Complex Curvatures in Form Theory and String Theory
The authors use new aesthetic criteria concerning structures and properties to explain parallel concepts within theoretical astroparticle physics and contemporary form/compositional research. These aesthetic criteria stem from complex curvature models developed both in string theory and in artistic perceptual research on transitional surfaces and concavities. The authors compare the complex curvatures of the mathematically derived Calabi-Yau manifold with one of Akner Koler's sculptures, which explores an organic interpretation of the looping curvature of a Möbius strip. A goal of the collaboration is to gain experience and insight into the twisting paradoxical forces in the 3D world and to explore the properties of transparency as applied to the Calabi-Yau manifold and a point cloud translation of Akner Koler's sculpture.
Physicist Lars Bergström and sculptor Cheryl Akner Koler have worked together on a number of art-science projects over the past 6 years. Their exhibition at the Stockholm Art and Science Festival in September 2002 (together with two other physicists, Narendra Yamdagni and Per Olof Hulth), entitled "Infinity," focused on the concept of infinite cyclical processes and extra dimensions (Fig. 1). Bergström comes from the field of theoretical astroparticle physics and is an expert in dark matter [1]. Akner Koler is a sculptor with a background in constructivist art and is working on a theoretical model that strives to renew 3D compositional theory [2].
As a physicist one sees the world in terms of energy, matter, attracting forces, transformation, spatial dimensions, etc. This interest in an abstract descriptive language of movements and forces in spatial context is shared by artists working within a constructivist discipline. Our collaboration is based on a common interest in expanding this descriptive aesthetic language in order to explain complexities in physical phenomena. Support for this need for an art-science aesthetic language is given by physicist Brian Greene:
Maybe, deep down, the universe has a less elegant structure than our experience has led us to believe, or perhaps we will find out that our current aesthetic criteria need significant refining when applied in ever less familiar contexts [3].
Our art-science collaboration inspired a dialogue in a cross-disciplinary context that focused on questions that otherwise would not have been explored. We presented for each other our different views on the same or parallel phenomena and worked to bring these ideas into the perceptual world. An interesting area that developed during the 9 months we spent preparing for the 2002 festival involved exploring complex curvatures and hypersurfaces in space. Through a variety of different media—3D form, smoke, CAD models, drawings and kinesthetic movements—we studied conceptual and embodied ways to approach our common theme. This article is limited to the specific study of twisted sculptural curvatures and the Calabi-Yau manifold from string theory.
Symmetry Conference
We also presented our work in complex curvatures at the Symmetry Festival in Budapest, Hungary, 16-22 August 2003. The conference, "Symmetry and Dis-Symmetry: A Synthesis of Constancy and Change" [4], was a cross-disciplinary forum for those who work with morphological studies of symmetry, asymmetry, broken symmetry, antisymmetry, etc. The International Society for the Interdisciplinary Study of Symmetry (ISIS-Symmetr), which hosted the conference, supports work that goes beyond the traditional boundaries of disciplines and professions. Its members are from art, science, design and engineering.
A Short History of String Theory and The Calabi-Yau
In the micro world we are getting used to the idea of employing more than the usual three spatial dimensions plus time—considered by physicists to be the fourth dimension. In string theory and its underlying models it is held that there may be six or seven extra dimensions; these are usually believed to be curved into themselves. The step out of 3D and into something more complex was a very important one, first taken by a German, Theodore Kaluza, and a Swede, Oskar Klein, in the 1920s. This step is a cornerstone of contemporary particle theory and string theory. The existence of the extra dimensions, if proven scientifically, would mean a revolution in the way we consider space and time, a revolution even greater than the one brought about by relativity and quantum mechanics. The lowest energy state of string theory (the vacuum state) does not seem to be unique, or at least we have not found the guiding principle to select the ground state of the theory. Therefore we can only play with low energy state (vacuum) models that seem to have the right amount of symmetry and broken symmetry to encompass the standard model of particle physics. One such type of vacuum model is the Calabi-Yau manifold, given by a simple homogeneous equation in five complex variables. Despite the mathematical simplicity of the defining equation, the properties of the manifold are complex, yet symmetrical in an interesting way. [End Page 227]
Overall view of two rooms in the exhibition at the Stockholm Culture House in September 2002. Cheryl Akner Koler's installation is on the left, and Lars Bergström's installation is on the right.
Properties of 3D Compound Curvatures: Current Models in String Theory
In string theory, there is great uncertainty about the "true" vacuum, particularly the one that causes the universe to expand and even accelerate. Mathematical models built up of strings compactified in certain coordinates, called compact extra dimensions (hypersurfaces), have led us to appreciate the significance of extra dimensions. The Möbius strip [5] (Fig. 2) shows that what is locally a flat surface may globally be very complex, because its orientation in space is undetermined. The twist from inside out gives this area its peculiarity, expressing only one continual single surface and one continual edge.
An algorithm by Andrew Hanson of Indiana University, U.S.A. [6], has integrated hypersurfaces into the fascinating Calabi-Yau manifold. Bergström has used this algorithm to produce a number of different views and interpretations; an example is shown in Fig. 3. This model regards the compactified dimensions of that manifold as the true space where strings roam. This means that in every point in our universe, there is such a manifold—of an incredibly small size. Close to the time of the Big Bang the characteristic radius of our universe was of the order of the Calabi-Yau radius (or Planck radius).
Twisted Curvature
Akner Koler's concavo-convex aluminum sculpture, Twisted Curvature (see Article Frontispiece and Color Plate C No. 2, sculpture on center podium), expresses a twist from the inner to the outer surfaces
Mathematical computer model of a Möbius strip.
Calabi-Yau manifold computer visualization model.
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and vice versa. This type of complex curvature is a traditional sculptural phenomenon that can be found in figurative and non-figurative sculptures throughout history, such as Michelangelo's The Dying Slave[7], Henry Moore's Reclining Figure, Barbara Hepworth's Pierced Form[8] and Alexander Archipenko's Walking Woman[9]. The fact that the volumes and surfaces of Twisted Curvature make a cyclical loop intensifies this spatial twist and expresses the paradox of the Möbius strip. Max Bill, Helaman Ferguson and Brent Collin have worked in this area and have given both sculptural and theoretical insight into the mathematical parallels of the Möbius strip and similar structures [10]. Twisted Curvature was not intended to express a mathematical concept; there is only an "implicit" mathematical correlation. The aim of creating this twisted sculpture was to explore organic sculptural traditions in parallel with models of curvatures in string theory as well as to bring digital technology into this sculptural area.
Close-up of a twisted area on Lars Bergström's computer visualization model of the Calabi-Yau manifold, using the algorithm of [4].
Transforming the simple curved movement of a Möbius strip curvature into a volume that embodies 3D compound curvatures brings out a more complex organic sculptural experience. Rowena Reed explains this organic experience in detail in her descriptive analysis of convexity and concavity [11].
The variation of volume thickness, material density and added compound curves that are accented within the twist brings out organic double-curved features that blend together with the mathematical features explored by Möbius. The visual model of the Calabi-Yau (Fig. 4) implies this paradox at a high level of complexity in extra dimensions. One of the main purposes in our collaboration is to gain perceptual experience of these twisting paradoxical forces in the 3D world. Michael Biggs has stated the importance of learning through perceptual experience for the purpose of guiding our thought processes:
I am interested in investigations in which aesthetic judgements are made in relation to sensory objects and one might argue that this process as well as having an empirical basis, that could be examined through experimentation, actually arises through the experience of being confronted with these judgements and that therefore the identification of the initial problem, as well as its conduct through experimentation arises in the realm of experience rather than in the realm of cognition [12].
Transparent Dimensions of Form
Twisted Curvature was plotted through a 3D digital scanning as a "point cloud" volume, shown in Figs 5-6 and in Color Plate C No. 2. A point cloud volume is made by correlating points on the surface of a physical object with points on a digital mesh that map the spatial coordinates in a 3D virtual matrix [13]. To visualize this digital information, a selection of points is made by defining maximal distance and ± variance between points. Sixteen transparent projections of this point cloud were chosen, five of which are shown in Fig. 5. In the exhibition, these transparent projections were mounted on curved white surfaces, hanging from the ceiling and surrounding the sculpture (see Color Plate C No. 2). Each projection is unique because the sculpture is totally non-symmetrical and the projections are all taken from different viewpoints. The phenomena of transparency and the twisted, continual surfaces of the form allow the point coordinates to become superimposed, and the spatial positions of each point are obscured in depth.
To fully experience the properties of the point cloud volume through these pictures, it helps if one avoids focusing on the outer and inner contours and lines of the form. Instead a relaxed perception of depth, movement and volume should be explored, one that allows the inner concave surfaces to transpose into the outer convex surfaces simultaneously. This transposition suggests the undulating change of material density that moves through curvatures between dimensions. Finding ways to understand how to move between dimensions is one common experience we have shared throughout this project. Perhaps hidden somewhere in this transparent hypertwist lie the organizing conditions that give electrons, light particles, photons, etc. their unique properties.
Seeing "solid" form as made up of points or particles in space is a conceptual
Five different point cloud projections of the sculpture Twisted Curvature, developed through scanning digital technology at Nufoparts in Sweden, 2002.
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model physicists use to explain the laws of nature. This scientific concept has been difficult to understand, because our perceptual awareness and visual rhetoric has traditionally reinforced solidity and stability, which is the opposite of moving particles in space. Today digital technology has made it possible to virtually and physically observe forms being composed of points or particles. Through theories from physics and advancements in technology we are slowly reorienting our ways of conceiving, perceiving and aesthetically judging form, space and content. This reorientation can be done without divorcing the historical traditions of sculptural aesthetic knowledge.
Cheryl Akner Koler, Inverted Virtual Point Cloud Nr. 6 of Twisted Curvature, 2002. Developed through scanning digital technology at Nufoparts in Sweden.
Reassessing the Properties of the Space Surrounding Hypersurfaces
There is a problem in visualizing the hypersurfaces of a Calabi-Yau, which exists at each point in our conventional 3D space. Is it like a sphere contained within itself ? Or could it be that there are changes in density, or other properties within the hypersurfaces, that allow some sort of integration with the surrounding space? A transfer of properties happens around the hypertwisted areas, which allows the "form" (or rather, mass, color, charge and other physical properties) to dissipate into other dimensions. We have tried to invoke a feeling for these mechanisms by making the Calabi-Yau model transparent and thereby smoothly connected to our space-time.
Transparent Calabi-Yau
The Calabi-Yau manifold is interesting as a model for the compactification of string theory from 10 dimensions to four dimensions (counting time as a dimension). Bergström has taken a 6D algorithmic model of the Calabi-Yau manifold and transformed it into a transparent version (Fig. 7) in which the integration with real space appears vague. This is meant to express the unity of all these 10 dimensional spaces and the fact that the compact extra dimensions are everywhere, influencing all the properties of matter as we know it.
Both the Calabi-Yau manifold and the Twisted Curvature sculpture were presented as transparent images through computer technology. The role transparency played in the Calabi-Yau image was to dissolve the boundaries between the object and the surrounding space. Aesthetically this implies the transformation of properties of matter merged with space. In the case of the point cloud images, the transparency intensifies the depth dimension and sense of volume by showing the "hidden" rear surfaces through the front.
The art critic Jan Butterfield addresses issues of artists collaborating with science and technology in search of instability, transparency and the seeming immateriality of materials [14]. She traces this search back to the 1960s when artists such as James Turrell, Robert Irwin and Maria Nordman began to introduce their works of "Light and Space" to the art community. Butterfield's critical and supportive analysis of the development of immaterial aesthetics lends a background to the cross-disciplinary nature of our work.
Transparent Calabi-Yau manifold computer visualization model.
Conclusion
Compound curvatures and transparency have inherent properties that move "between" dimensions, yet they are still physical phenomena that can be experienced in the perceptual world as well as the virtual world. We suggest that current research in theoretical astroparticle physics [End Page 230] and development of the renewal of constructivist aesthetics have a great potential in working together because of their shared interest in aesthetic experience in the physical and the abstract world. Through such art-science collaborations we can open up the field of aesthetics to include questions from disciplines beyond the traditional fields of philosophy, art and design. In time aesthetic criteria can better reflect the conceptual and perceptual dynamics of our time and begin to play a more vital role in shaping our intellectual, emotional and physical environment.
Cheryl Akner Koler is a sculptor who presently is working on a doctoral thesis supervised by the Department of Architecture at Chalmers Institute of Technology in Gothenburg, Sweden, in collaboration with the Institute for Industrial Design (ID) at Konstfack University in Stockholm. She is a professor of theoretical and practical aesthetics at the Institute of ID at Konstfack, where she has developed a model for understanding the "evolution of form." Her book Three Dimensional Visual Analysis documents the use of geometric and organic aesthetical abstraction as an analytical tool for supporting innovative design processes. Presently she is working with Lars Bergström and Narendra Yamdagni on a project titled "Cross Disciplinary Study of Complexity and Transformation."
Lars Bergström is a professor of theoretical physics, specializing in astroparticle physics, at Stockholm University. He is involved in several international research projects, such as AMANDA and GLAST, the Gamma Ray Large Area Space Telescope, which is a collaborative project between researchers in the U.S.A., Italy, France, Japan and Sweden. Bergström is one of the first astrophysicists in Sweden and has played a major role in describing the theoretical aspects of this field. He is also the scientific secretary of the Nobel Committee for Physics.
References
Bibliography
Glossary
a theoretical model that concerns the geometry of the universe. Eugenio Calabi and Shing-Tung Yau developed the theory for this 10D space/form concept, which has greatly influenced string theory. The main application of Calabi-Yau spaces is in theoretical physics, which is presented in purely mathematical terms. There are also attempts to visualize this theory through computer-generated models that are referred to as Calabi-Yau manifolds or Calabi-Yau varieties.
the generalization of the concept of a surface to more than 3D. Various aspects of a hypersurface can be visualized by projecting down to two or three dimensions.
a form that can be made by taking a rectangular strip of paper and joining the two ends of the strip together so that it has a 180° twist. This twist brings out the paradoxical property of a twisted surface with only one surface and one edge. The Möbius strip was discovered in 1858 by August Möbius.
a digital technology based on scanning physical models and translating the information derived from the surface of the model to points in a virtual spatial matrix. Each point correlates directly to a coordinate on the model. The digital image looks like a white cloud. [End Page 231]
