The Pattern Book: Fractals, Art, and Nature

The following sections are included:

• The Pattern Book: Fractals, Art, and Nature

• Contents

This pattern represents the emergence of heavenly bodies from primitive celestial vortices as imagined by Gabriel Daniel, a late seventeenth-century commentator on Descartes. More than a few controversialists of the early-modern era made their careers by reviewing, revising, and interpreting Descartes' remarkable vortex theory. These overly ambitious scribblers applied a plethora of methods to the French physicist-philosopher, trying everything from science to satire. Descartes had argued that the universe was a plenum, an infinitely large container completely filled with extended matter and uncorrupted by empty spaces. To explain the differences in the densities of objects in a universe allegedly packed as tightly as possible, Descartes conjectured that his cosmos might be pock-marked with “vortices”. Energetic, unstable, and even dangerous, these whirlpools might draw some collections of matter into tighter quarters than others. Turbulence reigns supreme in a Cartesian universe. According to vortecticians, matter begins as a plasma-like whirlpool, condenses into a furious sun, calms into a jiggling “terrela” (a luminous, gaseous, whirling, or incomplete planet like Jupiter or Saturn), and, at last, settles into a planetary lump. The pattern shown here portrays the dynamic, even terrifying interference of the great, originative vortices that were eventually to become the components of our solar system. Gabriel Daniel was himself a rather unstable fellow. After writing several hundred pages of satire against Descartes' theories, he apparently developed an affection for the great geometer, consoling himself that Descartes had never really died, but had only been carried up into some higher vortex by means of his own whirling tobacco smoke [1]

Describe here is a macroscopic chemical pattern occurring on a 1 mm scale in a chemical excitable solution which is maintained under conditions far from thermodynamic equilibrium. The Belousov-Zhabotinskii reaction has become the most prominent chemical model example of an excitable medium displaying spatio-temporal self-organization. In this reaction due to the nonlinear interaction of complex reaction kinetics with molecular diffusion, spiral-shaped waves of chemical activity are readily observed [1, 2]. For quantitative pattern analysis a computerized, video-based two-dimensional spectrophotometer has been designed which yields digitized high-resolution data of the spatio-temporal evolution of the patterns [3, 4]. Picture A of the figure shows a complex wave pattern in a thin solution layer of this reaction, as observed in transmitted light of appropriately selected wavelength and recorded by a video camera…

The figure visualizes an ensemble of three-dimensional conformations of the protein, Avian Pancreatic Polypeptide (APP), obtained by a method of optimization which is based on an analogy of the Traveling Salesman Problem …

These patterns, called “biomorphs”, are all generated by the same recursive tree algorithm, familiar from computer science textbooks and most easily understood with reference to the simple tree in the middle of the figure…

Wood patterns are infinitely varied, offering myriad opportunities for novel designs. Shown here is a photograph of a cross-section of an apple tree burl. Infrared film with A25 filter is used to eliminate the gray tones and to accent the harder wood lines.

Plant forms are characterized by arrangements of leaves around a stem: alternating, whorled, or spiraling. The various arrangements, or phyllotaxies, form patterns with different degrees of visual symmetry. Though these patterns may not be explicitly considered by gardeners and artists, a plant derives much of its beauty and its visual “feel” — formal or free, ornate or simple — from its phyllotaxy. When the phyllotaxy along a single axis is duplicated over and over, spectacular patterns can be formed…

Described here is a pattern showing different trajectories in a space of all possible rhythms of four notes. This space is traversed by a quantization system that seeks a metrical interpretation of a performed rhythm. The durations of the notes are adjusted to an equilibrium state in which many of the adjacent induced time intervals have a small integer ratio. The system is implemented in a connectionist, distributed way. A network of cells — each with very simple individual behavior — is used, in which the cells that represent adjacent intervals interact with each other…

Represented here is the first visualization of a nucleon structure developed according to the principles of the Heart Single Field Theory. In this illustration, each small circle represents a spherical space containing an electron-like component closely bound to its neighbors, their number totalling 1836 to form a proton. These components are “electron-like” only in terms of that aspect of their structure which is responsible for their mass; that aspect of their structure, normally responsible for “charge”, has been altered into an intra-nucleonic weave. This model was meant as an example of a type of theoretical construct that could be sought within the context of a coherent all-encompassing theory on the nature of the universe. Uncertainty, simultaneity and dualities (such as “wave/particle”) may not be considered as absolute features of elementary reality for the purpose of such models; rather, the behavior of precise fundamental structures must give rise to apparent uncertainty, instantaneity and duality…

Wood patterns are infinitely varied, offering myriad opportunities for novel designs. Shown here are photomicrographs of cross sections for various woods including Urnday, Orey wood, and Paldo or Guinea wood.

The pattern described belongs to the Moiré patterns first investigated by Lord Raleigh in 1874.

The Moiré phenomenon is based on purely geometrical principles in as much as the image processing in the eye of the observer need not be taken into account: Two textures with black and white components are geometrically superposed by forming the union or the intersection of the black pointsets…

Fractals have the property of producing complexity from simple, iterated rules. In this variety of broccoli, the rule of self-similarity operates on a spiral pattern to create three levels of nested spiraling florets. The effect is as elaborate and fanciful as Moorish architecture, giving the plant its unbroccoli-like name of Minaret. But the fractal generation of rich complexity from simple rules appears everywhere in nature. What is most remarkable about Broccoli Minaret is that the fundamental spiral pattern on which the fractal rules operate is itself the product of a few simple rules…

Described here is a pattern showing a useful representation of the sequence of nucleotides in the DNA of a plant virus. The DNA is the genetic material of the virus and encodes all functions necessary for the virus to reproduce itself in host plants. A DNA strand is a linear polymer of four nucleotides: adenosine, guanosine, thymidine, and cytidine monophosphates. The information content of DNA lies in the order of the nucleotides. To depict the order, the upper case Roman character for the first letter of the English name of a nucleotide usually represents that nucleotide. Such representations are difficult to scan for interesting features, inefficient in the amount of space they occupy and in some type fonts lead to mistaken reading of “G” and “C”. In this pattern [1], the top line of the sequence row has circles for nucleotides containing purine bases (“A” and “G”), while circles on the bottom line of the sequence row indicate pyrimidine nucleotides (“T” and “C”). The middle line of the row contains an additional circle for “G” and “C” residues. Most DNA molecules, including that of this virus, contain two anti-parallel DNA strands. The sequence of one strand is complementary to that of the other, according to the rules that “A” pairs with “T” and “C” with “G”. The Puppy representation shows the sequence of both strands; reading the diagram upside down gives the complementary strand…

Described here is an algorithm for generating stone wall patterns. This algorithm requires only a few parameters as input data. The output data are a bump plane, which represents each stone's height data, and an attribute plane, which represents each stone's attributes. The method is an enhancement of C. I. Yessios' work [1]…

Various techniques for music visualization, music transcription, melody storage, and melody matching have been proposed in the past (see references). A few weeks ago, I was playing with an audio data acquisition board in conjunction with a music synthesizer — acquiring about 7 seconds each of various voices. After I adjusted the pitch so that the period was a multiple of 512 samples, I was able to display the time-varying waveform as a 512 × 512 × 8 greyscale image on my NTSC frame buffer. Some of the patterns I looked at are really quite beautiful (see figure)…

Ernst Heinrich Haeckel (1834–1919) was a German biologist and philospher interested in the beauty of natural forms. Throughout his career he made detailed drawings of a range of organisms. He seemed particularly interested in deep-sea and microscopic life. Shown here are some of his drawings of various species of sea-lilies (animals related to starfishes and sea-urchins).

Described here is a pattern produced by chaotic behavior arising from aggregating particles. “Aggregation” is a term used to describe growth arising from the agglomeration of diffusing particles. In 1981, Witten and Sander developed a computer model for aggregation starting with a single seed particle at the center of a space [1]. Their computer program introduces a new particle which moves randomly until it approaches another particle and sticks to it. The first particles attach to the seed, but soon a branched, fractal structure evolves with a dimension of about 1.7. This process is called diffusion limited aggregation (DLA). Since the introduction of the Witten-Sander model in 1981, considerable research has been devoted to the properties of DLA [2–6]. In the diagram here, the additional contour lines give extra information to scientists concerning the growth process.

This display represents the secondary structure of an RNA sequence based on the first 2000 prime numbers.

The relationship between prime numbers has previously been investigated by a large number of methods [1]. Here, we describe a new approach in which we designate pairs of primes by the symbols A, G, C or U, depending on the property of that prime pair as described below. These symbols were chosen because they are commonly used to designate nucleotide bases in RNA. The computer program FOLD has been designed to investigate the properties of ribonucleic acid sequences [2, 3]; here we implement this program to analyze the series of primes…

The two patterns shown here represent two different views of a control switch in the genetic material of a virus called T7. Like many other viruses, T7 is made of proteins that coat and protect its genetic material DNA (deoxyribonucleic acid). The virus looks like a lunar-lander landing on the moon. When a T7 particle contacts the surface of the bacterium Escherichia coli, it sticks there. Once it has landed, it injects its DNA into the cell, much like a hypodermic needle does. The DNA contains instructions for taking over the bacterial cell. The cellular machinery unwittingly copies these instructions into RNA. The RNA is then used to make proteins that stop the normal cell mechanisms. Other instructions tell the cell to make copies of the viral DNA and to make the proteins of the viral coat. These are assembled to produce perhaps a hundred new T7 particles. The final instruction causes the cell to burst open, just as in the movie “Alien”…

This pattern shows the distribution of fractions of Gaussian integers whose denominators are limited to a certain size. A Gaussian integer is a complex number whose real and imaginary parts are both integers. A fraction of Gaussian integers, A/B, is simply A divided by B using the rules of complex arithmetic. The size of a Gaussian integer referred to above is its complex absolute value, given by for the complex number x + iy. In the pattern presented here the size limit was 25. The individual fractions were plotted by using the real part as an x coordinate and the imaginary part as a y coordinate. The page corresponds to the complex plane as follows: the origin of the complex plane is at the center of the large dot in the lower left corner of the page. The number 1 + i0 would be the center of the large dot in the lower right corner and the number 0 + i1 would be the large dot in the upper left…

The lattice pattern here was produced by a computer and demonstrates various types of repetition and symmetry. To draw a lattice pattern, the computer takes an elementary set of line segments and arcs, and manipulates them using a space group (a sequence of reflections, rotations, and translations) into a tile. These tiles are then stacked in a regular lattice in twedimensional space, thus initiating further symmetries. Even random, nonsymmetric starting sets of lines and arcs produce beautiful symmetrical patterns.

Shown here is the number of prime pairs G(E) which can be found to sum to a given even number E, plotted as a function of E. (For example, for E = 10, G(E) is 2, since 10 can be expressed by just two prime pairs — as 3 + 7 or as 5 + 5.) This pattern is connected with the famous Goldbach Conjecture, named after the Russian mathematician, Christian Goldbach, who speculated in a letter to Leonhard Euler that every even number greater than 4 can be expressed in at least one way as the sum of two odd primes. Yes — at least. J. J. Sylvester was apparently the first to show that “irregularities” should appear in the function G(E) because, if E is divisible by distinct prime factors p1, p2, p3, …, then G(E) will be increased by the multiplier

over even numbers which have no such prime factors. That multiplier splits G(E) into bands, corresponding to values of E divisible by 3, 5, 7, 3 * 5, 3 * 7, etc. What we call G(E) was analyzed extensively by Hardy and Littleton [1]. Nevertheless, to this day, there is no strict proof from basic axioms of what appears so clearly in the graph shown here — that Goldbach was right.

Described here is a pattern showing a curve that I call J-curve. Increasing the recursion depth by one looks like adding a second J-curve to the first one, rotated at an angle of 90°.

The pattern is created with OLFRAC by Ton Hospel from LISTSERV at BLEKULL11…

Described here is a pattern showing 4 J-curves.

The pattern is created with OLFRAC by Ton Hospel from LISTSERV at BLEKUL11…

Described here is a pattern showing 3 cubes. The three cubes are the pattern 3D-CUBE. The 180 “+“ performs the rotation of 180 * π * 97°/360° = 180 * 180° * 97°/360° = 180 * 48.5° = 8730° = 90° + 24 * 360°. The digits in the axiom select the color: 1 (blue), 2 (red), 4 (green)…

Described here is a pattern showing 95 cubes building words. The 95 cubes look like the pattern 3D-CUBE. Three sides are shown but here you can see the six sides of the cubes. The 180 “+“ performs the rotation of 180 * π * 97°/360° = 180 * 180° * 97°/360° = 180 * 48.5° = 8730° = 90° + 24 * 360°…

A deep reservoir for striking images is the dynamical system. Dynamical systems are models containing rules describing the way some quantity undergoes a change through time. For example, the motion of planets about the sun can be modeled as a dynamical system in which the planets move according to Newton's laws. Generally, the pictures presented in this section track the behavior of mathematical expressions called differential equations. Think of a differential equation as a machine that takes in values for all the variables and then generates the new values at some later time. Just as one can track the path of a jet by the smoke path it leaves behind, computer graphics provides a way to follow paths of particles whose motion is determined by simple differential equations. The practical side of dynamical systems is that they can sometimes be used to describe the behavior of real-world things such as planetary motion, fluid flow, the diffusion of drugs, the behavior of inter-industry relationships, and the vibration of airplane wings. The pseudocode below describes how to produce the Ikeda pattern. Simply plot the position of variables j and k through the iteration. The variables scale, xoff, and yoff simply position and scale the image to fit on the graphics screen. The Ikeda attractor has been described by K. Ikeda (see references)…

Depicted here is a pattern obtained from a non-linear transformation applied to a Penrose tiling. Such tilings are generated using two different tile shapes — a “Kite” and a “Dart” and through imposing certain matching rules on their edges [1]. The properties of Penrose tilings are of wide-ranging interest. They give rise to non-periodic, self-similar patterns with fivefold symmetry which have a host of fascinating mathematical properties [2]. They offer a theoretical model for some newly discovered materials which have been called “Quasi-crystals” [3, 4]. They also offer a new structure on which to base aesthetically pleasing designs [5]…

The pattern depicted here is an example of manipulation of mathematically derived material to produce a design. I used a commercial program (“Doug's Math Aquarium” — for the Amiga) to produce a fractal pattern. This software allows the user to change values in a basic equation. Fortunately, several equations are supplied with the software, which is helpful for innumerates like me. It takes a bit of experimentation to arrive at a result I like and can use. I can then save the file in IFF format and load it into my paint program. I can then manipulate it and use it in any manner. In this case, I printed a 32-color image in black and white…

The patterns depicted here are examples of fractal horticulture, showing life forms grown from user-defined seeds. A type of fractal genetics, perhaps. I used the public domain program called “Fractal Generator” written by Doug Houck. It allows you to draw simple line segments to create a shape. These segments continue to be replaced with small copies of the entire shape following the principles of self-similarity. The Koch Snowflake is the best known example of this type of shape…

Described herein is a pattern showing a projection onto a piece of paper of a four-dimensional magic hypercube of order three. It is merely one out of 58 possible ones of this size, each of which may be shown in 384 different aspects due to rotations and/or reflections. Larger magic hypercubes, both in size and dimension, have also been constructed…

The patterns shown in the figure are snapshots from the time evolution of the reversible Greenberg-Hastings cellular automaton. The original (non-reversible) Greenberg-Hastings cellular automaton was studied by Greenberg and co-workers [1] as a discrete model for reaction-diffusion in excitable media. Each cell can take one of three states: resting or quiescent (0), active (1), and refractory (2). A cell remains in the resting state until it is activated by an active neighbor. Once active, it will become refractory at the next time step independent of the neighborhood. Once refractory, it will become resting at the next time step independent of the neighborhood. The model attempts to model diffusion in an excitable medium composed of discrete chemical oscillators. It produces rotating spirals reminiscent of the Belousov-Zhabotinsky reaction [2]…

The pattern depicted here is an example of manipulation of mathematically derived material to produce a design. I used a commercial program (“Doug's Math Aquarium” — for the Amiga) to produce a fractal pattern. This software allows the user to change values in a basic equation. Fortunately several equations are supplied with the software, which is helpful for innumerates like me. It takes a bit of experimentation to arrive at a result I like and can use. I can then save the file in IFF format and load it into my paint program. I can then manipulate it and use it in any manner. In this case, I printed a 32-color image in black and white…

Described here is a pattern produced on an AMIGA 1000, a 512K computer, and output to a Xerox 4020 color ink jet printer. The software is Doug's Math Aquarium by Seven Seas Software, a program that permits the artist to create images based solely on mathematical equations. The AMIGA graphics dump can simulate in print nearly all of the computer's 4,096 color range, though only 32 can be displayed on screen at one time. However, in the case of the image shown here, because of the black and white limitations, the graphics dump was set to reproduce in a range of 16 levels of gray…

Described here is a pattern produced on an AMIGA 1000, a 512K computer, and output to a Xerox 4020 color ink jet printer. The software is Doug's Math Aquarium by Seven Seas Software, a program that permits the artist to create images based solely on mathematical equations. The AMIGA graphics dump can simulate in print nearly all of the computer's 4,096 color range, though only 32 can be displayed on screen at one time. However, in the case of the image shown here, because of the black and white limitations, the graphics dump was set to reproduce in a range of 16 levels of gray…

Described here is a pattern produced on an AMIGA 1000, a 512K computer, and output to a Xerox 4020 color ink jet printer. The software is Doug's Math Aquarium by Seven Seas Software, a program that permits the artist to create images based solely on mathematical equations. The AMIGA graphics dump can simulate in print nearly all of the computer's 4,096 color range, though only 32 can be displayed on screen at one time. However, in the case of the image shown here, because of the black and white limitations, the graphics dump was set to reproduce in a range of 16 levels of gray…

Described here is a pattern produced on an AMIGA 1000, a 512K computer, and output to a Xerox 4020 color ink jet printer. The software is Doug's Math Aquarium by Seven Seas Software, a program that permits the artist to create images based solely on mathematical equations. The AMIGA graphics dump can simulate in print nearly all of the computer's 4,096 color range, though only 32 can be displayed on screen at one time. However, in the case of the image shown here, because of the black and white limitations, the graphics dump was set to reproduce in a range of 16 levels of gray…

Described here is a pattern generated by a program which draws “locked links”. The links are always regular polygons and they are “locked” together by weaving their edges over and under each other. This is best illustrated in Fig. 1. Input to the program which generates the locked links includes the total number of sides for the polygon in the center of each figure. Even numbered inputs provide figures with two separate intertwined links, each with half the number of sides originally requested. Odd numbered inputs generate a single link with the input number of sides that intertwines with itself, basically forming a knot. Figure 1 contains examples for initial inputs of 3, 4, 5, 6, and 8 sides. Other inputs to the program include the length of a side of the polygon in the center of the figure, thickness of a link, center of the figure, and angle at which to begin drawing. The program is an intense exercise in algebra and trigonometry but is actually quite compact when completed. (My version is about 100 lines of code.)…

Described here is a pattern generated by incrementing variable “s” by 1 from 0 to 750 in the following 2 pairs of parametric equations:

The first pair of equations generates a design with 5 arms spiraling clockwise from the center. The second pair generates the same design but with the 5 arms spiraling counterclockwise from the center. (If you are examining the color figure these are the blue and red spirals respectively.) The result, especially in color, looks like an exponential tunnel with spiral arms descending into infinity. Alternatively, some see the result as a 5-pelated flower. This is actually the more common interpretation when both spirals are the same color. Another interesting point is that the spirals are generated using only straight lines…

Described here is a generalized pattern of (4⁴), the regular tiling by congruent squares. Figure 1 shows a configuration consisting of two points X, Y and lines x_i and y_i (i and j are integers) passing through X and Y respectively, such that the quadrangle determined by x_i, x_i+1, y_j, y_j+1 has an incircle for any i and j [1]. Figure 2 shows a special case where both X and Y are points of infinity, and it also shows that Fig. 1 can be regarded as a generalized pattern of regular tiling (4⁴)…

Described here is a pattern by Fujita Configurations. Let ABCD be a parallelogram; E, a point on the segment CD; F and G, points on the segments CE and DA, respectively, and let FG meet BE at H. Suppose the quadrangles ABHG, BCFH, and DGHE have incircles (see Fig. 1). This figure seems to have been first considered by one of the Wasan mathematicians S. Fujita for the case where ABCD is a square (see Fig. 2) [1]. (Wasan refers to Japanese mathematics developed independently of Western science between the 17th and 19th centuries.) And we call this figure a Fujita configuration…

Described here is a pattern showing one example of the structures contained within the Mandelbrot Set [1]. This area is sometimes ignored by traditional computer graphics methods, which produce images from just outside the Mandelbrot Set [2, 3]…

Shown here is an image derived from complex dynamics. In particular, the figure represents a complex variable recursion produced by iterating z = z³ + μ forty times. μ is a complex constant. Diverging points whose magnitude is between 2 and 3 are plotted. The picture was computed on a micro-VAX and plotted on a Talaris laser printer.

Consider the broken-linear transformation with the parameter

which maps

Figure 1 shows the behavior of the sequences {x_n) and {y_n) produced by this transformation used iteratively with r = 1

…

Algebraic polynomials are functions of the form

where a_k, k = 0, 1,…,m are fixed real or complex numbers. The number m ≥ 1 is the degree of the algebraic polynomials. The fundamental theorem of algebra states that every algebraic equation f(z) = 0 has at least one root. More precisely, if a polynomial vanishes nowhere in the complex plane, then it is identically constant. An algebraic equation of degree m has exactly m roots, counting multiplicities. If the real polynomial f (i.e., the coefficients a_k, k = 0, 1,…, m are real numbers) has a complex zero r = α + iβ, then it has also the complex root r = α + iβ. Here, r denotes the conjugate of r, α and β are real numbers, and . In other words, the complex zeros of a real algebraic polynomial always appear in conjugate pairs. The zeros of an algebraic polynomial with complex coefficients may not occur in conjugate pairs…

Described here are patterns derived from the chessboard, where black and white squares become rectangles, and their uniform colors are replaced by other patterns.

Each mosaic is a repetition, in an M by N matrix, of a mosaic element. This element is generated by a single vector, whose coordinates are increasing. I have used arithmetic, geometric, Fibonacci, and also random generated sequences to control the rate of growth, regularity, etc. The mosaic element has four axes of symmetry. The drawing of each rectangle is repeated four times, except for the middle cross, where the number of repetitions is two — one at the nodes of the cross, and one at the middle of the cross…

Much computer graphics art is created as a result of the illustration of the behavior of the sequence {z_n} through some kind of iterative process. Usually this sequence is generated by the iteration of the form

where z₀ is a given initial value. The iterative function Φ defines the recurrence relation. Often the behavior of the sequence {z_n) is studied as a function of the initial values z₀. The same approach is used here…

Traub [1] presented the following self-accelerating recursion to solve the nonlinear equation f(ze) = 0

where z₀ is given and f_n = f(z_n). This method was proposed for use in the complex plane. Presented here is a pattern showing the behavior of the sequence {z_n} generated by the above iterative process. The sensitivity of {z_n} to the initial values z₀ is displayed. Figure 1 shows the number of executed iterations (modulo 2) by alternating colors. The equation f(z) = z⁴ − 1 = 0 is solved in the complex domain. This equation has four roots, which are 1, −1, i and −i, . C₀ = −1/f′(z₀). The stopping criterion of this self-accelerating algorithm is as follows: IF |z_n+1− z_n| < τ THEN stop, where τ is a constant number (τ = 10⁻⁶). Initial values are between −1.5 and 1.5 in the real and imaginary directions.

Described here is a pattern showing a fractal generated by recursive construction of Voronoi diagrams. This technique can be used to generate patterns resembling roadmaps, leaf veins, butterfly wings, or abstract patterns.

A Voronoi fractal is constructed by first drawing the Voronoi diagram of a set of points. (A Voronoi diagram [1] of a set of points divides the plane into regions; each region is closer to one point than any other.) Then, using a larger set of points, a smaller Voronoi diagram is drawn inside each of the original regions. This process continues, recursively subdividing each region by drawing smaller Voronoi diagrams. Voronoi fractals are discussed in more detail in [2]…

Presented here is a pattern obtained by repeating a simple mathematical operation, such as

where x₀ is a given initial value and r is a parameter between 0 and 4. The behavior of the sequence {x_k} is very sensitive to the value of the parameter r (see [1]). The obtained sequence may be attracted (converge) to one point, a group of points, or an infinite number of points composed by a fractal structure called a strange attractor…

Presented here is a pattern resulting from the iteration, or repeated application, of a simple mathematical operation. In particular, the pattern shows the behavior of the sequences {x_n} and {y_n} generated by the following formulae

…

Presented here is a pattern showing the behavior of the sequence {z_n} as a function of the variable c in the following nonlinear equation:

This sequence is generated by Newton's method which is used to localize a root of the above equation. The function f is considered in the complex plane…

Iterative processes are used very often to create computer graphics arts. The classic example is the iteration realized in the complex plane

where z₀ = 0 and c is varied over the complex plane. This iteration is the basis for the construction of the Mandelbrot set [1, 3]. In the above iterative process, the number of executions is controlled by the value of the norm z, i.e., where z = x + iy. These numbers are often displayed as color patterns. Another approach is to realize the iterative process for a fixed number of times and to show the obtained value for the sequence…

“The Starry Night” is a “mathematical Van Gogh”, giving us a glimpse of a nebular cluster which has been generated by mathematical, as opposed to natural, forces. The study of this pattern falls under the iteration theory of Fatou and Julia, which examines discrete dynamical systems on the Riemann Sphere Ĉ with a dynamic given by a non-linear holomorphic or merornorphic mapping from Ω ⊂ Ĉ to Ĉ (see [1–3])…

The two patterns in “From Asymmetry to Symmetry” are a study in contrast. Figure 1 is a window into an asymmetric world in which one has no sense of orientation, whereas Fig. 2 is an extremely complex pattern exhibiting symmetry with respect to both the x and y axes. The study of these patterns falls under the iteration theory of Fatou and Julia, which examines discrete dynamical systems on the Riemann Sphere Ĉ with a dynamic given by a non-linear holomorphic or meromofphic mapping from Ω ⊂ Ĉ to Ĉ (see [1–4])…

The pattern described here was generated by an efficient recursive algorithm developed by the authors which produces approximations of self-similar fractal sets (see [3]). Such sets are constructed by a repeated scaling, translation, reflection, and/or rotation of a fixed pattern or set of patterns. The procedure is a “pattern rewriting system” in which a given geometric pattern is drawn repeatedly after suitable mappings. The pattern used to generate the fractal set by the rewriting system will be called a seed. The base is the initial configuration. This particular pattern was produced by using the “monkey's tree” as a seed with the Gosper curve as a base. The procedure was iterated 4 times. Using their algorithm, the authors duplicated the self-similar fractal sets in Mandelbrot [4], which constitute approximately 45% of the graphics plates in the book…

The pattern described here was generated by an efficient recursive algorithm developed by the authors which produces approximations of self-similar fractal sets (see [3]). Such sets are constructed by a repeated scaling, translation, reflection, and/or rotation of a fixed pattern or set of patterns. The procedure is a “pattern rewriting system” in which a given geometric pattern is drawn repeatedly after suitable mappings. The pattern used to generate the fractal set by the rewriting system will be called a seed. The base is the initial configuration. The seed for this particular pattern consisted of three components. The base was a square drawn counter-clockwise. The procedure was iterated 5 times to produce the figure. Using their algorithm, the authors duplicated the self-similar fractal patterns in Mandelbrot's The Fractal Geometry of Nature [4], which constitute approximately 45% of the graphics plates in the book. This particular pattern is an original configuration…

The pattern described here was generated by an efficient recursive algorithm developed jointly by the author and N. F. Reingold. The algorithm is explained in [1]…

Described here is a pattern which grows fonts or words or fractal shapes by random walk techniques as described in the next TOPSY-TURVY pattern. This pattern however shows that the seeded points do not have to be regular geometric figures. To generate this pattern, the word “fractals“ was written using a drawing package. For effect, the word was also underscored with a wiggly line. The word “fractals” and its underscore were then used as seeded points. Each random walk was started by choosing a column (0, 1,…, 639) and a row (0, 1,…, 479) at random. The search would then begin in any one of the four major directions, the distance of one pixel at a time, looking for seeded points. When a seeded point was found the searching pixel stuck to it and another starting point was generated and the search repeated. To prevent overcrowding as more and more pixels aggregated on the screen, the program only allowed a starting point to occur if it was more than 9 pixels away from any other pixel on the screen. If overcrowding was detected, a new potential starting was generated. Using this overcrowding protection prevents the random starting points from occurring too close to the points which are already aggregated. This helps to maintain the fractal quality of the pattern. Without the overcrowding DENSITY function in the program, the growth could have lumpy and solid spots. Seeded points can be made with different colors and an ABSORB color function can cause searching pixels to take on the color of the seeded pixel. Thus, clusters of different colors may be grown on the screen. With seeded pixels broadly outlining anything from fonts to countries, interesting patterns emerge…

Described here is a pattern which was grown using a program developed by the author. The program generates fractal shapes by random walks of “sticky“ diffusing particles as described by Pickover [2].

In this Topsy-Turvy growth, a rectangle is drawn along the perimeter of the screen using conventional graphics and constitutes the seeded points. The starting point for the search starts anywhere on the screen as if it were dropped from the third dimension. To prevent overcrowding, a DENSITY function is used which rejects a starting point if it is within a certain distance, e.g., 9 pixels, of any pixel already on the screen…

Clifford Pickover has experimented with symmetrized dot-patterns, which resemble snowflakes [1, 2]. Generally speaking, symmetrized dot-patterns are constructed by placing randomly positioned dots on a plane and then mirror-reflecting the dots to produce a pattern with sixfold symmetry. One difference between these symmetrized dot-patterns and actual snowflakes is that the latter are connected while the former consist of discreet dots. I have tried two algorithms for expanding symmetrized dot-patterns so that they become connected. The resulting figures are often pleasing to the eye and have some resemblance to actual snowflakes…

The patterns Figs. 1 and 2 are respectively the Mandelbrot and modified Julia sets for the function z − z^z + c.

For iteration of polynomials zⁿ + c where n is an integer, the “Mandelbrot set” (M) consists of those complex numbers z such that the sequence z = 0, z₁ = z, z₂ = z², z₃ = (z² + z)²,…, never satisfies |z| > 2. A number of algorithms for carrying out this iterative process have been widely used to generate the M set mappings of many functions which were not polynomials [1]. Complex functions which are analytic and continuous would be expected to generate Julia sets on iteration. Those mapped from z^z + c are rather unattractive. The M set for a function can also be defined as the collection of points for which the Julia set of f_c is connected. Julia sets derived from points c close to the origin would then be expected to have outlines close to the shape of the lemniscate controlled by the escape test. In the case of z^z + c, very few points in the plane are bounded for values of c close to 0 + 0i. The Julia sets derived from such points are therefore not of interest. A reflection perhaps of the less attractive properties of the M set. Alternative escape tests [2] do however yield mappings from the iteration, which are visually exiting [3] such as Fig. 2. Figure 1 was first obtained by the standard Level Set Method [1] applied to z^z + c using lines to plot the divergent set points only where adjacent points differ in divergence rate. Selection of the point c (0.09 + 0i) allowed the speed of mapping to be halved as the pattern is symmetrical…

The two patterns Figs. 1 and 2 result from studying the Julia set mappings of the dynamic f(z) = z^z + z⁴ − z + c and identifying areas of the maps whic have disconnected fractal geometry resembling biological forms…

The patterns illustrate a Julia set derived by iteration of the function z → z⁵ + c in the complex plane using a non-standard escape test and additional tests utilizing the values of z and |z| at escape to control the point plotting.

Mappings in the complex plane derived from iteration of a wide range of algebraic functions have been explored by computer graphical methods in order to study their fascinating and often beautiful fractal geometry [1]. Of special interest is the way in which the iterative mathematics produce “chaos” patterns close to the divergent and bounded point boundaries. For higher polynomial functions such as z → z⁵ + c the mappings are not visually very appealing since the area of chaos is small and the “quasi” circle outlines which illustrate the different divergence rates are close to the bounded set outline. The use of alternate divergence tests, especially those that make direct use of the values of z or |2|, have been investigated [2]. They frequently enhance the mappings of particular functions…

The circular pattern in Fig. 1 illustrates a mapping of the polynomial f_c(z) = z⁴ − z + c in the complex z plane for which the divergent points in the z plane iteration of z + c were identified by the iteration value for which |z| < 2. This definition of the escape radius implemented by Mandelbrot [1] determines that the curve for iteration value k = 1 is a circle. Alternative tests for divergence of points in similar mappings have been used [2]. All the points in the bounded set were iterated to a limit of 150…

The pattern is an alternative mapping of a Julia set for the function z → z⁴ − z + c which utilizes an escape test |z_real| or |z_imag.| <sqrt 20 to determine the divergent points in the complex plane.

Julia sets for the dynamic f(z) = z² + c have been widely studied and many beautiful images resulting from a boundedness test |z_n| > 2 have been published [1]. Fewer studies have been made of the patterns which emerge from iteration of other dynamic functions when alternative tests or additional tests are applied. Of particular interest are mappings of transcendental functions [2] and also studies with alternative tests on iteration of higher polynomials and mixed algebraic functions [3]…

The following sections are included:

• The Figures

• Commentary

• Acknowledgments

The following sections are included:

• The Figures

• Commentary

• Acknowledgments

• Reference

The following sections are included:

• Figure 1

• Figure 2

• Commentary

• Acknowledgments

Described here is a pattern showing an approximate copy of a famous M. C. Escher drawing of lizards [1], using the pattern and tile-editing software Mosedit®. The program itself was jointly created by N. Chourot and J. Baracs of Montreal. Although the drawing could be constructed by hand as Escher did, its creation using the computer serves as a litmus test of the flexibility and effortlessness of the graphical interface of Mosedit®…

Described here are flower-like patterns derived from orthogonal projection into 3-space of typical 4-dimensional regular polytopes (120- and 600-cell) and semiregular polytopes which are derived from regular truncation around vertices of the 120- and 600-cell polytopes…

Described here is a pattern showing a sequence of period doubling bifurcations toward chaotic motion originating from a stable steady state destabilized by diffusion…

Described here is an open-ended class of mathematically-generated line drawings based on Lissajous figures, but overlaid and sometimes entirely disguised with a variety of embellishments. Lissajous figures constitute a family of curves well known to scientists and engineers…

Unlike the arts of other cultures, Islamic art sets out deliberately to shun anthropomorphic forms and concepts. It was led to concentrate on the exploration of symmetry in geometrical patterns, an enterprise which resulted in an extraordinarily large, complex and elegant collection of patterns [1, 2, 3, 4]. Apart from their aesthetic merit, these patterns offer a treasure house of symmetry which make them of great interest to a large number of scientists and educators [5, 6]…

The following is a means of drawing computer-generated swirling tendrils. The technique uses the following to generate a series of x, y points given any initial point x₀, y₀ and four constants a₁₁, a₁₂, a₂₁, a₂₂…

Described here is a pattern showing a combination of many circles. There are fourteen rows of circles which emanate from the center like petals in a flower. The twenty circles in each row are different in radius, line thickness, and position. This pattern was produced by a program, Circlegraph, written by the author and printed on an HP laserjet at 300 dots per inch…

Described here is a pattern showing a computer simulation of a flowing fluid. Thirteen points were selected, then the computer began to plot lines from left to right. The lines tend toward the closest point which was still to the right. Once that point is passed, the lines tend toward the next point. Moire patterns arise because the lines, which are close together, have ragged edges. This pattern was produced from a program, Flow, written by the author and printed on an HP laserjet at 300 dots per inch…

The pattern described here is a biomorph image of the type first described by Pickover [1, 2, 3]. The pattern arises from the iteration of a very simple expression, namely, z → z³ + μ, where μ and z are complex numbers. The image was rendered with the aid of a Julia set algorithm and the special convergence criterion required to reveal biomorphic forms [1]. According to this criterion, z is taken to be convergent and a point is plotted if either the real or the imaginary part of z is small after many iterations…

Described here is a pattern formed by transient values generated when the iterative logistic formula NEWX = K * OLDX * (1-OLDX) is repeatedly mapped over certain very small K intervals. The K interval for Transient Microstructure was 3.69925–3.7019. This interval is within the domain of the chaotic attractors, but I have found no clear relationship between chaos and the microstructure. The pattern varies with the value initially assigned to OLDX; here, it was 0.3…

The patterns described here show how Engel curves are generated and some interesting computer-generated Engel curve circuits. These curves were published in the February 1983 American Mathematical Monthly. In Fig. 1, a curve is generated by first creating a grid of m rows of n points, m and n coprime and only one of m and n even. Figure 1 has a 2 × 3 rectangular grid of 6 points, the grid then being divided into 3 pairs of points, each pair connected by a single line to form a net of valence 1. This pattern is then repeated 6 times in 3 × 2 array. The same array is then replicated, turned 90 degrees and superposed over itself to form a net of valence 2 composed of 1 or more closed curves. Patterns that form a hamilton cycle, a single closed curve using every point of the net like this one are not very common as the m × n pattern increases. Several investigators showed that only 42 of them exist for 3 × 4 nets, none symmetrical. In what follows, the m × n net is called the efactor and the final pattern the eproduct…

The following sections are included:

• Pseudo Code for Modified Level Set Method

• Appendix

• References

The two figures illustrate patterns which are obtained when f(z):−sinh(z)+ is iterated in the z plane. They differ markedly from the full mapping of points which diverge and of those which belong to the bounded set (the Mandelbrot set) for the polynomial z² + c derived patterns. Such patterns have been intensively studied and frequently generated [1]. The similar mappings for the transcendental function sinh(z) + c has received little attention. More recently, the strange behavior of mappings of the transcendental hyperbolic cosine function has been studied in both the z and c complex planes [2]. The dearth of reports on the z plane iterative mappings of the hyperbolic sine function prompted a study of this mapping which produces the two patterns, see Figs. 1 and 2. The Taylor expansion of sinh(z) contains only terms of uneven powers of z and so under recursion it was thought that the bounded set would not map to an outline shape similar to the topology of the Mandelbrot set for z² + c even if an approximate expansion was used…

The illustrated patterns result from the iteration of f(z) :→ cosh(z) + c for complex z and c planes. Previous studies of this function at iteration have led to important observations [1] about the morphology and behavior of the mapping. The bounded set is a single unconnected cardoid when the convergence test |z| < 2 is applied. For higher values of the escape radius this set is progressively distorted until it only resembles the z² + c bounded set for points in the plane < (−2, 0). The shape of the leminiscates are then no longer circle-like and the mapping becomes periodic (2πi). The more complex dynamics of the cosh(z) + c iteration are therefore only realized by mapping with a large escape radius. Of particular interest in studies of this mapping is the possibility that the main cardiod centered at −0.14, 0i is connected to all the points in the bounded set as has been established [2] mathematically for the polynomials zⁿ + c maps. In these mappings the morphology of the central cardoid is retained by the island miniatures, thus giving a high degree of self similarity to the fractal geometry of the maps. Greater variations in the patterns mapped to divergent points and the geometrical shapes of the islands which form the bounded set are observed for the function cosh(z) + c…

The patterns illustrate alternative behavior of the mapping fc(z) = z² + c in the complex plane. Mandelbrot's study [1] of this function stimulated the publication of numerous mappings showing their beautiful fractal patterns [2]. Popularising the mapping algorithm [3] has resulted in a plethora of studies of the iterative properties of other mathematical functions [4]. Alternative tests for divergence have been used to produce many new pattern variations [5]…

The following sections are included:

• Pseudo Code: Modified Level Set Method

• Appendix

• References

The pattern illustrates an elaborated Julia set fractal obtained from iteration of the function z⁷ + c in the complex plane.

Iterative complex plane maps of the function z² + c have been widely explored by graphical methods because of the fascinating fractals that can be generated [1]. The striking beauty of these maps and the relationship of their geometry to the corresponding z plane mappings has prompted the study of the similar behavior of higher polynomial functions. On iteration, such functions generate z(n) values approaching infinity within a few iterations (n) and very few of the points in the complex plane are part of the “bounded set” if the value of c is in the main cardoid of the corresponding Mandelbrot set…

Described here are patterns showing branching structures generated by several iteration schemes. Figure 1 shows a magnification of the Mandelbrot set; the picture is centered at c₀ = −1.74995643 and is in the main cardioid cusp of the 3-cycle midget. The height of the picture is 1.4 × 10⁻⁷. Figure 2 shows a magnification of the Julia set for z² + c₀. This picture is centered at z = 0 and has height 0.282. Both figures were obtained by iterating f(z) = z² + c for the Mandelbrot set, scanning across c values and starting each iteration with z = 0, while for the Julia set c is fixed at c₀ and the scan is across the z plane. Similarities between magnifications of the Mandelbrot set and the corresponding Julia sets are quite common. Indeed, a theorem of Tan Lei [2] guarantees that at any Misiurewicz point c, the Mandelbrot set magnified around c converges to an appropriately rotated and magnified portion of the Julia set of z² + c. Some of the power of Tan Lei's theorem lies in the fact that Misiurewicz points lie arbitrarily close to every point of the boundary of the Mandelbrot set [2]. Note, however, that c₀, is not a Misiurewicz point…

Shown here are two images showing fractal features. In other words, the patterns are quite intricate and show self-similar edges. I call these patterns Nevada sets. You may write to me for more details on how I created these pictures.

Described here is a pattern showing a “handmade” quilt. The very idea of computer-generated handmade patterns seems an oxymoron! The precise regularity characteristic of a machine would seem to work against the artistic quality of the product and deprive it of whatever it is that makes handmade patterns special…

Pictured here are different magnifications for a Julia-like set for the iteration z → z^1.5 − 0.2. Depending on one's point of view, Fig. 1 resembles either a tree with two infinite spiraling branches, or biomorphic cells about to escape through an orifice. Figure 2 shows the tip of the right spiraling branch…

Described here are two views of a Julia-like set computed using the complex sine function. Figure 1 shows a broad view with many “storms” Figure 2 is an enlargement of one of the “hurricanes”. Note that this fractal hurricane even has miniature tornadoes along many of its arms…

Pictured here are two views of a Julia-like set computed using the complex sine function.

Figure 1 shows “arms” from the top and bottom reaching out for each other but never touching. Figure 2 shows an enlargement of the tips of the arms. While the arms from the sin(z) Julia-like sets are “shy”, the arms from the cos(z) Julia-like sets are very direct and unite quickly as the Julia set parameter is varied in a similar way.

Pictured here are two different views of the Mandelbrot iteration z → z⁻² + c. The initial condition for this iteration was changed from the usual z₀ = 0 to z₁ = c to avoid division by zero. This provided an initial condition equivalent to the usual Mandelbrot set iteration, z → z² + c…

Pictured here are two different views of the Julia set iteration z → z⁻² + c (related to the previous article, Mandelbrot Iteration z → z⁻² + c, Fig. 2). Both of the parameters c for the Julia set calculations shown here were chosen from along the real axis in the corresponding Mandelbrot set. The “movie” of the transformation of Fig. 1 into Fig. 2 is quite interesting.

Shown here are two views of Julia sets formed from the J₀ Bessel function. Figure 1 shows “fireworks”, while Fig. 2 shows an interesting fractal “parallelogram”.

Pictured here is a view of the mandelbrot set,

where, is the complex conjugate of z…

The pattern illustrates the variability of shape among isogonal polygons, that is, polygons whose vertices are all equivalent under symmetries of the polygon. Shown is the case of decagons, which is representative of a great wealth of forms, but still fits reasonably on a single page…

Described here is “analytic computer art” consisting of geometrical designs based on explicit mathematical functions. A common motif in analytical computer art is the polar coordinate curve. This has the form R = f(A) where R is the radius, f is a mathematical function, and A is the angle. The angle parameter A is swept through some range of values, the radius R is calculated, and the computer polar coordinate points (R, A) are converted to rectangular coordinates and plotted. The resulting curves frequently (but not always) exhibit angular symmetry, that is, they look the same after being rotated through suitable angles. What is special here is that simple polar coordinate curves are swept through the interval 0 to 360 degrees and incremented by a fixed amount between sweeps. The pictures were generated on a Tektronix 4052 intelligent terminal.

Euler's Triangle is, essentially, a graph of Euler's Phi-function, which for a positive integer N is defined as the number of positive integers less than N which are relatively prime to N. For N = 2, 3, 4, …, the successive rows of the triangle display a circle where N and each of the numbers 1, 2, 3, …, N − 1 are relatively prime; no circle where the two numbers share a common factor greater than 1. Thus, the circle in the top row of Fig. 1 stands for (2, 1), the two in the second row for (3, 1), (3, 2), and so on. When N is prime, there is a solid row of N − 1 circles…

These patterns were created by a computer program called Carpet, which is based on one of the three basic concepts described in the article, “Wallpaper for the mind”, in the September 1986 Scientific American. The program selects representatives in a uniformly random way from a domain of hundreds of millions of patterns. The following is a slightly simplified description of the defining process for the domain…

This pattern was created by a computer program called Carpet, which is based on one of the three basic concepts described in the article, “Wallpaper for the mind”, in the September 1986 Scientzfic American. The program selects representatives in a uniformly random way from a domain of hundreds of millions of patterns. The following is a slightly simplified description of the defining process for the domain…

This pattern was created by a computer program called Carpet, which is based on one of the three basic concepts described in the article, “Wallpaper for the mind”, in the September 1986 Scientific American. The program selects representatives in a uniformly random way from a domain of hundreds of millions of patterns. The following is a slightly simplified description of the defining process for the domain…

This pattern was created by a computer program called Carpet, which is based on one of the three basic concepts described in the article, “Wallpaper for the mind”, in the September 1986 Scientific American. The program selects representatives in a uniformly random way from a domain of hundreds of millions of patterns. The following is a slightly simplified description of the defining process for the domain…

TESS is a program which generates repeating surface pattern designs spontaneously. The program first selects one of eighteen schemes, or sets of rules, for creating a small square pattern in the upper left corner of the computer screen. Then it displays the pattern and propagates it in both directions so as to fill the screen. As it displays or prints the rows, the program may sometimes indulge in some row transfomations, such as reversing and sliding, so as to add some more variety…

Skew Squares is a product of a program called TESS which creates huge domains of surface pattern designs by applying rules defining eighteen schemes, or classes, of designs to create a base pattern, then replicating that pattern to fill the computer screen. The parameters for this particular pattern are now lost in antiquity, since the file from which it was printed has been erased, and TESS was deliberately designed to sample a different part of its pattern domain each time it runs. The pattern was probably based on either a scheme called “Stripes” or one called “Circ/Sq”, producing an off-center square, then subjected to a transformation that reproduced the base square into 2 × 2 array while rotating it.

The patterns introduced here comprise a family generated by an extension of methods used in creating the well-known Sierpinski triangle or gasket (Fig. 1). These deterministic fractals can be created either by a standard recursive method [1] or by an adaptation of Barnsley's chaos game [2], which produces a randomly generated sequence of points having the required fractal shape as attractor…

The pattern illustrated is an octahedral fractal object generated in a way analogous to the technique used to produce a Sierpinski tetrahedron (Fig. 1). The tetrahedron is the simplest form of polyhedron, having four triangular faces and four vertices. In its regular form, it is one of the five Platonic solids, these being the only absolutely regular tetrahedral solids. To construct a Sierpinski tetrahedron, the original shape is replaced by four similar half linear scale tetrahedra, at the same orientation as the original, each being placed within the original tetrahedron adjacent to a vertex. This procedure is recursively continued. If repeated indefinitely, this produces an exactly self-replicating fractal object with fractal dimension [1]

as the object contains four copies of itself at half scale. This is an example of a fractal object which has an integer dimension, although the dimension is lower than that of the least dimensioned Cartesian space within which the object can exist. At each stage of recursion, one half of the existing volume of the object is eliminated. This generates a geometric progression which sums to unity, showing that the object is eventually reduced to zero volume. In producing images of the Sierpinski tetrahedron, recursion is stopped after a set number of stages, giving an approximate depiction, as in Fig. 1…

Described here is a pattern showing a three-dimensional packing of rhombicosi-dodecahedra, viewed along its fivefold axis of symmetry. The polyhedra are packed in such a way that the center of each polyhedron lies at the vertex of a larger rhombicosi-dodecahedron. The model is only one of a family of sixty-four packings [1], where each of seven fivefold polyhedra (along with a point) are placed at the vertices of seven fivefold polyhedra (and a point) so that the polyhedra meet each other at a face, an edge, or a vertex. In the example, each polyhedron shares a square with its immediate neighbor…

This image is based on the well-known Penrose tiling which uses two rhombii [1], a “thin” rhombus with an acute angle of 36° and a “thick” rhombus with an acute angle of 72°. Penrose's “matching rules” for tiling the plane nonperiodically, obtained by marking the two rhombii in a special way, are abstracted into portions of circles at appropriate vertices using a notation by De Bruijn [2]. Nonperiodicity is ensured by completing the circle at the vertex of the tiling. In the figure, the Penrose tiling can be identified by the encircled vertices…

The three images in Figs. 1–3 are related in a sequence. The broad study is based on the concept “Structures on Hyper-structures” [1, 2], where n-dimensional cubes provide an organizing lattice or framework for classifying and generating form. The framework has a Boolean structure. Though typical, the examples are taken from space structures of one kind or another, this conceptual modeling technique has broader implications that extend into other fields. As an example in the study of space structures, this technique is used for subdividing the surface of a polyhedron, say, a cube. From n basic subdivisions of the surface, 2ⁿ subdivisions are obtained from all combinations of n, and these can be arranged on the vertices of an n-cube which maps all the structures and all the relationships (transformations) between them…

Described here is a pattern showing the spatial-temporal configuration generated by applying a next-nearest-neighbor two-state cellular automaton rule to a random initial configuration. Cellular automata are discrete dynamical systems which update configurations by local coupling [1]. For example, the next-nearest-neighbor two-state cellular automata are rules of the following form [2]:

where is the variable value at site i and time t, and f() is the function that represents the rule…

The pattern described here illustrates binary matrix symmetry. Black signifies +1 and white −1, while gray indicates that a cell may be either +1 or −1. The upper pattern contains all of the possible 3 × 3 matrices of this type. The lower pattern illustrates the symmetry groups, S-groups, for the upper pattern. The members of an S-group are equivalent under kπ/2 rotation: τ, reflection: ρ, and complement: χ. Our interest in this problem developed when we discovered that there are fifty-one unique TriHadamards as described in the illustration entitled TriHadamards. Since then, we have been looking for an analytic solution to the number of N-Hadamards…

This pattern is based on matrices derived from the work of the French mathematician, Jacques Hadamard. He defined a class of orthogonal two-valued matrices with unusual symmetry properties [1]. These matrices are now known as the Hadamard matrices and are commonly used in optics, acoustics, signal processing, image analysis, and pattern recognition [2]. The matrices have many properties in common with the Fourier Series. Hence, the coefficients of the Hadamard matrix are called the sequencies, corresponding to the Fourier frequencies. A transformation, known as the Hadamard transform, is homomorphically invertable, such that H(H(I)) = 1. This property, which the Fourier transform also shares, has made the Hadarnard transformation popular for data compression, and many high-speed algorithms have been developed to transform signals into the Hadamard domain…

Described here is a pattern showing a simple Julia set seeded near zero with the interior features revealed as a level set. It demonstrates a haunting similarity to the Circle Limit series of works by the Dutch artist, M. C. Escher [1, 2]…

Described here is a pattern showing a Julia set with interior features revealed that demonstrate a striking similarity to the Circle Limit series by the Dutch artist, M. C. Escher. The set is projected in a vanishing point perspective to achieve both the interesting effect of hovering over a fractal landscape and a new way of looking at fractals. The detail in the foreground combined with a broader view in the distance gives the impression of an in situ zoom, as the eye tracks the shrinking horizon from top to bottom of the page…

The jungle canopy shown here (Fig. 1) lies within the Mandelbrot set which is generated by the iteration of z = z * z + c (to a maximum of 50 iterations with an escape radius of 2) where z = x + yi [1−3]. The jungle canopy image is located between x = −0.3984473 and −0.3742513, and between y = 0.1272181 and 0.1452246 in the complex plane. This gives the sky (white) and the foliage (light gray)…

Described here is a pattern showing a tiling of a 23 × 24 rectangle by the “Y” hexomino.

Hexominoes are formed by joining six equal squares edge to edge, and 35 different shapes are possible [1]. According to Gardner [2], the shape used here is the only hexomino for which it is undetermined whether it can tile a rectangle. The figure shown was obtained by exhaustive computer search which tested all relevant rectangles which would fit within a 23 × 29 area and also a few larger. No other solution was found…

Figure 1 illustrates some of the variety of functions that may be created with fractal interpolation. In both diagrams, a family of curves was created, graphed and reflected about the x axis. To create each curve, a set of interpolation points is chosen, say, {(x_i, y_i) : i = 0, 1,…, M} and a subset of these points is chosen as address points, say, …

Described here is a pattern showing the result of three iterations of the logistic difference equation, x_n + 1 = r * x_n * (1 − x_n). The logistic difference equation is the canonical example of period-doubling and chaos in a discrete system, and is described widely in the technical and popular literature. See, for example, Peterson [1] or Reitman [2]…

Described here is a pattern showing the result of six levels of recursion based on the “inflation rules” of Penrose tiles.

In 1977, British physicist and cosmologist Roger Penrose discovered a pair of quadrilaterals that, when joined according to certain rules, force a nonperiodic tiling of a plane. “Inflation rules” guarantee that the tiling will cover an infinite plane, because each of the tiles can be decomposed into smaller copies of the two. The two tiles have come to be known as “kites” and “darts”, and have been discussed widely in the popular literature. See, for example, Gardner [1] and Peterson [2]…

The starting point of the drawing shown herewith is a fourth-order magic square which Dürer judged worthy to present in his famous engraving “Die Melancholie”, among other objects of geometrical significance. The magic square, which fascinated so many minds for so long, is a grid square filled in apparent disorder, with consecutive whole numbers. A more thorough examination reveals the repetition of the same total in the main directions of the gird. (Horizontals, verticals, and diagonals.) The square chosen by Dürer has another particularity as well: the two middle consecutive numbers of the fourth line 15 14, correspond to the year this masterpiece was created…

Described here is a computer-generated image inspired by an article by A. K. Dewdney. The article appeared in [1]. It included a number of images called biomorphs. One in particular, Dewdney called a radiolarian. I was interested in obtaining a detailed view of the center of the radiolarian. Figure 1 is the result, using the program below with an EGA-EPSON screen dump. Dewdney provided a program to generate biomorphs which “follows Pickover's basic algorithm”. The program is suggestive and not in a ready-terun form. The program below runs in QUICKBASIC where screen 9 may be used for better resolution. The purpose of the aspect ratio, 5/6, at the end of line 1000 is to make the image come out square on the printer. The program will also run in BASIC using screen 2, using 100 in place of 175 in line 1000, and using the appropriate aspect ratio…

Described here is a pattern showing smooth frontiers invading on a fractal support into a basin of attraction. The formerly connected basin is split into a Feigenbaum sequence of 2, 4, 8, 16, … parts. To obtain this pattern, we started with an iterative mapping very similar to the complex logistic map:

x_n and y_n denote the variables (with index n = 0, 1, 2, …), while a, b, and c represent constants. Due to the term ax_n, Eq. (1) is not analytic any more. On the basis of this map, basins of attraction (filled-in Julia-like sets) and Mandelbrot-like sets can be calculated, reflecting self-similarity and fractal structures [1, 2]…

Described here is a pattern showing graphical entities, which undergo mitosis (cell division), as a parameter in a mathematical feedback loop is varied. The term “Biomorph” is a name given to a computer graphic representation which resembles a biological morphology, hence, the name “Biomorph” [1]. As the term implies, biomorphs look like simple life forms, and the natural extension of the biomorph would obviously be their reproduction and cell division…

The pattern in Fig. 1 demonstrates the nonrandom distribution of primes in the first 100,000 numbers when they are plotted as a triangular array. This is similar to Fig. 2, the square spiral display of primes in the first 65,000 numbers that Ulam [1] published in 1964…

Described here is a class of patterns showing interesting tessellations of the plane. The constituent micropatterns whose repetitions in space produce the tiling patterns were introduced in [1]. These micropatterns are shown separately at the extreme right in the figure. The three basic regular polygons that tessellate the plane are used: triangle (top), square (middle), and hexagon (bottom). These micropatterns all share the following common property, which will be illustrated for the isolated triangular shape, shown at the top right of the figure. This image can be perceived in two stable states: either as a solid black three-pronged object with three diverging arrowheads, seen against a white background, or as three white arrowheads converging towards the center of a solid black triangle that lies beneath. Similar bistable percepts exist for the single square and hexagonal images on the right of the figure…

Described here are two patterns, formed by white and black lines of either horizontal or vertical orientation against a gray background. It is remarkable that these images unexpectedly give rise to vivid three-dimensional (3D) percepts, as if they are renderings of anaglyph terrains. Both images are members of a class of stimuli conceived by the authors [1], for studying the interaction of visual attributes (color, luminance, orientation, spatial frequency, etc.) in motion and texture perception. The two attributes employed in this figure are orientation (horizontal and vertical lines) and luminance polarity, or simpler, polarity (black and white lines) on a gray background). These patterns were generated on a computer by the algorithm described in [1]; the algorithm can produce much more complex patterns, depending on the goal of the visual experiment, but its details need not concern us here. The top figure shows a pattern in which only two types of lines exist: white horizontal (WH) and black vertical (BV). These lines are joined in such a way so as to alternate and to form staircases. Unexpectedly, however, they give rise to a strong 3D percept; some observers describe the image as thin cardboard slabs, cut along staircase contours, stacked one behind the other; others perceive the scene as rows of seats, much like in an ancient stadium. In the bottom figure, which also appears in [3] in a different context, we have all four possible combinations of lines: WH, BV (as before), plus black horizontal (BH), and white vertical (WV). These lines form a regular orthogonal grid pattern which, quite unexpectedly, produces a strong depth percept, in which two types of squares can be seen, in a checkerboard format, i.e., half of the squares seem to pop out (in depth) in front of the rest…

Described here are patterns showing mosaics constructed from higher-order spiral patterns found at high magnifications within the Mandelbrot Set. They were conceived as possible selections for the endpapers of a book about the M-Set…

These pictures were made with Michael Freeman's (Vancouver, BC, Canada) Mandelbrot program, V63MBROT, which was written specifically for an IBM PC system with an ATI VGA Wonder video card. Four of the picture sections (1a, 1c, 1d and 3a) are pictures in the Mandelbrot Set; the remaining sections are Julia Sets. The parameters for each section are given in Table 1. The figures are calculated at a resolution of 800 × 600 pixels, with dwell values set in the range of 10000 to 44444. There are color editing routines in Freeman's program so one can edit the pictures to enhance the contrast when the pictures are printed in black and white on the HP II laser printer. My usual approach is to make the spiral arms black and the background, between the spiral arms, a very light color…

Described here are patterns showing tilings of the circle as a model of the hyperbolic plane. Some of M. C. Escher's famous wood cuts use similar tilings but beautiful shapes instead of our chessboard pattern. Figure 1 first appeared in 1897 in [1]…

Here are patterns produced by color contouring the height of functions that are the products of three harmonics; one harmonic in the x, y, and radial directions each. In particular, Fig. 1 is colored by the height of the function in the domain: −32 ≤ x ≤ 32, −24 ≤ y ≤ 24. The colors are chosen to vary through blue, light blue, light cyan, green, yellow, and white as the function height varies from −1 through 1. Figure 2 is a color contour plot for the function on the same domain with the same coloration scheme. Notice the vertical, horizontal and circular 0-level sets in both figures.

Described here are patterns that evolve from a single “defect” in bounded, one-dimensional, cellular automata in which the number of cells in the row is equal to the number of possible states, n.

Each cell's neighborhood consists of two adjacent cells, but the cell at each end of the row lacks one neighbor. The state of each cell in generation g is coded by an integer from 0 to (n − 1)…

Described here are art prints which are created by using the representation of the Mandelbrot Set in the 1/c-plane. Using different drawing programmes written for ATARI-ST computers, the prints are various combinations of the original set…

Described here are art prints which are created by using the representation of Barry Martin's sets of iteration of real numbers…

Described here is an ornament which was created by playing around with lines in a vector-oriented drawing programme. After various rotations, the vector-object was turned into a pixel-object, shaded differently and arranged to produce the final pattern.

Procedure GCONTOUR [1] produces contour plots, in which values of three variables x, y, z are represented in two dimensions. One of the variables is a contour variable. Presented here is a pattern showing the values

at the point (A, B). z is the contour variable. The result of the function mod(argument1, argument2) is the remainder when the quotient of argument1 divided by argument2 is calculated…

Presented here is a pattern showing the values

…

A surface can be described by a single relationship satisfied by the Cartesian coordinates of its points

For example, the sphere with radius R has the following equation:

…

Presented here is a pattern representing the partition of the rectangle by using curves. The partition is done by a simple algorithm realized by a computer (see [1]). The algorithm starts from a rectangular area limited by a frame composed of four lines. From the point (x, y) which lies on the line or curve, a curve is drawn until it meets another curve or line. The points (x, y) are chosen randomly. Figure 1 illustrates the rectangle with “colored” pieces.

Presented here are two patterns generated by using squares as a basic structural theme. The pattern (Fig. 1) illustrates what I call “constrained randomness”. It was created by randomly placing filled squares so they do not quite touch. Figure 2 shows a drastically different situation. A new placing square must have one common point with at least one previously located square. For more information, see [1]

Described here are patterns showing some new and interesting manifestations of chaotic behavior arising from parametric equations dynamics of the form

where a_n, b_n for n = 0, 1, 2,…, 5 are real constants…

Described here is a pattern which is interpreted as showing a galaxy launching a viral attack on an intruder…

The patterns of an oscillating quadratic membrane, first discovered by the German physicist E. Chladni in the early 19th century, can be calculated by solving the partial differential equation

…

A close friend once sighed, “How nice it would be, on cold winter evenings, to have a cat blanket.”

“What”, I asked, “is a cat blanket?”

“It's a blanket of cats that all come up on your bed, hundreds of cats. They arrange themselves into a blanket, leaving no spaces between them. Then they all go to sleep, purring through the night”. She seemed to purr at the idea…

This abstract pattern is composed of repeated shapes and parts. Try coloring the pattern in different ways to create your own unique art.

Described here is a pattern that I found when analyzing the geometry of traditional Tamil designs [1].

Tamil women of South India used a mnemonic device for the memorization of their standardized pictograms. After cleaning and smoothing the ground, they first set out an orthogonal net of equidistant points. Each design is normally monolinear, i.e., composed of one smooth line that embraces all points of the reference frame. Figure 1 shows a pavitram- or ring-pattern reported by Layard [2] that does not satisfy the Tamil standard as it is made out of three superimposed closed paths. In [1], I suggested that the reported pavitram design could be a “degraded” version of an originally monolinear pattern. The reconstructed design shown in Fig. 2 is probably the original pattern. Figure 3 displays a reduced version of Fig. 2, i.e., without “border ornamentation”. Applying the same geometric algorithm as used in Fig. 3, one always obtains monolinear patterns for dimensions (4m + 1) × (4n + 1) of the rectangular reference frame, where 4m + 1 denotes the number of dots in the first row and 4n + 1 the number of dots in the first column (m and n represent arbitrary natural numbers). Figure 4 illustrates this geometric alogrithm in the case 9 × 9 and Fig. 5 shows the 17 × 17 version with a rotational symmetry of 90°.

These designs were created for the Leading Part Company of Osaka. Designs in this series employ lines, circles, dots, squares, and ellipses. Some patterns seem to recede and shimmer in the manner of “op” art. Hopefully, designers and graphic artists will find these patterns of interest.

This etching (Fig. 1), which I made in 1978, originates in the experience of seeing large-scale patterns in the Greek landscape. When I was first in Greece, in 1967, I was struck by the apparent struggle between order and disorder and by the tension between the horizontal (the terraces, stone walls) and the vertical (cypresses and pines). For three months, as I was traveling through Crete, I spent all idle time covering a piece of paper with tiny new hieroglyphs, reinventing the landscape as a tapestry by seriation of basic elements such as suns, trees, birds, and eyes in many variations. Ten years later I transferred this design — somewhat modified — onto a wax-covered zinc plate, using the finest needle I could find. To convey something of the richness of a landscape, I found I had to create a texture with features too small to be resolved by the eye in the normal viewing distance. Only in this case, the experience of viewing the etching could invoke the experience of viewing a nature scene with fractal qualities. I made the etching in an art class and printed an edition of 10. Apart from using translational symmetries, the pattern explores themes of texture and the creative role of variations in a design…

Shown here are Persian wall tiles with metallic colors (from the 13th and 14th centuries). For a millennium, Iranian artists have attempted to state their feelings about the world in terms of harmonious and imaginative patterns, every sketch and design being an implement for worship, and a reflection of inner strength.

Since the earliest times (of, for example, Aristophanes) the ultimate demand made by a man, dominant in a social group, for recognition by his sycophants (the word itself being an obvious euphemism), has been “Kiss my arse”. Almost all European languages have equivalent expressions [1]. The Gaelic version, “Pogue Mahone”, is widely known as it is used as the name of a British pop group…

Interest in astrology and astronomy began thousands of years ago in Mesopotamia, Egypt, Central America, and the Far East. Today, we still use Assyrian-Babylonian and Egyptian astronomical symbols for sun, moon, and stars. Greek and Roman names signify the planets and constellations, while zodiac signs are Chaldean.

In the past, decorative letters were used in books, serving as the initial letters of paragraphs, chapters, and headings. Modern printers and designers no longer make extensive use of decorative letters. Letters are now generally employed to create special effects in advertising or striking book cover designs. Shown here is a sample from an unusual alphabet integrating human body parts and architectural themes.

The patterns presented here are based on ambiguous, cubist shapes. Ambiguous features, visual illusions, and the depiction of impossible objects have fascinated artists and lay people through the centuries. Famous past artists who have experimented with these visually interesting art forms come easily to mind: Escher, Magritte, da Vinci, Picasso, Dali, Albers, Vasarely, and most recently, Shepard [1]. See [1–6] for more information. Past researchers have even used such geometrical art as a probe of the human perceptual-cognitive system [1]…

Diaper ornaments may be generally defined as designs in which features occur at regular intervals, and which are enclosed or connected by geometrical flowing lines. Shown here are Japanese diaper ornaments.

Described here is a pattern showing the shapes of a flower as translated into the electronic environment. It differs from some of the other computer-based designs, in that a drawing of a flower was first made and then placed into the computer by the use of a mouse and the software program “MacPoint”. The basic pattern has been then manipulated by the computer-based software to form a set of variations on a simple theme, in this case, the “Satanic Flower”. The final results are printed out on a laser printer. I used the MacPoint part of Hypercard and employed a Mac SE computer to create these images.

Celtic art often consists of intricate knots, interlacements, and spirals. One form of Celtic art called anthropomorphic ornaments are those based on the forms of the human body. The made an early appearance in the art of Bronzeage Britain and Ireland. They are usually in conjunction with spiral ornaments, and the leg-joints and rib-forms of the animals in ornamental rendering have spiral terminal treatments. Abstract representations of human male figures with interlacing limbs, bodies, hair, top knots, and beards are used to decorate the sacred pages of the Gospels of the Book of Kells. Here, I show how to design some of the ornaments found on the stone at Clonmacnoise and the Book of Kells.

Described here is a pattern consisting of identically formed figures which cover a plane in a tile-like manner (Fig. 1). There is no blankspace between them. Figure 2 shows a single element of this pattern, a little smiling clown. This element has three different orientations in the pattern. You will notice some symmetries. For example, rotate the pattern at certain points (e.g., where the three hands meet at one point) by 120 degrees. In addition, three colours are sufficient to differentiate the clowns so that two neighbors are not of the same color…

Celtic art often consists of intricate knots, interlacements, and spirals. Here, are some examples from the Book of Kells.

It was a small discovery in my childhood that there were so many similar patterns existing in nature. This realization has since evolved into the concept of what I call the “law of similarity” in the natural world.

My long speculation on this topic eventually brought me to an idea — evolution of light — which became the title of my visual book completed in 1979. As illustrated in the volume, patterns of transformative motion are the basis of a theory of evolution based on the law of similarity. In this process, patterns oscillate and bind together to form matter which is actually only the temporary consequence of light transformations…

Art Deco, with its bold color schemes and strong geometrical patterning, has recently been rediscovered with tremendous impact, and is of major importance to designers today. Art Deco is the name now generally applied to the most typical artistic production of the 1920s and 1930s. This name comes from the large exhibition held in Paris in 1925, called the Exposition Internationale des Arts Decoratifs et Industriels Modernes. The art of these years had many sources, including turn-of-the-century art nouveau and such archeological interests of the day as ancient Egyptian and Mayan Art.

Shown here is a pattern belonging to an infinite series of periodic patterns. These are generally produced in the tables of f values of quadratic and higher degree binary forms when the multiples of a number are selected. The patterns result from the overlay of nets of points with integer coordinates…

Described here is a pattern showing multiple copies of my logo and printers' chop mark, assembled in a nonstatic pattern. This image was originally created by me as my artists' logo and is composed of the following significant elements…

The following sections are included:

• Glossary

• Index