The focus of this paper is the Vesica Piscis, a symbol made from the intersection of two circles of the same radius and where the centre of each circle lies on the circumference of the other. The origin of the Vesica Piscis is uncertain, but it can be found in different cultures throughout many historical periods. The Christian religion was most likely responsible for its spread, first as a fish symbol, then as an architectural niche surrounding sculptures and drawings of Christ, and finally as the Gothic pointed arch. A related geometric construction is the Reuleaux Triangle, which uses three intersected circumferences. In the second half of the twentieth century several architects rediscovered both types of geometrical constructions, producing variations of each. This paper commences with an overview of the history and construction of these geometric forms, and then analyses existing buildings which use them, before discussing different design strategies to develop new mathematical models based on ancient designs. Keywords Vesica piscis Á Reuleaux triangle Á Circle intersection Á Structural expressionism Vesica Piscis The Vesica Piscis is a geometric composition formed by the intersection of two circles with the same radius, intersecting in such a way that the centre of each circle lies on the circumference of the other. This geometric form can be expanded to
Abstract. Almost all industrialized materials commonly employed at engineering and building construction are approximately unstretchable, metal sheets, plywood or glass. So there is no doubt about the advantages offered, mainly from the economic point of view, by the processes that use flat or developable surfaces in the resolution of doubly curved ones. Using two prototypical kinds of developable surfaces, an adaptation method of double curvature surfaces is formulated using either developable strips or planar quadrilateral surfaces. Through the geometric concept of apparent contours and by the systematization of a process inspired by traditional projective geometry an algorithm is built, using the newest and outstanding CAD processes. They systematize and allow obtaining single-curvature strips or flat facets, in order to be able to address its construction, using materials that can be bent in one direction or rigid material with no possibility of being bent at all by simple and economical procedures. These strips obtained are from the geometrical viewpoint absolutely developable. They are patches extracted from cones or cylinders. The main subject to be developed in this paper is on one hand showing of the system and its geometric basis and on the other the exhibition of the results that are being obtained over physical models and prototypes built in diverse materials.
La investigación que se presenta forma parte de la exploración de las posibilidades que ofrecen las superficies desarrollables en la construcción de Arquitectura enfocadas principalmente a la fabricación digital. Este tipo de superficies han sido profusamente tratadas en los últimos años, asociadas tanto a la búsqueda de sistemas de discretización de superficies libres y formas complejas con facetas planas o tiras desarrollables como a la utilización de estas formas geométricas como elemento de composición proyectual. Esta asociación de geometría y construcción no es nueva y se puede encontrar en los propios fundamentos que llevaron a la formulación de la Geometría Descriptiva como herramienta aplicada a la construcción. A través de una profunda revisión de la geometría tradicional pre-computacional basada en la Geometría Descriptiva clásica, se ha procedido a reinterpretar los procedimientos gráficos tradicionales. Mediante la instrumentalización a través de los nuevos procesos computacionales de trabajo es posible alcanzar niveles de desarrollo que previamente quedaban meramente planteados a nivel de formulación teórica. Concretamente se desarrolla y reformula la idea de las superficies desarrollables que se apoyan en dos directrices en el espacio, es decir, es decir tiras desarrollables,que son conocidas en la bibliografía tradicional española como superficies convolutas.
The use of geodesic curves of surfaces has enormous potential in architecture due to their multiple properties and easy geometric control using digital graphic tools. Among their numerous properties, the geodesic curves of a surface are the paths along which straight strips can be placed tangentially to the surface. On this basis, a graphical method is proposed to discretize surfaces using straight strips, which optimizes material consumption since rectangular straight strips take advantage of 100% of the material in the cutting process. The contribution of the article consists of presenting the geometric constraints that characterize this type of panelling from the idea of "rectifying surface", considering the material inextensible. Experimental prototypes that have been part of the research are also described and the final theoretical results are presented.
Reciprocal structures are constructed by using a wide variety of patterns. These designs are a good source of inspiration when working with laminar constructions (sheets). Using the same formal schemes, laminar constructions feature better bending of their elements along the shape of the model and the application of extra pressure on the connection joints. Many of the geometric constructions made at the University of the Basque Country and presented in this paper show structural, constructive and formal improvements in many reciprocal structures assembled using sheets instead of linear elements. Keywords Reciprocal frame structures Á Leonardo grids Á Laminar constructions Á Complex shapes Á Developable surfaces Reciprocal Structures Although reciprocal structures have probably been known since antiquity, the first detailed sketches are found in the folio 899v of Leonardo da Vinci's Codex Atlanticus (Williams 2008). In recent years the scientific community has begun to consider these structures as something more than a historical curiosity. Some scholars attribute to Villard de Honnecourt the first drawing of a reciprocal
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