We investigate computational methods for seeking complex behavior among 2-dimensional Boolean automata. We consider general automata and those that respect various symmetries on local 3 by 3 Moore neighborhoods. We use the standard deviation of input entropy to guide Monte Carlo searches. We also use genetic algorithms for finding complex automata from other complex automata. We also use searches that require the simple glider that arises in Life to be viable and do likewise for other simple gliders that arise in general symmetric searches. Indeed, our richest searches for complex automata utilized symmetry, required very simple gliders to be viable, and used bounds on input entropy in Monte Carlo searches. While complex behavior may be rare in rule-space, it appears that there are many symmetric cellular automata with complex behavior that can be found using the techniques that we discuss.

In 1984, solid-state materials were discovered that had diffraction pattern symmetries that are forbidden for three-dimensional crystalline structures. The structures of these materials have become known as quasicrystals. A considerable literature has developed for both physical and mathematical quasicrystals. There are essentially three forms of mathematical three-dimensional quasicrystals. In this paper, we will develop computational techniques to construct these three forms of quasicrystals as point sets, and then organize those point sets in order to visualize patches of quasicrystals using parallelepipeds.

. A perfect parallelepiped has edges, face diagonals, and body di-agonals all of integer length. We prove the existence of an infinite family of dissimilar perfect parallelepipeds with two nonparallel rectangular faces. We also show that we can obtain perfect parallelepipeds of this form with the angle of the nonrectangular face arbitrarily close to 90 ◦ . Finally, we discuss the implications that this family has on the famous open problem concerning the existence of a perfect cuboid. This leads to two conjectures that would imply no perfect cuboid exists.

J is an open source programming language with a rich collection of well-designed primitives and a consistent, compact, mathematics-like syntax. It has amazing array facilities, superb numeric and data processing capabilities, and graceful features allowing functions to be defined compactly.

Cyclic cellular automata in two dimensions have long been intriguing because they self organize into spirals and that behavior can be analyzed. The form for the patterns that develop is highly dependent upon the form of the neighborhood. We extend this work to three dimensional cyclic cellular automata and observe self organization dependent upon the neighborhood type. This includes neighborhood types intermediate between Von Neumann and Moore neighborhoods. We also observe that the patterns include nested shells with the appropriate forms but that the nesting is far more complex than the spirals that occur in two dimensions. (C) 2011 Elsevier Ltd. All rights reserved.

There are parallelepipeds with edge lengths, face diagonal lengths and body diagonal lengths that are all positive integers. In particular, there is a parallelepiped with edge lengths $271$, $106$, $103$, minor face diagonal lengths $101$, $266$, $255$, major face diagonal lengths $183$, $312$, $323$, and body diagonal lengths $374$, $300$, $278$, $272$. Focused brute force searches give dozens of primitive perfect parallelepipeds. Examples include parallellepipeds with up to two rectangular faces.

Perfect parallelograms have edge lengths and diagonal lengths that are all positive integers. These generalize Pythagorean triples which are perfect rectangles. We consider the distribution of perfect parallelograms and show they satisfy a quadratic Diophantine equation. The solutions to that Diophantine equation can be generated by a finite collection of matrices that generalizes the matrix based tree of Pythagorean triples.

Interacting populations exhibit complex behavior in nature. Classic quadratic iteration models with two or three populations exhibit some of the features seen in nature, but fail to account for spatial variation. Indeed, the diversity paradox is that many classic population models predict one species dominates while nature exhibits diversity. While various schemes have been presented to address this dilemma, we present a simple, deterministic cellular model that incorporates classic iteration schemes and a spatial migration component that provides for self-organizing and rich behavior. Our visualization method allows us to observe dynamically changing predominance of species, global diversity, waves of species progression, and highly organized spiral structures.

Cyclic cellular automata on the integer planar lattice are known to typically evolve through distinct phases ending with minimal periodic terminal states that usually appear as intertwined spirals. Here we explore the diversity of spirals that arise from nonstandard neighborhoods on the integer lattice and from looking at the automata on quasi-crystalline arrangements of cells. We see that phase transitions and development of spirals are almost ubiquitous yet the particular form of the spirals is very dependent upon the particulars of the underlying neighborhoods; in fact the spiral forms echo the neighborhoods. The quasi-crystalline illustrations provide much more subtle echoes in the spiral forms that show artifacts from the non-periodic local symmetry that occurs.

The Binet formula provides a mechanism for the Fibonacci numbers to be viewed as a function of a complex variable. The Binet formula may be generalized by using other bases and multiplicative parameters that also give functions of a complex variable. Thus, filled-in Julia sets that exhibit escape time may be constructed. Moreover, these functions have computable critical points and hence we can create escape time images of the critical point based upon the underlying multiplicative parameter. Like the classic Mandelbrot set, these parameter space images provide an atlas of Julia sets.

It is unknown whether there are perfect parallelepipeds, that is, parallelepipeds with integer-length edges, face diagonals and body diagonals. A stronger version of the problem also requires the coordinates to be integer. In that case, the vectors are integer-length integer vectors. We will show how to extend integer length integer vectors in any dimension to one higher dimension and utilize that construction to present three parametric families of parallelepipeds that are nearly perfect in the sense that only two conditions need be satisfied in order for the parallelepiped to be perfect. Computer searches show many examples where either, but not both, of those conditions may be satisfied.

Snow crystals are intriguing because they exhibit both symmetry and remarkable diversity. Previous studies have used two-dimensional models to approximate snow crystal growth. Here generalizations to three dimensions are considered. In particular, a cellular arrangement of cells is updated according to local rules that involve the identification of receptive cells and averaging values from non-receptive sites. The cellular arrangement is the same as the configuration of water molecules in ice that occurs under ordinary conditions on Earth.

Identifying the historical data that is the best analog with a pattern from which a forecast is sought allows time series data to be extrapolated. That technique of best analogs is most effective when the data contains underlying deterministic chaos. Here we apply similar techniques, modified to use two space dimensions instead of one time dimension, to fill-in and extrapolate missing image data. The technique is successful at replacing significant amounts of missing data with reasonable data derived from the image itself.

Pythagorean Triples are well-known examples of integer solutions to sums of two squares giving another square. It is well known that Pythagorean Triples may be generated parametrically. It is somewhat less well known that they may also be generated via matrices. In this note we describe how matrix generators may be used to produce all the Diophantine solutions of a square being a sum of squares when the number of squares in the sum is between 3 and 9. For 3 <= n <= 8 all the Diophantine solutions may be obtained via matrix multiplication from a single type of initial solution. For n = 9 two different types of initial solutions are required.

Fractals is a hands-on introduction to the visualization of mathematical objects, algorithms and data. The book gives a self-contained introduction to J as raster and object graphics are introduced. Fractal times series illustrate techniques for dealing with one-dimensional data. Studies of iterated function systems and chaotic attractors offer experience with visualizing two dimensional phenomena while investigating fractals and symmetry. Finite automata and fast Fourier transforms are used for playful things like the game of life and serious topics such as image processing. Similar techniques are used for studying classical complex dynamics illustrated by the Mandelbrot set and Julia sets. The mathematics of three dimensional projection is discussed and visualization of three dimensional data is the focus of scenes created by rendering J created objects with raytracers and OpenGL.

Computational exercises and demonstrations appear frequently in the classes that I teach at Lafayette College. The software that I use for in-class demonstrations is usually Mathematica [4] or J [3]. I focus on one language or the other throughout the course when I expect the students to develop the ability to make substantial explorations and do independent experimentation. There is an ongoing expectation at Lafayette that Mathematica be used in Calculus Classes [2]; I also use Mathematica for my Numerical Analysis and Techniques of Math Modeling classes. I use J for teaching Number Theory and Linear Algebra and it is central to my mathematical visualization course that follows the text [6]. However, I always have interest in how the explorations that I do in one language would be done in the other.

Computational exercises and demonstrations appear frequently in the classes that I teach at Lafayette College. The software that I use for in-class demonstrations is usually Mathematica [4] or J [3]. I focus on one language or the other throughout the course when I expect the students to develop the ability to make substantial explorations and do independent experimentation. There is an ongoing expectation at Lafayette that Mathematica be used in Calculus Classes [2]; I also use Mathematica for my Numerical Analysis and Techniques of Math Modeling classes. I use J for teaching Number Theory and Linear Algebra and it is central to my mathematical visualization course that follows the text [6]. However, I always have interest in how the explorations that I do in one language would be done in the other.

Whether there exists a parallelepiped with edges, face diagonals, and main diagonals all of integer length is an open question. This is equivalent to thirteen linked quadratic diophantine equations. We look at the basic diophantine equation: the structure of integer length vectors in dimensions two, three, and four and give matrix generators for producing all the 3-dimensional integer length integer vectors. Parametric families of parallelepipeds that have good properties and the results of computer searches for perfect parallelepipeds are also described.

The growth of real valued cellular automata using a deterministic algorithm on 2-dimensional quasicrystalline structures is investigated. Quasicrystals are intermediate between the rigid organization of crystals and disorganized random structures. Since the quasicrystalline structures may be highly symmetric or not, we are able to obtain highly organized and relatively random growth patterns. This deterministic growth produces dendrite, sector, stellar, regular polygons, round, and random DLA-like structures.

Snow crystals have a rich diversity of forms with striking hexagonal symmetry. The two-dimensional types include dendrite, stellar, sector and plate forms. Physical studies have shown that the particular form of a snow crystal is dependent upon the temperature and saturation in the growth environment. We investigate a simple, local, two-dimensional model that has just two parameters exhibiting the desired diversity. The dendrite growth is especially realistic. We use the model to investigate the impact of varying conditions of growth.