Γενικό Τμήμα, Πολυτεχνική Σχολή, ΑΠΘ Ευθειογενείς Επιφάνειες Θαν. Κεχαγιάς Γενικό Τμήμα, Πολυτεχνική Σχολή, ΑΠΘ

Ταινία Moebius

4πλη Ταινία Moebius

(Ανάλογο σε 3 διαστάσεις) Φιάλη Klein (Ανάλογο σε 3 διαστάσεις)

Moebius House

Moebius House

http://courses. arch. hku http://courses.arch.hku.hk/ComGraphics/01-02/themes/topological/3_10_2001presentation_files http://ftp.pku.edu.cn/www.rice.edu/www.arch.rice.edu/working/housearrest/felicity.htm http://courses.arch.hku.hk/ComGraphics/01-02/themes/

Uncertainty with respect to positional theory and hierarchy characterizes the graphics system as well as many of the projects. There's too much preoccupation with the neoclassical proclivity to occupy the center, to bisect it, or fill it -- Le Corbusier thought of solid and void as "full" and "empty" -- versus the Roman and modern proclivity to define it or empty it (like a Rothko or Jefferson's Lawn) or counter-pose it or deny it or recomplicate it or simply downplay it (like UN Studio/Van Berkel & Bos in their Mobius House in this show). Many of these projects could benefit from the advanced researches into form and space by the celebrated twentieth-century painters upstairs in the same museum, whose paintings teach remarkable lessons about the center and countless related fundamental problems in the twentieth-century chess-game of positional -- juxtapositional, repositional, subpositional, superpositional, macropositional, micropositional, oppositional, counterpositional -- structures. http://www.thearchitectpainter.com/MadisonGray/deep_SIGHT/reviews/moma_KnightMoves.htm

1997 - Toolo Open Arena http://www.ocean-north.net/architecture/toolo/01.html

Ταινία Moebius

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In fresh H2O eXPO architecture and installation act as an integrated structure in order to get the best of both worlds. On the one hand the immersive capacities of architecture, and on the other hand the possiblity of informing the public about the latest policies of Dutch water management. The commision was accompanied by two related data; to follow the hydrological cycle as a conceptual guideline and to have a building consisting of two interconnected parts, a fresh water part and a salt water part, designed by two different architects, NOX and Kas Oosterhuis. http://ihong.hypermart.net/meta.htm

Sagrada Familia by Antoni Gauda The Sagrada Familia has not yet completed before Gaudi’s death in 1926, but the construction work had continued according to his proposals. However, detail drawings and models had been destroyed during the Civil War(1936-39). His successors had finding difficulties in following Gaudi’s design just according his sketches. Parametric design has been used to interpret part of a relatively small, but elaborate, unrealized design by Antoni Gaudi. In this case study, parametric design will be shown to have assisted the search for an effective solution to design problem. This evidence will be used to support the potential of parametric design as a valuable intermediary in a range of architectural design issues.

The curvy plastic morphology of the columns do not give any obvious sense of their composition. Therefore, a selection of ruled surfaces (hyperbolic paraboloids) are joined at their seams, which a lines, not curves. The individual column is comprised of four full surfaces and a further four which are bisected by a plane which in reality the triforium.

The explicit statements were made into parametric inputs in the 3D modeling software from ComputerVision’s CADDS5. The difference between the new method and conventional AUTOCAD modeling is that the model in CADDS5 is governed by the parameters shown as intersections between entities, dimensions or configurations such as four equally spaced. As each notation or number is selection, a calculation appears in which a new value or arrangement is typed, and the model regenerates itself attempting to accommodate the changes. If there is a solution, the user is asked to accept or to return to the unmodified version. If there is not, the user is informed at what point the model became impossible to reconfigure, providing an insight intor the limits of the various parametric changes. Experimental configurations, tested iteratively, finally produced the desired result which can restore Gaudi’s sense of beauty towards the Sagrada Familia.

Κεντρικές Ιδέες Ολα αυτά γίνονται με καμπύλες και γραμμές Μπορούμε να δουλέψουμε με βασική αναλυτική γεωμετρία ‘Η να θεωρήσουμε τις γραμμές ως σημεία ενός γενικευμένου χώρου Χρήσιμη είναι και η παραστατική γεωμετρία (γιατι?) Προβλήματα Προσέγγιση μιας γενικής επιφάνειας από ευθειογενή επιφάνεια Προσέγγιση ενός «νέφους» σημείων από ευθειογενή επιφάνεια Εφαρμογές σε morphing Computer Art με χρήση ευθειογενών επιφανειών (?)

Θεωρία Αναλυτική και Διαφορική Γεωμετρία Γεωμετρία των Γραμμών (Line Geometry) Προβολική / Παραστατική Γεωμετρία Computer Graphics, Animation, Morphing ... Πηγές Computational Line Geometry (Pottman+Wallner, Springer, 2000) Computer Aided Design (Περιοδικό) Computer Aided Geometric Design (Περιοδικό) J. of Visualization and Computer Animation(Περιοδικό)

M.C. Escher, Mobius II, 1931 http://www.mcescher.com/ http://users.erols.com/ziring/escher.htm http://www.worldofescher.com/

M.C. Escher, Convex and Concave, 1955

M.C. Escher, Relativity, 1953