Graph Rotation Systems for Physical Construction of Large Structures

In this dissertation, I present an approach for physical construction of large structures. The approach is based on the graph rotation system framework. I propose two kinds of physical structures to represent the shape of design models. I have developed techniques to generate developable panels from any input polygonal mesh, which can be easily assembled to get the shape of the input polygonal mesh. The first structure is called plain woven structures. I have developed the ?projection method? to convert mathematical weaving cycles on any given polygonal mesh to developable strip panels. The width of weaving strips varies so that the surface of the input model can be covered almost completely. When these strip panels are assembled together, resulting shape resembles to a weaving in 3-space. The second structure is called band decomposition structures. I have developed a method to convert any given polygonal mesh into star-like developable elements, which we call vertex panels. Assembling vertex panels results in band decomposition structures. These band decomposition structures correspond to 2D-thickening of graphs embedded on surfaces. These band decompositions are contractible to their original graph. In a 2D-thickening, each vertex thickens to a polygon and each edge thickens to a band. Within the resulting band decomposition, each polygon corresponds to a vertex and each band corresponds to an edge that connects two vertex polygons. Since the approach is based on graph rotation system framework, the two structures do not have restrictions on design models. The input mesh can be of any genus. The faces in the input mesh can be triangle, quadrilateral, and any polygon. The advantages of this kind of large physical structure construction are low-cost material and prefabrication, easy assemble. Our techniques take the digital fabrication in a new direction and create complex and organic 3D forms. Along the theme of architecture this research has great implication for structure design and makes the more difficult task of construction techniques easier to understand for the fabricator. It has implications to the sculpture world as well as architecture

Multi-Panel Unfolding with Physical Mesh Data Structures

In this thesis, I demonstrate that existing mesh data structures in computer graphics can be used to categorize and construct physical polygonal models. In this work, I present several methods based on mesh data structures for transforming 3D polygonal meshes into developable multi-panels that can be used in physical construction. Using mesh data structures, I developed a system which provides a variety of construction methods. In order to demonstrate that mesh data structures can be used to categorize and construct physical polygonal models, this system visualizes the mathematical theory and generates developable multi-panels that can be printed and assembled to shapes similar to original virtual shapes. The mesh data structures include ones that are orientable: Quad-Edge, Half-Edge, Winged-Edge; and also one that is non-orientable: Extended GRS. The advantages of using mesh data structures as guides for physical construction include: There is no restriction on input design model as long as it is manifold, it can be of any genus with n-sided polygon faces; Different mesh data structures provide more options to better fit the input design while taking the physical constraints and material properties in consideration; Developable panels are easy to obtain from thin planar materials using a laser-cutter; When we use mesh data structures, it is also intuitive to assemble such planar panels using mesh information. Laser-cut developable panels based on mesh data structures provide, therefore, a cost-efficient alternative to 3D printing when dealing with large structures

A Topological Theory of Weaving and Its Applications in Computer Graphics

Recent advances in the computer graphics of woven images on surfaces in 3-space motivate the development of weavings for arbitrary genus surfaces. We present herein a general framework for weaving structures on general surfaces in 3-space, and through it, we demonstrate how weavings on such surfaces are inducible from connected graph imbeddings on the same surfaces. The necessary and sufficient conditions to identify the inducible weavings in our framework are also given. For low genus surfaces, like plane and torus, we extend our framework to the weavings which are inducible from disconnected imbedded graphs. In particular, we show all weavings on a plane are inducible in our framework, including most Celtic Knots. Moreover, we study different weaving structures on general surfaces in 3-space based on our framework. We show that any weaving inducible in our framework can be converted into an alternating weaving by appropriately changing the strand orders at some crossings. By applying a topological surgery operation, called doubling operation, we can refine a weaving or convert certain non-twillable weavings into twillable weavings on the same surfaces. Interestingly, two important subdivision algorithms on graphs imbeddings, the Catmull-Clark and Doo-Sabin algorithms, correspond nicely to our doubling operation on induced weavings. Another technique we used in studying weaving structures is repetitive patterns. A weaving that can be converted into a twillable weaving by our doubling operation has a highly-symmetric structure, which consists of only two repetitive patterns. An extension of the symmetric structure leads to Quad-Pattern Coverable meshes, which can be seamlessly covered with only one periodic pattern. Both of these two topological structures can be represented with simple Permutation Voltage graphs. A considerable advantage of our model is that it is topological. This permits the graphic designer to superimpose strand colors and geometric attributes — distances, angles, and curvatures — that conform to manufacturing or artistic criteria. We also give a software example for plane weaving construction. A benefit of the software is that it supports plane weaving reconstructions from an image of a plane weaving, which could be useful for recording and modifying existing weavings in real life

Satin Non-Woven Fabrics for Designing of Self-Regulating Breathable Building Skins

In this paper, we introduce the concept of 2-way 2-fold genus-1 non-woven fabrics that can be used to design self-regulating breathable building skins. The advantage of non-woven structures over woven structures for breathable skin design is that they can completely be closed to stop air exchange. We have developed a theoretical framework for such non-woven structures starting from the mathematical theory of biaxial 2-fold Genus-1 woven fabrics. By re-purposing a mathematical notation that is used to describe 2-fold 2-way 2-fold genus-1 woven fabrics, we identify and classify non-woven fabrics. Within this classification, we have identified a special subset that corresponds to satin woven fabrics and allows for maximum air exchange. Any other subset of non-woven structures that correspond to other classical 2-way 2-fold genus-1 fabrics, such as plain or twill, will allow for less air exchange. We also show that there exists another subset of satin non-woven fabrics that can provide the biggest openings.Comment: 10 page

Low-dimensional Topology and Number Theory

The workshop brought together topologists and number theorists with the intent of exploring the many tantalizing connections between these areas

Matter in Loop Quantum Gravity

Loop quantum gravity, a non-perturbative and manifestly background free, quantum theory of gravity implies that at the kinematical level the spatial geometry is discrete in a specific sense. The spirit of background independence also requires a non-standard quantum representation of matter. While loop quantization of standard model fields has been proposed, detail study of its implications is not yet available. This review aims to survey the various efforts and results

トポロジー的織込み構造の構成と分類

Tohoku University小谷元子課

Mathematical Simulations in Topology and Their Role in Mathematics Education

This thesis presents and discusses several software projects related to the learning of mathematics in general and topological concepts in particular, collecting the results from several publications in this field. It approaches mathematics education by construction of mathematical learning environments, which can be used for the learning of mathematics, as well as by contributing insights gained during the development and use of these learning environments. It should be noted that the presented software environments were not built for the use in schools or other settings, but to provide proofs of concepts and to act as a basis for research into mathematics and its education and communication. The first developed and analyzed environment is Ariadne, a software for the interactive visualization of dots, paths, and homotopies of paths. Ariadne is used as an example of a “mathematical simulation”, capable of supporting argumentation in a way that may be characterized as proving. The software was extended from two to three dimensions, making possible the investigation of two-dimensional manifolds, such as the torus or the sphere, using virtual reality. Another extension, KnotPortal, enables the exploration of three-dimensional manifolds represented as branched covers of knots, after an idea by Bill Thurston to portray these branched covers of knots as knotted portals between worlds. This software was the motivation for and was used in an investigation into embodied mathematics learning, as this virtual reality environment challenges users to determine the structure of the covering by moving their body. Also presented are some unpublished projects that were not completed during the doctorate. This includes work on concept images in topology as well as software for various purposes. One such software was intended for the construction of closed orientable surfaces, while another was focused on the interactive visualization of the uniformization theorem. The thesis concludes with a meta-discussion on the role of design in mathematics education research. While design plays an important role in mathematics education, designing seems to not to be recognized as research in itself, but only as part of theory building or, in most cases, an empirical study. The presented argumentation challenges this view and points out the dangers and obstacles involved