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Human creativity relies to a large part on our ability to recognize and match patterns, to transpose these patterns into different domains, and to find analogies in new domains to known facts in old domains. In the realm of geometrical proofs and geometrical art, such analogies can carry concepts and methods from spaces that are easy to deal with,...
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... tightly intertwined rings form a simple yet intriguing configuration that seems to have symbolic meaning in several cultures (Fig.5a). An attempt to place three loops in space as compactly and as symmet- rically as possible, leads to an arrangement known as the Borromean rings (Fig.1a, Fig.5b). ...
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... tightly intertwined rings form a simple yet intriguing configuration that seems to have symbolic meaning in several cultures (Fig.5a). An attempt to place three loops in space as compactly and as symmet- rically as possible, leads to an arrangement known as the Borromean rings (Fig.1a, Fig.5b). Individual pairs of rings are not actually interlocked; the configuration only holds together when all three rings are present. However, when we try to place three perfectly toroidal rings into a constellation of high symmetry, the result is a pairwise interlocking configuration with three-fold symmetry (Fig.5c). Next, we aim to ...
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... pairs of rings are not actually interlocked; the configuration only holds together when all three rings are present. However, when we try to place three perfectly toroidal rings into a constellation of high symmetry, the result is a pairwise interlocking configuration with three-fold symmetry (Fig.5c). Next, we aim to cluster more than three rings around the origin, without much concern whether indi- vidual pairs of rings mutually interlock. ...
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... boundary constraints to guarantee smooth continuation of the patches are not hard to derive; and the inner control points of the cubic patches are adjusted to minimize any apparent bumps. Finally, Jane Yen added highlights and shadows and rendered that surface (Fig.15a) with the Blue-Moon Rendering Tools [4]. ...
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Relational reasoning ability relies upon by both cognitive and social factors. We compared analogical reasoning performance in healthy controls (HC) to performance in individuals with Autism Spectrum Disorder (ASD), and individuals with schizophrenia (SZ). The experimental task required participants to find correspondences between drawings of scene...
Citations
... Spiral-like curves are a common occurrence in nature. They appear in various forms in a wide range of living beings and processes; e.g., mollusk shells, hurricanes, or galaxies (Cook, 1979; Séquin, 1999). Depending on the form of these curves, the Yin-Yang symbol can take on different shapes. ...
... Other methods to develop forms have been mathematically based. These include fractals (Yessios [10]), space curves (Krawczyk [3]), and complex surfaces (Sequin [8,9]). Still others have investigated agent based autonomous systems which are capable of collaborating, building, degenerating, and transforming, using genetic algorithms, Krause [2]. ...
With the increase use of algorithms to develop images, as well as, three-dimensional sculptural and architectural forms, additional focus should be placed on methods to teach students how to approach such a design technique. This paper reviews one such method. Architectural students are looking for inspiration for new forms to better realize their own designs and computer methods are offering them another tool to expand their investigations. This paper outlines one method used as part of a programming course developed for architectural students. The method highlights a very sequential approach to form investigation in using a common starting geometry. This approach stresses the development of rules and evaluating their results as a method to determine the next step to investigate. Equal importance is placed on the anticipated, as well as, the unexpected. 1. Introduction As commercial Computer-Aided Drafting and Design systems have grown in power to handle both twodimensional ...
In the 1970s and 1980s, Alan Holden described symmetric arrangements of linked polygons that he called regular polylinks and constructed many cardboard and stick models. The fundamental geometric idea of symmetrically rotating and translating the faces of a Platonic solid is applicable to both sculpture and puzzles. The insight has been independently discovered or adapted by others, but the concept has not been widely used because no closed-form method is known for calculating the dimensions of snugly fitting parts. This paper describes a software tool for the design and visualization of these forms that allows the dimensions to be determined. The software also outputs geometry description files for solid freeform fabrication and image files for printing paper templates. Paper templates make it easy to teach the concepts in a hands-on manner. Examples and variations are presented in the form of computer images, paper, wood, metal, and solid freeform fabrication models.
