of Protopterus and other lungfishes there exists a spiral fold. On a smaller scale, many unicellular animals have incorporated spiral shapes; for example, the infusoria are creatures with spherical form and have screw-like spiralling motions for locomotion [14]. On even smaller scales, spirals also abound: for example, the a helix, the supercoiled a helix, the helical packing of subunits in viruses and in DNA, and the coiling of DNA strands in a nucleosome [15, 16]. Still smaller yet, subatomic particles, such as electrons, have been postulated to be composed of spiral structures called 'spirons' [17] (Fig. lc).

Other examples in nature include the

Fig. 5. Random-segment spirals. These 'fuzzy' spirals were created by the author by connecting pairs of points with straight lines. The eye perceives spiral patterns by observing the correlated pairs.


%

I 1.1

!i

LT Tc

Fig. 6. Dynamical systems involving real numbers. These pictures were created by the author by plotting trajectories through time. (a) Phase portrait of an RLC circuit. (b) Phase portrait of an oscillator.

(c) Phase portrait of coupled differential equations.

Fig. 7. Iteration of complex functions. These plots, created by the author, describe the behavior of complex functions and show how complicated behavior can arise in systems from simple rules. (a) Complex map of a single function, z = sin z + M.(b) Complex map of a network of equations. (c) Complex map of a single function, z= z2 + ,U. The different shades of gray indicate different rates at which

the function explodes.


.II

L

patterns on Spiriferida (certain fossilized shells), spiral cleavage (a pattern char- acterized by formation of an embryo- logical cell mass showing spiral symmetry), the spiral valve (a spiral fold of mucous membrane in the small intestine of sharks which increases the area for

absorption), fingerprint whorls, and the axonme (bundle of fibers in a flagellum) [18].

Examples of natural non-living spiral motifs are the vortices of water waves and air-including vertical spiralling columns of air which carry gliding birds to greater heights and the whirling path of air in organ pipes-and galactic formations (see below).

IV. SPIRALS MADE BY HUMANS

In modern technological society, spiral devices play an invaluable role. Examples include the spiral bevel gear (a gear with oblique teeth), spiral chutes for conveying material, the spiral cutterhead (a rotary digging device), spiral mold cooling (to cool an injection mold by passing liquid through a spiral cavity) and the spiral staircase. Other spiral devices include a variety of gauges, heat exchangers, thermometers, pipes, distillation equip-

Fig. 8. Trigonometric 'bushy' spirals. Here, an infinite variety of spiral-like patterns were created by the author using a complicated

hierarchy of trigonometric functions.

ment and intrauterine devices for birth control. The spiral spring (an Archi- medean spiral) is particularly interesting in its unique ability to respond to both torsional and translational force. In musical instruments, spirals also abound. Examples include the posthorn and many modern brass instruments. Children's toys often comprise spirals (e.g. the Slinky and Chinese yo-yos), and spiral forms also are used in vending machine dispensers, record grooves, ornamental alphabets (Fig. 2) [19], and for the curse-word symbols that car- toonists use. Extremely intricate spirals can be made chemically by coating a glass sheet with carbon black and various liquids [20].

As mentioned in the Introduction, written language sometimes contains spirals, and Thai printing (Fig. 3a) is one example [21]. Another example is the ornate design that represents the word for spiral in Farsi (pronounced 'mar-peach', which literally means snake curl) shown in Fig. 3b. Finally, Tamil script (from southern India) is famous for its spiral forms (Fig. 3c).

Doodles-those designs we make by idle, unconscious scribbling when our thoughts are occupied elsewhere-often contain a remarkable number of spiral forms. Just what this unconscious tendency to draw spirals tells us about the unconscious human mind and its inner images is not clear, but from ancient times the spiral has been used in art and dance to induce a state of ecstasy [22]. The actual mental and physical ability permitting humans to draw spirals probably begins around age 3. Today, psychologists use the spiral for deepening relaxation and bringing about a 'deeper level of mind' [23].

Ancient examples of spirals include prehistoric spiral mazes [24], Stone Age animal ornaments [25], terracotta pot spiral designs from the sixth century B.C. [26], decorations from ancient Altaic works (middle of the first millennium B.C.) [27], engravings on threshold stones of initiation chambers in the Bronze Age in Ireland [28], Tibetan Tanka artwork [29] and scrollworks for Irish manuscripts [30]. Spirals also often appear in artists' drawings of the creatures of mythology, legend or religion (Fig. 4a-f) [31]. A beautiful example of an eighteenth- century engraving depicting a spiral musical instrument is shown in Fig. 4g. This 'voice amplifier' appeared in Jesuit Filippo Bonanni's book Gabinetto Armonico. Spirals also are found in modern surrealist art (for famous examples, see the works of Joan Mir6, Matta and Alfred Jarry [32]).

V. STRANGE SPIRALS

As mentioned in the Introduction, 'strange' spiral is a term coined to denote entities that have many of the visual properties one normally associates with

spirals as defined above, yet have other novel features such as being infinitely convoluted, being infinitely discontinuous or simply having more complicated generating formulas (see below).

Galaxies

The logarithmic spiral's most im- pressive appearance is in the arms of many galaxies. Although it is clear that the overall mass distribution and motions of the components of a galaxy are determined by gravity, it has not been clear what is responsible for the striking spiral morphology of galaxies [33]. Spiral arms extend over 20,000 parsecs (1 pc = 3.26 light years), and the contemporary view is that it is necessary to have a long- range interaction like gravity to create such long-range order. In spiral galaxies, the spiral arms are sites of active star formation.

Spiral galactic shapes are relatively easy to program on the computer, as they consist of dots forming a double logarithmic spiral. One arm is 180? out of

phase with the other. To obtain a picture of a galactic distribution of dots, simply plot dots at (r, 0) according to:

r = e[6 tan 0] (12)

r2 = e[(7r + 0) tan 4] (13)

where r, and r2 correspond to the two intertwined spiral arms. The curvature of the arms is controlled by 0 which should be about 0.2 radians for realistic results. In addition, 0 < 0 < 1000 radians. For extra realism, a small amount of random jitter may be added to the final points.

Fuzzy Spirals If a pattern of random dots is

superimposed on itself and rotated by a small angle, concentric circles are perceived about the point of rotation [34]. If the angle of rotation is increased, the perceived circles gradually disappear until an unstructured dot display is seen. This effect demonstrates the ability of the human visual system to detect local autocorrelations and may suggest a physiological basis of form perception in higher animals.

Figure 5 shows 'fuzzy' spirals created by a procedure related to the random-dot spirals. To reproduce these figures, first


plot about 5000 randomly positioned dots. These same 5000 dots are then rotated by 2? to 30?, expanded slightly and then connected to their respective original dots. The eye perceives spiral patterns by observing the correlated

pairs. By varying the angle and specific type of random numbers used, one can generate a variety of visually exciting patterns. For some of the figures, Gaussian white noise is used. This can be

generated by:

n

Noise= (-) , 6i (14) n n i=

where 6 is a random number and n is about 6.

Spiral-Like Forms from Differential Equations

RLC Circuit

The differential equation

Lx+Rx +( )x=O (15) C

governs the behavior of a simple RLC electric circuit where x represents the charge on the capacitor and x represents the current in the loop [35]. R is the resistance, L the inductance and C the capacitance. The equivalent first order system is:

i =y (16)

y - (cx- (L ) y (17) This can be easily implemented on a

computer:

XI = X,i + y,t- (18)

Yt= Y-+ [( )tLC-

R \ - -.j (19)

where X > 0 is a constant known as the

'step size' of the numerical solution. X is

kept small (X - 0.1). Also, -3 < x < 3, and -3 <y < 3. R, L and C can be set to 1.0. Figure 6a reveals that all initial points (x,y) have spiral trajectories into the fixed

point in the center of the figure. Watching this shape dynamically unfold on the cathode ray tube (CRT) is especially useful for revealing this 'damped' behavior.

Self-Sustained Oscillations

In this class of problems, we have a nonlinear vibration where the damping is also nonlinear. One of the best-known cases encountered in practice is that of the triode vacuum tube. An electrical circuit containing such a tube [36] is shown in Fig. 6b with its accompanying 'phase portrait'. Figure 6b was generated from the differential equation governing the current in an oscillator:

CLX + F(x) + x = 0 (20)

The function F(x) is a nonlinear function (caused by the nonlinear relation between current and voltage in the tube). The function I have used was suggested by Stoker [37]:

F(x)=e [- x+ (-) ] (21)

I have implemented this system on a

computer by deriving the following coupled discretization using the forward Euler approximation [38]:

Xt = x_-i + Xy,_l (22)

y, = y,t- + x,- R (- Yt-

+ )(23) 3

where X < 0. X is kept small (X - 0.05). Also: -3 < x < 3, and -3 < y < 3, and

= 1.0. From a mathematical standpoint, the

graph of Fig. 6b is fascinating. As it evolves on the CRT, one can see that all initial points (x,y) contained within the bounding ovoid shape (known as an attractor) spiral outward to the attractor surface, while all points outside the attractor spiral inward toward it. Once a point in the path meets the attractor, it circulates along the attractor forever. As X is increased, more stochastic behavior is observed. The reader is encouraged to experiment with this parameter and to observe the results.

Other Spiral Trajectories As the reader may imagine, an infinite

variety of spiral patterns can be computed from differential equations. Figure 6c is but one example of another system [39].

Complex Iteration Spirals

Computers with graphics have

played a critical role in the study of iteration and in helping mathematicians form the intuitions needed to prove new theorems about convergence of sequences of points in the complex plane [40]. Today, there are several scientific fields devoted to investigating how complicated behavior can arise in systems from simple rules and how minute changes in the input of a nonlinear system can lead to large differences in the output; such fields include chaos and cellular automata theory [41].

One example of complex iterations is given by the simple mapping

z, = sin(z,_ ) + ti (24)

where z is a complex number and , is a complex constant. To use this technique, an initial z point is selected and the equation is iterated. In other words, the resulting value is recycled in the equation, producing a mathematical feedback loop. For certain initial values of z, this sequence of points explodes; these are colored black in Fig. 7a, which represents a map of the z-plane. Here ,u = (0.1, 0.1) [42]. Figure 7b was computed using a network of equations described elsewhere [43]. Figure 7c was computed using z = z2

+ /.

Trigonometric Iteration ('Bushy' Spirals) An infinite variety of spiral-like

patterns can be created using a com-

plicated hierarchy of trigonometric functions. The user enters a series of eight parameters: 1, d, g, k, u,, u2, wI, w2. In order to produce Fig. 8, first parameters s and t are computed for the pictures:

s= tan[ wI ] W2

(25)

t= 360[w, sin(s)+ w2 cos(s)] (26)

Parameter t controls the window of the

graphics screen-the lower left-hand corner is set to (-t, -t), and the upper right-hand corner is set to (t,t). The following sets of equations are then iterated for 0 < bi < 360 and 0 < aj < 360. The step size for the i loop is d, and the

step size for the j loop is g. Nested inside these two loops are the following equations:

z = kr [ul sin(laj) + u2 cos(laj)] (27)

r = bi [wl sin(z) + w2 cos(z)]

x = rcos(aj) + f

(28)

(29)


y rsin(ai) + 3 (30)

where

fp=0.1Xrx r, (31)

and y is a random number on the interval (0,1). Figure 8 results from plotting (x,y) coordinates. The eight parameter values are all set to values near 3 for Fig. 8. This system of equations follows that of Jacobson [44], the significant difference being the /f parameter, which gives the resulting patterns a much more 'natural' look. The spiral nature of these patterns is clearer when the viewer watches the figures plotted dynamically on the CRT.

VI. CONCLUSION

It is indeed a surprising and fortunate fact that nature can be expressed by relatively low-order mathematical functions.

-Rudolf Carnap (science lecturer)

Among the methods available for the characterization of complicated mathematical and physical phenomena, computers with graphics are emerging as an important tool [45]. Computer graphics also provides a way to represent both natural and artistic shapes which include mountains [46], shells [47], trees [48], wood grains [49], leaves [50] and stone walls [51]. Recently an algorithm was introduced that creates very complicated forms resembling invertebrate organisms [52] using simple polynomial equations.

From an artistic standpoint, spiral equations provide a vast and deep reservoir from which artists can draw. The computer is a machine which, when guided by an artist, can render images of captivating power and beauty. New 'recipes', such as those outlined here, can be used with such traditional elements as form, shading and color to produce futuristic images and effects. The recipes function as the artist's helper, quickly taking care of much of the repetitive and sometimes tedious detail. By creating an environment of advanced computer graphics, artists with access to computers will gradually change our perception of art.

As far as the physical universe is concerned, spiral shapes are one of nature's most fundamental forms. As science writer Kathleen Stein once pointed out, spirals appear early in the chain of animal evolution: cilia, worms' gills, fly larvae and some shark-egg capsules have them. The cochlea in the ear of every mammal is screw-shaped, and so are many corals. Many of the mathe-

matical shapes in the present paper are magnificently complicated structures which no one could have appreciated fully or even suspected before the age of the computer. The richness of the resultant forms often contrasts with the simplicity of the generating formulas.

Precisely why the spiral is ubiquitous in nature and in civilization is a profound question. Whether one is considering the movement of stars, the development of an embryo, the motion of a pencil on a page or many of the phenomena that make up the fabric of our universe, it is clear that symmetry operations are often nature's guiding hand. Spiral patterns often occur spontaneously in matter that is organized through symmetry transformations: change of size (growth) and rotation. Form follows function, and the spiral form can allow for the compaction of a relatively long length. Long-yet-compact tubes are useful in spiral molds, brass instruments, mollusks and cochleas for obvious reasons including physical strength and increased surface area. For some phenomena-such as in doodles, written languages and spiral galaxies- the precise 'reason' for spiral forms is less clear.

The mathematical concept of similarity holds one of the keys to understanding the processes of growth in the natural world. As a member of a species grows to maturity, it generally transforms in such a way that its parts maintain approxi- mately the same proportion with respect to each other [53], and this is probably a reason why nature is often constrained to exhibit self-similar spiral growth. Through time, humans have imitated the spiral motifs around them in their art forms and sciences, and occasionally they invent new spirals not known to have specific counterparts in the natural world. It is probable that this multitude of 'traditional' and 'nontraditional' spiral formulas will help scientists better understand the fundamental rules underlying the apparent spiral repetition of nature, since they can now generate, define and predict these patterns in precise scientific terms.

A report such as this can only be viewed as introductory; however, it is hoped that the techniques, equations and systems will provide a useful tool and stimulate future studies in the graphic characterization of the morphologically rich spirals of extreme complexity produced by relatively simple generating formulas.

Acknowledgements-I would like to thank Jacques Boivin, author of The Heart Single Field Theory, for permission to reproduce his

hypothetical 'spiron' shapes. I also thank Elahe Khorasani and Padmanabhan San- thanam for samples of Farsi and Tamil writing, respectively. Finally, I thank Adam for his repeated careful attempts to draw a spiral.

REFERENCES AND NOTES

1. L. Young, The Mystery of Matter (New York: Oxford University Press, 1965).

2. B. Mandelbrot, The Fractal Geometry of Nature (San Francisco: Freeman, 1982); H. Peitgen and P. Richter, The Beauty of Fractals (New York: Springer-Verlag, 1986).

3. D. Postle, The Fabric of the Universe (New York: Crown Publishers Inc., 1976).

4. The present paper is the second in the author's "Mathematics and Beauty" series which presents aesthetically ap- pealing and mathematically interesting patterns derived from simple functions. The resulting pictures should be of interest to a range of scientists as well as to home-computer artists. Articles in the series include: C. Pickover, "Mathematics and Beauty: Time-Discrete Phase Planes Associated with the Cyclic System, {x(t)

-fly(t)), (t) =(x(t))}", Computers and Graphics 11, No. 2, 217-226 (1987); "Blooming Integers (Mathematics and Beauty III)", Computer Graphics World 10, No. 3, 54-57 (1987); "Computer Graphics and Wild Monopodial Tendril Plant Growth (Mathematics and Beauty IV)", Computer Graphics World 10, No. 7, 143-145 (1987); "Mathematics and Beauty V: Turbulent Complex Curls", Computers and Graphics 11, No. 4, 499-508; The Journal of Chaos and Graphics, Vol. 1, IBM Research Report RA 186 (1987); Computers, Pattern, Chaos, andBeauty, IBM Research Report RC 12281 (1986). Order IBM research reports from: IBM Distribution Services, Stormytown, IBM Watson Lab, York- town Hts, NY 10598, U.S.A. For other descriptions on this series, see: I. Peterson, "Picture This" (and cover picture), Science News 131, No. 25, 392- 395 (1987); I. Peterson, "Portraits of Equations" (and cover picture), Science News 132, No. 12, 184-186 (1987); R. Rivlin, "Computer Graphics: The Arts", Omni Magazine 8 (1986) p. 30.

5. M. Gardner, In the Unexpected Hanging (New York: Simon and Schuster, 1969).

6. Gardner [5]. 7. Y. Kawaguchi, "A Morphological Study

of the Form of Nature", Computer Graphics (A CM-SIGGRAPH) (1982).

8. E. Haeckel, Art Forms in Nature (New York: Dover, 1974); J. Harter, Animals (New York: Dover, 1979).

9. S. Braun, "Botany with a Twist", Science 86 7, 63-64 (1986).

10. R. Dixon, "The Mathematics and Computer Graphics of Spirals in Plants", Leonardo 16, No. 2, 86-90 (1983).

11. Kawaguchi [7]. 12. Gardner [5]. 13. T. Schwenk, Sensitive Chaos (New York:

Schocken Books, 1976). 14. Schwenk [13]. 15. B. Vainshtein, "Symmetry of Biological

Macromolecules and Their Associa- tions", Comp. and Maths. with Appls. 12B, 237-269 (1986).


16. C. Pickover, "DNA Vectorgrams: Re- presentation of Cancer Gene Sequences as Movements along a 2-D Cellular Lattice", IBM J. Res. Dev. 31, 111-119 (1987).

17. J. Boivin, The Heart Single Field Theory (1978) (available for $2.50 from J. Boivin, 4531 Bordeaux, Montreal, Canada H2H 1Z9); reprinted in Specula- tions in Science and Technology 3, 185- 204 (1980).

18. For examples of fingerprint whorls and axoneme, see G. Doczi, "Seen and Unseen Symmetries", Comp. and Maths. with Appls. 12B, 39-62 (1986).

19. C. Grafton, Bizarre and Ornamental Alphabets (New York: Dover, 1981).

20. J. Zvilna, "Colored Symmetries in Space-Time", Comp. and Maths. with Appls. 12B, 895-911 (1986).

21. M. Haas, The Thai System of Writing (New York: Graphic Arts Press, 1956).

22. M. Samuels, Seeing with the Mind's Eye: The History, Techniques, and Uses of Visualization (New York: Random House, 1982).

23. Samuels [22]. 24. Doczi [18]. 25. Schwenk [13]. 26. E. Rozsa, "Symmetry in Muslim Arts",

Comp. and Maths. with Appls. 12B, 725- 750 (1986).

27. K. Mamedov, "Crystallographic Pat- terns", Comp. and Maths. with Appls. 12B, 511-529 (1986).

28. Schwenk [13]. 29. Samuels [22]. 30. E. Makovicky, "Symmetrology of Art:

Colored and Generalized Symmetries", Comp. and Maths. with Appls. 12B, 949- 980 (1986).

31. R. Huber, Treasury of Fantastic and Mythological Creatures (New York: Dover, 1981).

32. W. Rubin, Dada and Surrealist Art (New York: Abrams, 1975).

33. L. Schulman and P. Seiden, "Percolation and Galaxies", Science 233, 425-431 (1986).

34. C. Pickover, "The Use of Random-Dot Displays in the Study of Biomolecular Conformation", Journal of Molecular Graphics 2, 34 (1984).

35. R. Finney and D. Ostberg, Elementary Differential Equations with Linear Algebra (Reading, MA: Addison-Wesley, 1976).

36. J. Stoker, "Mathematical Methods in Nonlinear Vibration Theory", in Pro- ceedings of the Symposium on Nonlinear Circuit Analysis (New York: Interscience Publishers, 1953) pp. 28-55.

37. Stoker [36]. 38. Finney and Ostberg [35]. 39. For a detailed description of the

mathematics of its creation, see Pickover, "Mathematics and Beauty: Time-Discrete Phase Planes" [4].

40. C. Pickover and E. Khorasani, "Com- puter Graphics Generated from the

Iteration of Algebraic Transformations in the Complex Plane", Computers and Graphics 9, 147-151 (1985).

41. C. Pickover, "Pattern Formation and Chaos in Networks", Commun. of the ACM (1988) 31, No. 2, 136-151.

42. For more specific information on how to create figures such as these, see Pickover and Khorasani [40].

43. Pickover [41]. 44. J. Jacobson, "Analytic Computer Art",

Proc. 2nd Symp. on Small Computers in the Arts (1982) pp. 47-60.

45. C. Pickover, "The Use of Computer- Drawn Faces as an Educational Aid in the Presentation of Statistical Concepts", Computers and Graphics 8, 163-166 (1984); "The Use of Symmetrized-Dot Patterns Characterizing Speech Wave- forms", J. Acoust. Soc. Am. 80, No. 3, 955-960 (1986); "On the Educational Uses of Computer-Generated Cartoon Faces", J. Educational Tech. Sys. 13, 185-198 (1985); "Frequency Representations of DNA Sequences: Application to a Bladder Cancer Gene", J. Molec. Graphics 2, 50 (1984); "Representation of Melody Patterns Using Topographic Spectral Distribution Maps", Computer Music Journal 10, No. 3, 72-78 (1986); "Computer-drawn Faces Characterizing Nucleic Acid Sequences", J. Molec. Graphics 2, 107-110 (1985); "A Monte Carlo Approach for e Placement in Waveform Fractal-Dimension Calcula- tion", Computer Graphics Forum 5, No. 3, 203-209 (1986); "Graphics, Bifurca- tion, Order and Chaos", Computer Graphics Forum 6, 26-33 (1987); "The Use of Random-Dot Displays in the Study of Biomolecular Conformation", Journal of Molecular Graphics 2, 34 (1984); "Spectrographic Representations of Globular Protein Breathing Motions", Science 223, 181 (1984); and L. Cohen and C. Pickover, "A Comparison of Joint Time Frequency Distribution for Speech Signals", IEEE International Conference on Circuits & Systems 1, 42-45 (1986).

46. Mandelbrot [2]. 47. Kawaguchi [7]. 48. M. Aono and L. Kunii, "Botanical Tree

Image Generation", IEEE Computer Graphics and Applications 4, 10-34 (1984).

49. C. Yessios, "Computer Drafting of Stones, Wood, Plant and Ground Materials", Computer Graphics (ACM- SIGGRAPH) (1979).

50. G. Kolata "Esoteric Math Has Practical Result", Science 225, 494-495 (1984).

51. Yessios [49]. 52. C. Pickover, "Biomorphs: Computer

Displays of Biological Forms Generated from Mathematical Feedback Loops", Computer Graphics Forum 5, No. 4, 313- 316(1987).

53. J. Kappraff, "A Course in the Math-

ematics of Design", Comp. and Maths. with Appls. 12B, 913-948 (1986).

GLOSSARY

Altaic-belonging to the Altai mountains of central Asia.

attractor-the behavior that a system settles down to.

cellular automata-a class of simple mathematical systems that are becoming important as models for a variety of physical processes. Though the rules governing the creation of cellular automata are simple, the patterns they produce are complicated and sometimes seem almost random, like a turbulent fluid flow or the output of a cryptographic system.

chaos-irregular behavior displaying sensitive dependence on initial conditions. Interestingly, chaotic behavior can sometimes be defined by a simple formula.

complex number-a number containing a real and an imaginary part, and of the form a + bi where i = -1.

damp-to cause a decrease in amplitude of successive oscillations.

dynamical systems-models containing the rules describing the way a given quantity undergoes a change over time. For example, the motion of planets about the sun can be modelled as a dynamical system in which the planets move according to Newton's laws.

Fibonacci sequence-the sequence 1,1,2,3,5, 8,13 ... (Un = Un-2 + Un-_), which governs many patterns in the plant world.

fractals-intricate curves that exhibit increasing detail ('bumpiness') with increasing magnification.

Gaussian white noise-a random number sequence shaped so that the distribution of values follows a bell-shaped curve.

helix-a space curve lying on a cylinder (or sphere, or cone) which maintains a constant distance from a central line (i.e. a 'spiral extended in space').

iteration-repetition of an operation or set of operations. In mathematics, composing a function with itself, such as in flfx)), can represent an iteration.

linear transformation-a function satisfying these two conditions: 1) F(p+ ~) = F(p) + F() and 2) F(rp) = rF(p), r e R.

transformation-the operation of changing (as by rotation or mapping) one configuration or expression into another in accordance with a mathematical rule.


mathematics and beauty: a sampling of spirals and...

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