Sliding the Trinoid's Edge  
 Topological Slide:  Artist's Statement
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Michael Scroggins: 

The original grant application to the Art and Virtual Environments Project proposed the use of a tiltable platform as a kinesthetic interface for navigating upon topological surfaces. It went on to describe the project in the following words: 
"The 'rider' will wear a head mounted display enabling an interactive wide-angle stereo view of a three-dimensional space.  The space will consist of a model of a topological surface to which the platform is bound and upon which it is free to slide.  The 'rider' may traverse the model's surface by leaning in the direction in which she desires to move.  The amount of lean in a given direction will determine the rate of sliding." 

This simple concept was the heart of the proposal and there was no conscious attempt to deal directly with the many interesting and complex conceptual issues surrounding VR. The Topological Slide was conceived to be a direct sensual experience of surface with overt handles for the intellect located in the mathematical concepts integral to the surfaces formation. 

In the early stages of the project, I tried to discourage any associations with surfing preferring instead to invoke the more general model of sliding. My attempts to avoid the surfing tag were doomed however, in light of the prevalent use of surfing as a metaphor. No one seemed to be speaking of "channel skating" or "internet skiing"! 

Ironically, the germinal idea for the Topological Slide did arise in an idle daydream of a VR surfing simulator. A constant tubular wave generator could allow the visual illusion of riding inside the most intense part of a wave. Getting tubed can be a transcendent experience as the level of concentration required to remain inside the tube can propel one into a state of mind where there is no past and no future, just a lucid present. Unfortunately, the surfing simulator idea unavoidably degenerated into the annoying notion of a kind of VR karaoke where one could replace Frankie and Annette as surf heroes. 

An attempt to recuperate some aspect of the surfing model led me to consider utilizing the continuous looping flow of a Mobius strip as a substitute for a simulated wave. Reflecting on the topological properties of the Mobius strip led to consideration of the Klien bottle and then on to the wide range of minimal surfaces which Stewart Dickson had brought to my attention years earlier. 

My modernist indoctrination gave weight to the idea that traversing mathematical objects was much more interesting than riding a weak replica of an ocean wave and that experiencing the abstract made concrete rather than the actual made virtual was clearly a more valuable process. There is a long and rich history to the linkage of art and mathematics and the Topological Slide project may be viewed from several positions in this stream.1  One way of thinking of the project is to consider the topological surfaces as a priori objects presented as art. Jung remarked that number might be seen to be as much discovered as invented by man,2 and Duchamp established that the found object could take on a power equivalent to that of the crafted art object.3  

A contributing factor to my personal fascination with the Topological Slide is my mathematical naivete. In addition to the sensual delight of navigating the surfaces in a very physical way, there is the drive to gain an understanding of the principles by which those surfaces are formed and the underlying concepts that make them interesting to mathematicians. In the course of dealing with these objects, my curiosity has been aroused enough to desire further education in mathematics. I have been forced to come to terms with how little I understand of non-Euclidian geometries and thus how limited my concepts of space have been. It is my hope that this project will also stimulate others --particularly children-- to expand their mathematical understanding. 

I asked my old friend and colleague, Stewart Dickson, to collaborate with me on the project because of his deep interest in visual mathematics and experience designing topological object databases. I was responsible for the design and fabrication of the tilting platform, and as the person who conceived and initiated the project, my primary role was to coordinate its overall realization. Stewart provided the code for the topological objects and was responsible for writing and implementing the original platform interface routines. Subsequent programming necessary to realize the project was performed by Graham Lindgren, John Harrison, and Glen Frazer. Further work in optimizing the Topological Slide code and implementing edge control algorithms was performed by Sean Halliday.  
Stewart Dickson: 

The construction of minimal surfaces by contemporary mathematicians like David Hoffman and William Meeks is an exercise of obtaining a specific topology with the geometry constrained to be the purest possible expression of the form. The breakthrough work they did used the computer to make pictures of the forms they were studying. The pictures showed them aspects of the objects they could not deduce from the equations alone. They began doing a kind of experimental mathematics which revolutionized their field of work. They began designing mathematical systems which were driven by a visually-derived structural model. They then had a higher- level point-of-view from which they could advance their formal rigor. Documentation of this work can be found at 

The objects we have chosen to use in the Topological Slide are artifacts of the historical legacy of cyberspace. The general problem of placing the rider on a mathematically- specified surface is fairly complex. First the (X,Y,Z) coordinate in three-space on the mathematical surface must be calculated as a function of the (U,V) coordinate in parametric space of the rider's location. This location is the cumulative effect of the two acceleration vectors specified by the platform tilt axes and a constant friction or drag coefficient. The rider must then be oriented in a natural manner to the surface at the correct three-dimensional address. 

The parametric surface I am using in this piece is generally represented by a polynomial of fourth degree or higher. Costa's Genus One, Three-Ended Minimal Surface is specified as the real parts of three integrals of expressions involving the Weierstrass Elliptic P-function over regions in the complex plane. In general, solutions to an arbitrary polynomial equation of degree five or more cannot necessarily be written as algebraic expressions. For equations of degree exactly five, the solutions can in principle be written in a complicated way in terms of elliptic functions; for higher-degree equations, even this is not possible. This is a fundamental difficulty in the state of contemporary mathematics.4  The surfaces we are exploring are indeed on the fringe of human knowledge. 

In short, these calculations are non-trivial. I rely on Mathematica to obtain graphical representations of the surfaces. Even if I could derive C-language implementations of the calculations, it would not be advantageous to calculate them on the fly; therefore, we used pre-computed CGI-style geometry description records to represent the surfaces. 

I had originally intended that the Topological Slide contain an object which is in metamorphosis from its simplest form (disk, sphere, torus, etc.) into "full bloom". This action was intended to better illustrate the nature of the shape as well as to add dynamism to the riding experience.  

Another class of surfaces we intended to illustrate was the three-dimensional representation of an entity which is defined in a higher-dimensional space. I have prepared pairs of immersions in three-space of the Real Projective Plane which metamorphose one into another: The Veronese surface which metamorphoses into the Steiner Roman surface and the Apery cylindrical parameterization of the Roman surface which metamorphoses into the Werner Boy surface. I also have two different (orthogonal) parameterizations of the Klein bottle which transform one into the other. Further development time would have allowed me to realize these goals. 


1. An excellent collection of essays on the subject can be found in Michelle Emmer's The Visual Mind:  Art and Mathematics (Cambridge, Massachusetts: MIT Press, 1993) 
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2. Carl Jung, The Collected Works of C. G. Jung, Vol. 8, "The Structure and Dynamics of the Psyche", (Princeton University Press, Princeton, New Jersey, 1969),  457. 
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3. The ironic  inversion inherent in this reading can be discerned in Calvin Tomlin's The Bride and the Bachelors  (London: Weidenfeld and Nicholson, 1965), 26 
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4. Paraphrased from Stephen Wolfram, Mathematica: A System for Doing Mathematics by Computer. 2d ed. (Reading, Massachusets: Addison-Wesley, 1990). 
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