DIMACS Workshop on Algorithmic Mathematical Art: Special Cases and Their Applications

May 11 - 13, 2009
DIMACS Center, CoRE Building, Rutgers University

Bahman Kalantari, Rutgers University, kalantari at cs.rutgers.edu
Dirk Huylebrouck, Sint Lucas (Brussels), dirk.huylebrouck at architectuur.sintlucas.wenk.be
Radmila Sazdanovic, The George Washington University, radmila@gwmail.gwu.edu


Fedor Andreev, Western Illinois University

Title: From Continuous to Fractal: Exploring and Root Finding

We follow the metaphor of the game of hide-and-seek to discuss various methods of finding root locations. I present some polynomiography algorithms derived from these methods. The level of the talk requires mostly high school algebra although we look at some connections with the fastest-descent algorithm. A series of images varying from rather mundane to fairly complicated and intriguing is presented.

Christopher Bartlett, Towson University

Title: The Fibonacci Series As An Algorithmic Organizing Principle In The Composition Of Figurative Painting.

This talk examines the use of the Fibonacci numbers as an algorithmic procedure or formula for designing visually harmonious self-referential areas of a realist painting which guide the positioning of the primary verticals, horizontals, and key focal elements.

Paul Burdick, New England Conservatory of Music, John Kiehl, Soundtrack Recording Studios, NYC

Title: Making Explicit The Implicit Intersections of Art & Science

This presentation will explore the links between Art and Science by examining the similarities and differences between the "perceptual inquiry" of the Arts and the "conceptual inquiry" of the Sciences. Based on this framework, a methodological model for Art/Science integration will be also be presented. These ideas will be perceptually and conceptually presented as well using algorithmic music software (authored by the presenters) as a point of mutual experience and reference.

Robert Bosch, Oberlin College

Title: TSPortraits of Knots and Link

We will give a detailed description of how we use the Traveling Salesman Problem, and some of its variants, to convert pictures of knots and links into simple closed curves.

Scott Carter, University of South Alabama

Title: Illustrator sketches obtained from projecting from 4-space to 2-space

I will show a series of sketches that I have prepared using three ideas. The first idea is that of the triangle times the triangle. It is a four dimensional figure with nine vertices and nine edges. The second idea is a convenient projection of 4-space onto the drawing plane. Then I tile portions of directions parallel to the xy, xz, xw, yz, yw, and zw planes with parts of the original figure. Color choices, layers, and filters are used to create planar images which I hope are pleasing. The results are in homage to Tony Robbin.

Jean-Marie Dendoncker, Ghent University, Belgium

Title: Algorithms through the eyes of an educator

The talk gives some examples of arithmetical en geometrical algorithms used in mathematics and art education. First, we focus on their use as tools to help deprived and underprivileged children: cusped curves (the Cremona representations of a cardioid, nephroid: links to multiplication tables), the Lamé curve (visualized by shrinking wavefronts of an ellipse) and some three dimensional surfaces. Next, we show applications on a more explorative level: finite geometry models (the Headwoodgraph, the Desargues configuration, quadrangles of orders (2,2) and (4,2)) and applications of the plastic number in the work of Hans Van der Laan, a Dutch architect.

Doug Dunham, University of Minnesota Duluth

Title: Repeating Hyperbolic Pattern Algorithms - Special Cases

About 50 years ago M.C. Escher created his four "Circle Limit" patterns, which were repeating patterns in the Poincare circle model of hyperbolic geometry. They were based on the regular tessellations {6,4} and {8,3} of the hyperbolic plane. In general, {p,q} represents a tessellation by regular p-sided polygons with q of them meeting at each vertex; this tiling is hyperbolic if (p-2)(q-2) > 4. About 30 years ago two students and I came up with an algorithm to draw hyperbolic Escher patterns. The basic algorithm worked if p > 3 and q > 3, but the cases p = 3 and q = 3 required special algorithms, which we will discuss. Also we will discuss special cases of a more general algorithm not based on regular tessellations.

Helaman Ferguson, Mathematician, Sculptor

Title: Theorems in Stone and Bronze

Helaman Ferguson will discuss his mathematical sculpture and give many instances of his publication of theorems in stone and theorems in bronze. He will show how he carves billion year old stone with his various tools, diamond chain saw included, as well as how he does millenia old lost wax bronze casting to celebrate mathematics and its algorithms.

HANDS ON ACTIVITY: The oldest (29Kyrs) recorded mathematical algorithm; how to do this iterative algorithm, and its profound significance in our culture and technology in this 21st century. We will instantiate this algorithm with many recycled materials in this activity.

Greg N. Frederickson, Purdue University

Title: Dissecting and Folding Stacked Geometric Figures

A geometric dissection is a cutting of one or more geometric figures into pieces that we can rearrange to form other geometric figures. A dissection is hingeable if we can attach the pieces together with hinges, so that we can swing the pieces one way on their hinges to form one figure, and swing them the other way to form another figure. A piano-hinge is a type of hinge that connects two pieces along a shared edge and allows a folding motion. We define a stack-folding dissection to be a dissection of a figure that is p levels thick to a second figure that is q levels thick, such that all pieces are connected into a single assemblage using piano-hinges.

We explore a variety of stack-folding dissections that involve regular polygons, stars, and "well-formed" polyominoes, with the goal of using as few pieces as possible. We focus on general methods (i.e., algorithms) to produce stack-folding dissections of any 1-high well-formed polyomino to the 2-high version of the same polyomino. We identify a set of just nine shapes that suffice for such dissections. We also present general methods to produce stack-folding dissections of 1-high and 2-high equilateral triangles to n-high equilateral triangles, for various values of n. We illustrate our dissections with video and animations that emphasize their symmetry and their beauty.

Georg Glaeser, University of Applied Arts, Vienna, Austria

Title: 1001 Images of Mathematics

A series of partly new mathematical visualizations is introduced. A wide spectrum is covered, e.g., functions, formulas, curves and knots, planar geometry, topology, tessellations, fractals, non-Euclidean and higher dimensional geometry, kinematics, mapping theory and minimal surfaces.

The over 1000 images are published in Georg Glaeser and Konrad Polthier: Bilder der Mathematik (Images of Mathematics), Spektrum Akad. Verlag / Springer, Heidelberg, May 2009.

Ted Goranson, Earl Research

Title: The "Kutachi" Project

The talk will describe a forthcoming project which in part will invite participation from the community by expected, sponsored workshops.

There have been many descriptions of a "form" language or shape grammar. We seek one that takes advantage of new, rather radical results in mathematical logic that allow for a geometry of narrative and logic. Such a grammar would use a variety of related visual techniques to allow visual comprehension of the nature of "situations" that include unknowns, emotional and intuitive aspects. It would, as well, be mathematically based, leveraging these new category theoretic results.

Contributors from many disciplines are sought. The forthcoming website for the Symmetry Society will be used for collaboration.

Dirk Huylebrouck, St-Lucas School of Architecture, Brussels, Belgium

Title: Unusual Methods of Mathematical Visualisation

When proposing visualizations for mathematics, the usual media are computer drawings or 3D models. There is a rational for more surprising visualization methods, as good teaching partially relies on a surprise effect. The author proposes a tryout of 3 "unusual" visualization methods: visualizing Pappus' theorem using laser beamers (based upon a set-up made by two students, Annelore Vercouter and Leila Lavens), or Pascal's generalization for conic sections, visualizing transformations of polyhedral objects using black light, and visualizing mathematics through gastronomy (called an "art" in Belgium and France). Examples will be demonstrated live, using, respectively smoke, gloves and ice cream.

Petra Ilias, Walter Lunzer, Ruth Mateus-Berr, University of Applied Arts, Vienna, Austria

Title: The Way Polynomiography Things Go, You real-eyes what you in habit

As artists and designers we work in different fields in the Art/Design education department of the University of applied arts in Vienna. We have been working with Bahman Kalantari´s POLYNOMIOGRAPHY© project. The aim of our research was to find out its inspiring aspects in our design- and art work. Our central question was, if, and how POLYNOMIOGRAPHY© stimulates our creativity and where it would lead us.

In these three months working with this software, we limited ourselves down to working on one algebraic formula per person. We intertwined this process by exploring abstract shapes and converted them into objects. We worked in a democratic co-operation and aesthetic simplicity. POLYNOMIOGRAPHY© led us on a journey, drifting from neo-situationist experiences concerning iterative lines and patterns from our daily travel from home to work, such as from Vienna to New Jersey. We followed the concepts of the metamorphosis of Maria Sibylla Merian and discovered the voyage from larvae to butterflies, such as Schönbrunn to Surinam. We found out the way things go in POLYNOMIOGRAPHY© and designed hyper-real landscapes composed of polys and blobs. The solution was proved to be the journey of the process.

David Hume suggests that the world might have been designed by a baby god, in the act of play. Only artists that retained a closeness to childhood engaging in playfulness would have dreamt about POLYNOMIOGRAPHY© and would make it happen the way we did. We approached this topic by story-telling, in the end the presentation portrays a performance of our recent work.

  1.  "The Way Things Go" adapted citation from Fischli & Weiss 1987
  2.  The Situationiste International (S.I.) were an european artist goup of the 60´s.
  3.  David Hume, Dialogues Concerning Natural Religion 1776, Arthur
      C. Danto, "Fischli and Weiss; Play/Things", in Peter Fischli and
      David Weiss: In a Restless World, Walker Art Center, Minneapolis 1996

Daniel Lordick, Technical University, Dresden, Germany

Title: Architectural Fractals

Structures of stochastic self-similarities can sometimes be found in architecture. Frequently mentioned examples are Gothic cathedrals, the Eiffel Tower, and a class of Indian temples. Due to construction constraints and usability, self-similarity in buildings is often limited to two or thee steps. Furthermore, the usage of fractal ideas - by geometric means - happens most commonly unconsciously. In this talk we are not looking for more or less convincing examples of fractal logic in architecture. On the contrary, we go the opposite direction. The object is to develop well-defined fractal structures that might be able to fit into an architectural environment. The approach is to derive architectural elements from observation and to refine them to the needs of a clear fractal pattern. The examples shown will be three-dimensional but without practical use. We would like to present a few examples, explain the algorithms behind them, and show the corresponding models produced by a rapid prototyping system. Some of the models have been part of the exhibition "Good Vibrations - Geometry and Fine Arts", that has been on show from May to August 2008 at the ALTANA-Gallery in Dresden.

Jay Kappraff, New Jersey Institute of Technology

Title: A New Course in the Mathematics of Design

I am developing, for next spring, a course in the Mathematics of Design for students from the new School of Design at NJIT that is part of the School of Architecture. The course is modeled after a course that I used to teach to students from the NJIT School of Architecture and that was the basis of my book Connections. The course will also be modeled after a course at University of Belgrade developed by Slavik Jablan and Radmila Sazdanovic. The course will include elements of Modular tilings, graph theory, theory of knots, fractals, 3-D geometry and Polyhedra, symmetry and music, theories of proportion, and projective geometry. The course will be project oriented with no examinations. Grades will be based on projects, scrapbooks and journals. The approach is informal and constructive with an emphasis on algorithms needed to carry out the projects rather than basic theory. My presentation in this conference includes images from various aspects of the course with special emphasis on an algorithmic and constructive approach to fractals and projective geometry.

Bahman Kalantari, Rutgers University

Title: Is Popularization of Polynomiography Possible?

Harry Potter books are said to encourage reading among the youth. The Rubik's Cube with its intricate design and mathematical connections became a best-selling puzzle. Can the algorithmic visualization of polynomial equations through polynomiography turn into a creative activity that would popularize math among the youth, bring innovations to art and design, bridge art and math, inspire mathematicians and educators, but also engage the general public?

Even today the historic problem of solving a polynomial equation remains to be a profound and fundamental task in science, math, and education. Polynomiography reveals magnificent visual beauty behind solving a polynomial equation, turning the problem upside down and into a medium of expression, art and design, education, creativity and discovery, play and fun. It liberates the appreciation of polynomials for a much larger audience than ever before and in doing so it drastically widens their scope of application, even for the specialists. What is visually revealed through polynomiography is not limited to the unveiling of mathematical properties, rather the polynomiographer has the choice to compose and instill human emotions in creating artistic polynomiographs, images that are computer assisted as opposed to computer generated. Polynomiography in its full development can truly turn into a multidisciplinary medium to be enjoyed by all.

Iraj Kalantari, Western Illinois University

Title: Media for Play, Expression, Curiosity, and Learning: Mathematics through Polynomiography

Studying of mathematics and its machinery by students in K-12 competes with distractions of infinitely many layers. The 'X-rays' of formal algebra, which provide 'light' in making the invisible visible, is seen as a medicine and cannot easily compete in the modern world sometimes intoxicated by visual feasts projected at every turn. When we can provide a platform, teaching and learning may benefit much from development of visually rich and expressive media for directed activity.

We argue for benefits afforded by immersion into a medium, such as Polynomiography, and report on some experiences of fifth- and six-graders with the software.

Carolyn A. Maher, Rutgers University, Kevin Merges, Rutgers Preparatory School

Title: Polynomiography as a Visual Tool: Building Meaning from Images

Two decades of NSF-funded research conducted by Maher and colleagues on how children build mathematical ideas over time has yielded findings on the important role of representations in learning mathematics with conceptual understanding. This research-based perspective guides our approach to teaching and learning mathematics as a sense-making activity. Exploring graphical representations of mathematical ideas is key in student learning. Building images that foster the connection of meaning to symbols provides opportunity to extend and enrich student understanding. We will discuss ideas on how student investigations of polynomials in secondary mathematics were enhanced using polynomiography software. We will share some ideas built by high-school students using the software in student investigations of real and complex roots of polynomials with real coefficients. Directions for future research will be explored.

Dimitris Metaxas, Rutgers University

Title: Physics-Based Methods for Modeling Open Surface Fluid Phenomena and Soft Tissue Strains

We will present state-of-the-art methods for open surface fluid simulation, developed at the Center for Computational Biomedicine, Imaging and Modeling of Rutgers University. The first is a Marker Level Set method that enables the simulation of both small and large scale fluid dynamics, and our methodology for simulating multiphase boiling phenomena. We will show boiling simulations in 0 and 1-gravity and animations of interaction between animated bodies and liquids, with strong generation of droplets and bubbles. The second method involves the use of meshless finite methods, for the accurate estimation of cardiac strain from MRI-tagged data. These methods, coupled with learning methods allow the accurate analysis of the cardiac motion and the strain-based comparison of different imaging modalities such as ultrasound and MRI.

This is joint work with Viorel Mihalef and Xiaoxu Wang

Amadeo Monreal, Technical University of Catalonia, Spain; Dirk Huylebrouck, St-Lucas School of Architecture, Brussels, Belgium

Title: Catenary or parabola, who will tell?

Gaudí is known for using catenaries in his designs. However, he used conics as well, and some confusion stills occurs whether a Gaudi curve is a catenary or a parabola. The co-authors turned their attention to this problem. Monreal, a mathematician from Gaudi's region, wanted to find out which arch is used in a given architectural picture through a quick and easy method, developing an easy procedure based on a high definition picture of an arch introduced it in a CAD software. Co-author D. Huylebrouck was fed up with the common architectural practice of embellishing images with all kinds of geometric figures, including catenaries, and he thought the well-known celestial mechanics' curve fitting method should be extended to architecture as well. Fortunately, both authors agree on the Palau Güell's doors interpretation, but some differences in the thoroughness of the mathematical approach remain.

Andrew Nealen, Rutgers University

Title: Contemporary Video Game Design: Challenges in Visualization, Interaction and Dynamic Simulation

Video game design has experienced a renaissance over the past decade. Parallel to the multi-million dollar games industry, independent game developers have been a constant and growing source of innovation: instead of taking years and big budgets, such developers create small prototypes and game experiments, and focus on what it takes to make games enjoyable. In this talk I will demonstrate some recent notable examples, and argue that it is the simplicity of the abstraction that makes these advances possible. Furthermore, I will discuss the concept of the "interaction uncanny valley", by which I describe how developments in visualization and animation have overtaken virtual interaction by orders of magnitude. Along these lines I will present some suitable abstractions of visuals, interactions and simulations that can be leveraged to alleviate this problem. The talk will conclude with some open problems and possible avenues for further research.

Ken Perlin, New York University

Title: Pseudo-randomness in procedural design

In procedural construction there is a tension between the calm of order and symmetry on the one hand, and the fire and chaos of randomness on the other. Somewhere between these two extremes lies an ideal aesthetic balance. In this talk we explore how pseudo-random techniques can be used in a controllable and effective manner within a number of domains of creative design, including texture, sculpted forms, music and animation.

Ron Resch, Artist

Title: Poly-Plodes, Polyhedra that kinematic implode and explode.

Poly-Plodes, (a term created by Resch for the continuous, kinematic imploding and exploding of polyhedra) were begun by Resch in the early 60's. In this talk the geometrical methods and parameters to transform all polyhedra so that they can become Poly-Plodes will be described and animations and films of models will be shown.

Rinus Roelofs, Sculptor, Hengelo, The Netherlands

Title: Sculptures and Structures.

The subject of my work as a sculptor is my fascination for mathematics. And especially for mathematical structures. Often I am amazed about all the possibilities to create interesting structures out of simple basic shapes. You can think of tilings, polyhedra or space frames. Mathematics offers us many techniques to describe the possible structures, which can lead to a better understanding. Some of the mathematical techniques are also suitable to create new structures. For me the mathematical transformations are very important. Not only the geometrical transformations like translation, rotation and mirror image, but also transformations as inversion, complement and duality. Some of them are also known in visual arts. You can think of complementary colors or of shape and rest shape. And for instance when we think about perspective, to take another point of view is also a kind of transformation. The many different kinds of transformation that mathematics offer us are now my main tools to develop my own ideas about structures. The process that leads from fascination to understanding often offers me ideas for sculptures. Sculptures in which I try to express this fascination. So my work can be described as art about mathematics, rather then as mathematical art. In my presentation I will try to illustrate this.

Tony Robbin, Artist; New York and Gilboa

Title: Many Spaces in the Same Space

Both our objective experience of physics and our subjective experience of the world encourage us to see many spaces in the same space at the same time. My talk presents 40 years of work to realize that vision.

Radmila Sazdanovic, George Washington University

Title: To cut or to knot

I will illustrate ideas for creating mathematical objects from other mathematical objects, such as tessellations, families of graphs and families of knots given in Conway notation. In addition, I will illustrate how to visualize the corresponding families of Jones and Chromatic polynomials using Polynomiography software.

Nathan Selikoff, Artist

Title: Aesthetic Explorations of Algorithmic Space

I love to experiment in the fuzzy overlap between art, mathematics, and programming. The computer is my canvas, and this is algorithmic artwork-a partnership mediated not by the brush or pencil but by the shared language of software. Seeking to extract and visualize the beauty that I glimpse beneath the surface of equations and systems, I create custom interactive programs and use them to explore algorithms, and ultimately to generate artwork.

In the world of chaotic dynamical systems, minute changes in initial conditions produce radically different results. The interface of my software gives me hooks into the algorithms and allows me to exert some control. But there is always tension-between the computer and me, between simplicity and complexity, and between problem solving and spontaneity.

Art and mathematics, the right brain and the left, are inextricably linked in this work. My art depends on mathematics, yet simultaneously illuminates and unravels its beauty. I am the explorer who uncovers something extraordinary, bringing into view that which was always there to be discovered.

Lillian Schwartz, Visiting Scholar, New York University

Title: Leonardo's Choice for the Model in Creating the Mystical Image on the Shroud of Turin

Questions asked over the centuries concerning the Shroud of Turin: is the image on the Shroud the true Christ? A painting? Photograph? Some special method? And how old was it? The latter was answered easily enough when a few threads were analyzed. But the identity of the face remained a puzzle until this past January. Just as I analyzed Leonardo with regard to the identity of model for the Mona Lisa, I applied computer techniques to the Last Shroud and came to a conclusion. Part of that conclusion was presented in a recent documentary aired on the Discovery Channel in the United States and other stations abroad, but now I will provide a more detailed analysis that will return us to that bridge which is born of creativity and science parsing time and space in a manner not mystical but nonetheless true.

Dmitri Tymoczko, Princeton University

Title: The Geometry of Music

In my talk, I will explain how to translate basic concepts of music theory into the language of contemporary geometry. I will show that musicians commonly abstract away from five types of musical transformations, the "OPTIC transformations," to form equivalence classes of musical objects. Examples include "chord," "chord type," "chord progression," "voice leading," and "pitch class." These equivalence classes can be represented as points in a family of singular quotient spaces, or orbifolds: for example, two-note chords live on a Mobius strip whose boundary acts like a mirror, while four-note chord-types live on a cone over the real projective plane. Understanding the structure of these spaces can help us to understand general constraints on musical style, as well as specific pieces. The talk will be accessible to non-musicians, and will exploit interactive 3D computer models that allow us to see and hear music simultaneously.

Gunter Weiss, Technical University, Dresden, Germany

Title: Categories of Algorithmic Aesthetics: Obvious < Hidden < Secret < Geometric

The lecture aims at an analysis of different viewpoints to geometric works provided by colleagues (Hans Dirnböck, Lazar Dovnikovic, Hirotaka Ebisui, Georg Glaeser, Fritz Hohenberg, Daniel Lordick, Jun Mitani, Laszlo Vörös, Robert Wiggs a.o.). "Beauty is a matter of taste." - And culture! A discussion about aesthetics of Algorithmic Computer Generated Art should consider both, the obvious visible part of an art object and the more or less hidden mathematics of it. A mathematical idea has a beauty of its own, leads to actively programmed algorithms and, via computer graphics tools - algorithms, too, (which by themselves might be a "secret" even to the artist-mathematician), finally leads to an arbitrary realisation. Only the connoisseur and expert will discover the full spectre of aesthetics, while others must be content with the aesthetics of the result alone. Some examples will show both, that the aesthetic value of an object increases and even decreases, when having understood the geometric idea behind it.

Eva Wohlleben, Artist

Title: Corpuscle Geometry

In the year 2006 I built paper models of several previously unknown polyhedra. These geometric bodies are highly symmetric and consist of almost regular triangles only. Since the building blocks they are made of -- the so-called corpuscle -- are slightly irregular, the structures are elaborate to calculate, although easily realized as models.

The corpuscle is a flexible building block. It allows for building three-dimensional periodic structures. Algorithms can be used to design open ended chains. Closed rings and networks are tough to calculate and so far have only been realized materially, not virtually.

Some corpuscle polyhedra, and all open-ended chains, show inevitable mobility. Whenever an impulse is brought into the system at any point, the whole system moves in a defined way. During the movement several corpuscles decrease their volume, while simultaneously other corpuscles increase theirs. The periodicity of three, which is to be observed in that motion, might lead to new applications in the art of engineering. Yet it is best observed by directly experiencing the physical models.

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Document last modified on May 6, 2009.