Architectural Geometry

This paper discusses results that arise in specific configurations pertaining to invariance under isoconjugation. The results lead to revolutionary theorems and crucial properties in both Euclidean and Projective geometry. After discussion of important theorems and properties of associated configurations, the authors present and prove an important, new result and its application in difficult geometrical configurations.

Tartaria (originally pronounced “Tataria” without the first “r”) is the name of the pre Mongolian empire that originated in northern Asia before spanning the entire northern hemisphere. Great Tartaria was the largest empire during its time and would have still been the largest empire today. The Tartarian empire flourished due in part of the civilization being a leader in advanced technology, free energy, and grand architecture.
Tartaria is also where the Greek word “Tartarus” originated from.
The denotation of lost souls spending eternity in Tartarus the underworld is due to the Tartarian Empire having been buried and wiped out during the mud flood. The world of the Tartarians is literally the world under our world. The underworld. Tartarus wasn’t the underworld and will never be the underworld. Tartaria is the underworld. The religionists used the Mud Flood deluge and how it wiped out the Tartarian Empire to confuse the populous into thinking they would join those in the “underworld” if they challenged the religious authority.
Who were the Tartarians?
The Tartarians (or Tartars) were the indigenous people (quite possibly founded by Noah’s sons) making up the world renowned Tartarian empire. The Tartarians were a tall people, averaging some eight to twelve feet in height. They would have been considered giants to our current average height of only around six feet; however, back then the height of ten feet was the average. Like the civilizations before them whose heights averaged twelve feet, fifty feet, two+ miles, etc…each preceding civilization had an average taller height than the civilizations coming after them and each succeeding civilization had an average shorter height than the civilizations predating them. Statures are diminishing after each deluge and with each new astrological age we enter.
Civilizations in the current astrological age (the Age of Pisces) are shorter in height than civilizations that existed in the previous astrological age (the Age of Aries) and will be taller in height than civilizations that will exist in the next astrological age (the Age of Aquarius).
The Tartarians are thought to have been “Breatherians,” a being who does not rely on the digestion and burning of calories from food/water, but instead rather receives energy straight from the aether. The aether is thought to be the very fabric of the space time continuum that some would associate with electrons, the wind, the holy spirit, the atmosphere, and the gasses in the atmosphere such as oxygen, nitrogen, and hydrogen. Since the The Tartarians quite possibly had a completely altered digestion system than the one we now have, they had no need for toilets in their bathrooms. Which could hint at one of the reasons why bathrooms in the recent past have been used quite extensively as social gathering powder rooms to not only freshen oneself up but to hear and spread local gossip.
The Tartarians were masters of masonry, brickwork, steam punk style technology, universally free energy, and grand architecture. Their Roman/Gothic esque style architecture can still be found in the current guise as water ducts, city halls, banks, water stations, cathedrals, churches, hospitals and similar public and city works type buildings.

Turnstile is a digital artwork that translates the traces of pedestrians on a public square into a virtual geometric architecture. On the front wall of the Schadowstraße underground station in Düsseldorf, visitors can observe pedestrians within this generative architecture on an LED wall. The installation can be seen as the outcome of an artistic research process investigating swarm behaviour as a manifestation of collective togetherness.

The work presented below is part of the results of a research project at UPM.
The objective of the research consisted in the reconstruction of a selection of architectural projects chosen for both, its importance in the historical context and its geometry; so that the development of non-uniform rational Bsplines (NURBS) could be implemented as a tool for the graphical representation of architecture.

NURBS can represent complex geometries with precision (such as the hyperbolic paraboloid), either in the form of models or virtual models. Thus one can hypothesize and even complete information gaps -in the absence or loss of documentation-, of old or not constructed architectural projects.
During the communication it is intended to show the results achieved in one of the specific cases analyzed: the non-built market for Mexico in 1956, by Félix Candela.

The product of decades of teaching and research, this unique and inspirational design resource presents a history of the intimate relationships between geometry, culture and design throughout human history, from the Neolithic period through the Indian, Egyptian, Babylonian, Chinese, Greek, Celtic, Islamic, pre-Columbian and Renaissance cultures, to the present and the possible future. Explaining key principles that can be applied across all design disciplines, Roger Burrows reveals fresh insights into how geometry as a visual language has evolved to meet our needs, initiated new technologies, solved problems and changed the way we think about the world around us.

The fabrication and construction of curved beams along freeform skins pose many challenges related to their individual and complex geometry. One strategy to simplify the fabrication process uses elastic deformation to construct curved beams from flat elements. Controlling the curvature of the design surface and beams has the additional potential to create repetitive building parts with beneficial beam orientation. We aim for strained gridshells built entirely from straight or circular lamellas of the same radius and with orthogonal nodes. The lamellas are aligned normal to a reference surface enabling an elastic assembly via their weak axis and a local transfer of loads via their strong axis. We show that the corresponding reference surfaces are of constant mean curvature and that the network of beams bisects principal curvature directions. We introduce a new discretization of these networks as quadrilateral meshes with spherical vertex stars and present a computational workflow for the design of such structures. The geometric advantages of these networks were key for the fabrication and assembly of a prototype structure, the Asymptotic Gridshell. We describe the complete process from design to construction, presenting further insights on the symbiosis of geometry, fabrication and load-bearing behavior.

Claude Perrault defined stereotomy as “the art of using the weight of stone against itself so as to hold it up thanks to the very weight that pulls it down”. It is indeed a sophisticated ancient technique
highly integrated with knowledge of both practical and erudite geometry as well as mathematics. With knowledge being a defining factor of stereotomy, a few questions are raised as to its democratic
distribution and accessibility as well as the social groups that have controlled it in a chronological context. Taking to consideration the emergence of Géométrie descriptive by Gaspard Monge  shortly
before 1800, one would wonder about the very existence of complex stereotomic buildings dating before the 18th century, one such example being the Chateau d’Anet, thus questioning the historical factors behind its development, geographical movement and the intentions of the people being in possession of such knowledge.

This is the sixth of a series aimed at providing the existing details of the Madrasa al-Ashrafiyya built by Sultan Qaytbay in 887/1482 and referred to as the Third Jewel of al-Haram, the other jewels being the Qubbat as-Sakhra (Dome of the Rock) and the Al-Aqsa Mosque.  The madrasa although badly damaged by an earthquake in 1545  still contains many interesting details that directly relate it to Sultan Qaytbay’s Cairo foundations.

My intention is to illustrate the architectural evidence with minimal text partly because I have previously described and detailed many of the features , but, and perhaps more importantly, I believe that it should be the norm to make one’s research notes available to enable others to see the details and make their own comparisons and conclusions .

[ES] La cabecera de la iglesia de Santa María del Salvador de Chinchilla, construida a mediados del siglo XVI por Jerónimo Quijano, maestro mayor del Obispado de Cartagena, ha despertado un gran interés como pieza clave del Renacimiento en el sur de España. Consta de dos partes: un ábside cubierto por una bóveda de horno avenerada, y un crucero de planta rectangular que lo precede. Frente a las bóvedas ovaladas o tóricas empleadas en otras ocasiones por Quijano, aquí parte de una imposta definida por un rectángulo central rematado por dos semicircunferencias en sus extremos, que se cubre mediante una bóveda de cañón rematada por dos rectángulos en los extremos. Dado que los tratados y manuscritos del período no exponen con detalle la solución completa, nos  proponemos abordar el análisis geométrico del conjunto a partir de un levantamiento riguroso llevado a cabo mediante fotogrametría de imágenes cruzadas para pasar después a su estudio constructivo y la comparación con las soluciones parciales del manuscrito de Alonso de Vandelvira. El dibujo a línea desempeñarán un papel central en toda la investigación, desde el levantamiento hasta la comparación con el texto vandelviriano; de esta manera, esperamos mostrar además el notable potencial de los medios gráficos en la investigación en Historia de la Construcción
[EN] The ensemble of the presbytery and crossing of the church of Our Lady in Chinchilla, built in the mid sixteenth century by Jerónimo Quijano, master mason of the bishopric of Cartagena, has fostered a remarkable interest as a key piece of Renaissance architecture in Southern Spain. It consists of two parts: an apse covered by a halfdome, and a rectangular crossing before it. In the crossing, instead of employing oval domes or torus vaults, as in previous works by Quijano, he used a plan defined by a central rectangle with two semicircles at the ends, which is covered by a combination of a barrel vault and two half-domes. Since treatises and manuscripts of the period do not discuss in detail the complete solution, in this paper we present out a geometrical analysis of the whole through a rigorous survey carried out using crossed-images photogrammetry. This survey will furnish a solid foundation for the study of the vault construction, confronting it with the partial solutions shown in Alonso de Vandelvira’s manuscript. Line drawing will play a central role throughout the study, from the survey to the comparison with Vandelvira’s text; in this way, we hope also to show the significant potential of graphic media as a research resource in the field of Construction History.

structural architectures
geometry, code and design I
Erasmus Intensive Programme in Milano 2011
Technische Universität Kaiserslautern, Germany 2012
ISBN 978-3-941438-84-2

The international summer school “Structural Architectures“, funded by the European Commission in the frame of Erasmus Intensive Programmes, focused on structural thinking in architecture and its application in architectural design processes.
The background of this approach was the Symposium ‘Structural Thinking in Architecture’ at the University of Kaiserslautern, Germany in June 2010, where the partner universities of the summer school started their discourse on this research topic together with other architects and researchers.
A critical analyses of structuralist concepts was related to actual methods of parametric and rule-based design with digital tools. Based on this shared research on structural thinking in architecture the idea grew to develop a summer school for students during which the different aspects could be experienced, related to each other and applied to architectural projects.

This paper reflects on the role of computation in speculative design. It suggests that found, unexpected traces of computational processes can amplify designers’ imagination. This theme is considered through a reflection on a practical workflow that pays close attention to the artifacts of algorithmically generated mesh geometries. The resulting interpretation of found artifacts as active participants in design processes is innovative in the field where computational objects (such as meshes) are typically thought of as neutral tools. Reconsideration of meshes as objects with agency can be extended to other computational entities, resulting in significant implications for design thinking and design craftsmanship.

The purpose of this paper is to outline an innovative modular housing system based on geometric regularities, which responds to the issues raised by urbanization, the principles of environmental awareness, and the model of a sustainable city. The starting point of the analysis of wood-based systems is analysing the variations of elements obtained by repeating and assembling regular flat joints for filling the planar and the spatial. The following step is to model real spatial situations. According to the principal idea, geometric principle based logic games provide a pattern for designing modular housing units or systems. Although inherent in the classification and variation of the same elements, the system has a high degree of freedom and flexibility due to its nature. Multidimensional vertical spaces provide the opportunity to maximize the space given, while the openness and the flexibility of the system, the current character of the city, neighbourhood features remain-moreover, its implementation is flexible, adaptive, and customizable.

Written for architectural educators, this paper discusses whether digital fabrication can be usefully employed in early architectural education. The paper uses examples from a course that aims to introduce the fundamentals of digital architectural design to first-year students. To achieve this, the course integrates digital fabrication as the core element of the production workflow. Challenging but rewarding, early adoption of digital fabrication exposes students to the process- and material-based thinking of contemporary architecture at a time when they form lasting attitudes to designing.

'Herring bone' is a procedure of construction, already known to Brunelleschi and to the Sangallo, which enables the dome to be vaulted without the use of reinforcements. This technique was adopted to cover the octagonal rooms surrounding the dome of St. Peter's. Assisted by analytical photogrammetric techniques and computer simulation of the constructive process, the Authors show that the 'herring bone' construction in these domes is achieved with the simple aid of one main support (namely a spar or a cable fixed at the centre of the dome) and without any other guide tracings. The curve thus described by the bricks of the 'herring bone' on the intrados of the dome is a spiral with equal angles (or rhumb line) similar to the spirals found in nature in mollusc shells.

Choir and tower of Dominican church in Košice (Kaschau, Kassa) in Slovakia preserved a rare collection of medieval stonemasonry elements. Capitals, consoles, baldachins and a boss with figural, zoomorphic, floral and architectural motives come from two style epochs: from the end of 13th century and the end of 14th century. Building unity of church, proved by recent sounding research, and predominance of younger elements (14 of all 19 preserved) forces us to regard the minority older elements as spolia. Dating of church to 13th century by previous scholarship is disproved in addition by recent finds of younger architectural components in situ, not known earlier (south portal, choir rose-window), and iconographic analyses of stonemasonry fill, never carried out before.
A headstone of analyses is a capital describing motif of "Domini canes". First art depiction of this self-reflection of Dominicans as stalwart "dogs of Lord" originated in Florence in the years 1366-8, what is terminus a quo for its appearance in Košice. Church choir and tower-base arcade, serving as a portico for friars entering the choir, generate an ideological unit whereby special position of its every element. Spolia, incorporated within the most sacramental east part of interior, remained not only passive antiquities, but were adapted and arranged to depict the sanctification of the church and whole monastery – Blessing of Virgin Mary.
Paper analyzes all preserved elements of decorative program, explained as a stone display of Dominican order dogma, produced by artistic resources of late 14th century Košice.

The work presented below is part of the results of a research project at "Universidad Politecnica de Madrid". In this paper we analyze the project for Murcia airport, which was done by Fernando Higueras under the guidance of Felix Candela. Some of the unpublished original documents of the project are included in the paper. The source of them are the Official Colleges of Architects of Madrid and Catalufia. In addition, the authors of the paper have done a 3D reconstruction that allowed to analyze the airport. The project is compounded by hyperbolic-paraboloid surfaces, such as those that Candela developed during the 50s. If it had been built, it would have become the largest cantilever done by Felix Candela using the "umbrella" structure.

Previously titled "A Dynamic Geometry for Designers, Artists and Architects." This dynamic geometry provides a means to create golden ratio and other proportional two-and three-dimensional designs for applications in architecture, design and art. The geometry is dynamic and is based on circles or spheres that change size and position to form close-packing cells and clusters that fill space. Space filling lattices are generated from the cells and clusters by connecting contact points of the circles or spheres or by adding clipped tangent lines and planes at circle or sphere contact points. The lattices serve as 'ideation lattices' from which two-and three-dimensional forms can be extracted-of all sorts and of many types-limited only by the extent of one's imagination. The software so far used to generate the space filling lattices has been Rhino 3D and SolidWorks, so outputs can range from CNC cut components, or molds, to complete 3D printed, or 3D fused, forms. The author is currently developing a concept of micro and modular human habitats for high-density urban areas based on space-frames extracted from one of the geometry's lattices.

This paper offers a geometrical perspective on the design of Strasbourg Cathedral’s west façade block, and on the creation of the associated drawings, known as Strasbourg Plans A, B, and B1.  It makes several principal arguments:
 That Plan A was composed out of squares and equilateral triangles, much like the cross section of the cathedral’s nave, but in different combinations that resulted in slightly different proportions;
 That Plan B was composed principally of octagons based on the ground plan of the anticipated spires, which were already foreseen by its original author, even if his followers inked the spire zone;
 That the author of Plan B1 misread the intentions of the author of Plan B;
 That even the lowest levels of the current façade block were built in accord with a new design, rather than being based directly on Plan B;
 That the spire type in Plan B differs fundamentally from the openwork spire type seen at Freiburg Minster, and in the drawing known as Cologne Plan F.
On these bases, this paper argues against the notion that a single designer--even the widely praised but poorly documented Erwin von Steinbach--could have been responsible for Plan B, Plan B1, the current design of the Strasbourg façade block, and the invention of the openwork spire type.

Las reglas usadas por los arquitectos para diseñar el espacio se basan en modelos y prototipos que, con los años, se han adaptado para resolver problemas espaciales y funcionales específicos. El autor muestra que es posible proyectar edificios, incluso si se encuentran geográfica y cronológicamente distantes, con un solo modelo. Edificios similares en cuanto a composición, en los que pequeñas variaciones en el desarrollo conducen a resultados distintos. Este estudio continua las investigaciones de Michel Ecochard sobre edificios octogonales.

A number of essays and interviews published between 1997 and 2008, revised for this edition, which V2_Publishing kindly made available as a complete pdf. The book contains a memorable introduction by the late Detlef Mertins, a number of early essays that were strongly influenced by Paul Virilio, and many discussions of architectural and art works combined with theoretical analyses.

This research proposes a method to prefigure responsive models by investigating the form's spatial movement in the step from one configuration to another. Folded surfaces have been chosen for investigation. Contemporary architecture proposes buildings that are no longer simply juxtaposed volumes marked by light, i.e., surfaces drawn by shadows that change with the position of the Sun throughout the day. Today, buildings interact with the surrounding environment, changing the morphological characteristics of their exteriors in relation to various environmental parameters. Buildings change in relation to external conditions, but they also communicate via the possibilities expressed by a skin that shimmers in relation to the message to externalize. We are faced with responsive forms that the designer should control through design. The design of forms in movement is seen widely in the study of objects called " folded surfaces ". This area addresses the different means of prefigurement offered by the design used to initially investigate the layout, which is drawn according to canonical rules of symmetry, balance, and the ability to tessellate flat space. The next step is to translate the signs into mountain and valley folds, studying the different spatial confor-mations assumed by tiles of the folded surface rotating around the pattern lines. Spatial control of the model is ensured by geometrically controlling the basic entities that compose the folded surfaces: the surface/face of the pattern, the zipper lines between the faces, the point/vertex where the zipper lines meet. By applying the basic rules of descriptive geometry, an algorithm that controls the entire form can be constructed. This is the fundamental contribution of nodal programming tools for modelling support. Today almost all design programs include nodal networks, which allow the geometric algorithms used in the research to be specified and some variable dimensions to be extrapolated from the definition. Acting on the variables, it is possible to control the step from the folded surface of one conformation to another. In this way, the " fourth dimension " , the form's movement, can be anticipated and designed

The Ushtogai Square, a massive neolithic Kazahkstan geoglyph discovered in 2007, shows the same exact geometry and numbers as seen in the megalithic monuments at Carnac and Stonehenge. It is also perfectly orientated according to the cardinal directions.

144 pp. Presses universitaires de Franche-Comté – Geometria practica est le titre de nombreux traités mathématiques rédigés du Moyen Âge à la fin de l'époque classique. Branche peu étudiée par l'histoire des sciences et des techniques, à la marge des mathématiques savantes et des traditions techniques, la géométrie pratique pose des questions historiographiques et épistémologiques propre. Cet ouvrage collectif vise à élucider certains de ces problèmes comme le rapport à l'utilité, l'importance des solutions approchées ou la limite de la mathématisation. Il montre la place qui revient aux instruments de mesure, de relevé ou de tracé, et étudie les modalités de l’enseignement de la géométrie aux praticiens.
Avec les contributions de Samuel Gessner, Eberhard Knobloch, Jorge Galindo Diaz, Joël Sakarovitch et Dominique Raynaud.

One of the central issues of contemporary architecture is the possibility of combining the potentialities of industrialized production and the quality of the artisanal work. The process of systematizing architecture allows the coexistence and balance between these two principles. This concept has its antecedents throughout the history of architecture and, from a few decades, it extends to other components of everyday space, such as household appliances. In the present, there are several ways that explore the world of systematization of architecture, aligning with past episodes such as electrification and mechanization of it. An example of these is the work of Fran Silvestre Arquitectos that, thanks to a deep and meticulous study of the field of geometry, has developed its own particular systematization process through the concepts of seriation and modulation: some abstract geometries - the so-called types or systems - enable to generate a better distributions of the different parts and, furthermore, they establish a relationship between formal coherence, resistant structure and manifestation of the concept of order. So systematization involve the different instances of the project - from the ideation to the construction - with the aim of the achieve the efficient beauty.

"NEXUS NETWORK JOURNAL

Volume 7 Number 1 Orietta Pedemonte and Laura Tedeschini Lalli, Guest Editors, Spring 2005

The experience of teaching mathematics in schools of architecture attempts to bridge the gap that separates the humanistic and the scientific. STUDIO 1032's article, The Geometry of Frank Lloyd Wright, builds on the legacy of geometric patterning found throughout different periods of Wright's work with its basis in proportions from nature. This research, conducted over a decade at the University of Wisconsin Milwaukee was the groundwork for Wrightscape: The Geometry of Wright high school edition. "

This paper addresses the aesthetics of ’how’ a building is designed. Through the morphological analysis in 2-D and 3-D drawings, both relational and constructive, of the Smith House (1965-1967), the paper aims to read the intellectual form of the house within its projected form. This reading proceeds by reconstructing the logic of organization of the material construction, accordingly subdividing the space of the building into a pattern and generating its formal properties. As such, the paper argues that the aesthetics of architectural form can be read as an interaction between formal elements and abstract spatial motifs directed by design strategies and tactics, in the distinctive final form. The Smith House is chosen because it represents the beginnings of the formal vocabulary and formal logic of Richard Meier, a logic that endures throughout his later career. The paper concludes that through the tracing of ‘how’ the house was designed, a deeper aesthetic appreciation of the work emerges and furthermore, the development of a terminology and understanding of logical form not only serves as basis for communication, whether between teacher and pupil, architect and client, or critic and public, but can also be used as a didactic method for instructing design studios.

This paper investigates the aesthetics of architectural form by addressing ‘when’ this form becomes aesthetically distinctive. The theoretical foundation of this paper is based on Nelson Goodman’s exemplification, density and repleteness as necessary symptoms for any form to function as a work of art. The aim is to introduce this philosophical view to architecture in order to understand how the aesthetics of architectural forms can be inferred from these three symptoms. To pursue this aim, this paper looks at Richard Meier’s Barcelona Museum of Contemporary Art where besides its architectural function, the design of the museum with its formal language is considered a work of art in itself. Through morphological analysis and analytical drawings the three symptoms of aesthetics are investigated within the form of the building and identified. The paper concluded that architectural work is aesthetically pleasing when formal design language is syntactically dense, its content is semantically dense, and the design process is replete. The findings of this research encourage a rational understanding of the aesthetic judgment in architecture and provide a major asset in architectural pedagogy. Therefore, when students and instructors go through it, students’ right to develop their own aesthetics is preserved.

This dissertation aims to introduce the geometric figure of the Vesica Pisces and the Mandorla that derives from it. The Vesica is built with only two circumferences. This figure allows you to create geometric figures from three to twelve equal sides. The study was carried out for the purpose of revealing Vesica Pisces´ full potential and to unveil its use throughout history.
Research was conducted in two fields:  theoretical, which focuses on authors who refer to the figure of the Vesica Pisces directly or indirectly; and practical, where with the ruler and compass we expose exhaustively (but certainly not all) the possible constructions starting only with two circumferences or two arches. Namely the Vesica Pisces and the Mandorla.
Throughout several examples presented in this dissertation is corroborated the simplicity of Vesica's construction which leads to the compositional complexity of geometrical figures. This simple method of construction of more complex geometrical figures gets us closer to the artist's intuition and useful for its ease of implementation in the teaching compendium.

El presente trabajo aspira a ofrecer una serie de fundamentos matemáticos, geométricos y aritméticos, que se aplicaron y/o aplican, con la finalidad de llegar a la Aeuritmia. Además, se pretende mostrar la racionalidad de los arquitectos griegos al vincular matemática y filosofía con arquitectura, obteniendo excelentes resultados en los templos dóricos.
De ahí que la primera parte del trabajo se haya destinado a contextualizar brevemente la Grecia de los períodos arcaico, clásico y helenístico; cronológica y geográficamente. Ahondando más tarde en el ambiente intelectual de la época y algunos avances en el campo de estas ciencias. Finalmente se ha aportado, en esta investigación, una breve evolución del dórico reflejado en los templos, desde los primitivos lignarios a su práctica desaparición en el siglo II a.C; con el fin de crear una aproximación mental de la extensión del dórico geográficamente.

La segunda parte del presente estudio se ciñe a la presentación de los
principales refinamientos, correcciones o formas de construir de la Antigüedad. Es decir, la aplicación de una matemática imperfecta, sobre planos, de una matemática pitagórica para su posterior adaptación a la óptica humana; una visión imperfecta. Añadiendo una serie de elementos, sistemas de proporciones e ilusiones que hoy en día pueden aplicarse a la arquitectura griega. Llegándose a producir disputas entre autores que defienden que los arquitectos los conocían y realizaban a propósito; mientras que otros grupos niegan estas cuestiones, abriendo así, la veda de la casualidad.

Geometry lies at the core of the architectural design process. It is omnipresent, from the initial form-finding stages to the final construction.
Modern geometric computing provides a variety of tools for the efficient design, analysis, and manufacturing of complex shapes. On the one hand this opens up new horizons for architecture.
On the other hand, the architectural context also poses new problems to geometry. Around these problems the research area of Architectural Geometry is emerging. It is situated at the border of applied geometry and architecture.
A research area being so interdisciplinary as Architectural Geometry and involving fields as diverse as art and design, computer science, engineering and mathematics, needs scientific exchange.
Advances in Architectural Geometry 2008 is our first attempt in bringing together researchers from these fields to discuss recent advances in research and practice and to identify and address the most challenging problems. We aim at connecting researchers from architectural practices and academia.
AAG 2008 received 56 submissions which have been reviewed by the members of the Scientific Programme Committee. The specific cultures in the involved scientific communities often resulted in a high variation of grades the same paper received by reviewers from different areas. This made the selection process particularly difficult.
We decided to accept only a small number of papers for oral presentation, since a low number of presentations offers sufficient time for discussion, which is essential for this first conference on Architectural Geometry. At this point, we would also like to thank the members of the Scientific Programme Committee for their excellent and mostly timely reviews which had to be performed during a quite short time span. Extended abstracts of the 14 accepted oral presentations form the content of the proceedings. The copyrights remain with the authors and thus publication of the full version of the papers elsewhere poses no problems.
The variety of research areas and the balance between academia and architectural practice is also reflected in our choice of invited speakers: Konrad Polthier (FU Berlin), Carlo Séquin (UC Berkeley), Dennis R. Shelden (Gehry Technologies, LA), Charles Walker (Zaha Hadid Architects, London) and Arnold Walz (designtoproduction, Zürich). We are very grateful to them for accepting our invitation and for sharing their most recent research results.
This event would not have been possible without the help of many friends and colleagues. Special thanks belong to our conference secretary Natalie Klement for her help with all organizational issues and to our conference technician Ronald Haidvogl for his technology support. We greatly appreciate the important support in early organizational issues by Werner Purgathofer and Anita Mayerhofer.
Furthermore we would like to thank Martina Milletich from the Austrian Academy of Sciences for a very smooth handling of all our requests.
Last but not least we would like to express our sincere gratitude to the sponsors of this conference: RFR (Paris) and Waagner Biro Stahlbau AG (Vienna). Their generous support greatly reduced the registration fees and thus helped to attract many more, especially young people to this first conference on Architectural Geometry.

P r a c t i c a l a n D s y m b o l i c G e o m e t r y i n b o r r o m i n i ' s s a n c a r l o a l l e Q u at t r o F o n ta n e 557 Figure 2 Francesco Borromini, plan of San Carlo alle Quattro Fontane, Rome, early 1660s (Albertina Museum, Vienna, Graphische Sammlung, Az. Rom. 173)

Com este artigo pretendo dar a conhecer, de forma mais extensa, o vasto património disperso pelas paredes do Mosteiro de Santa Maria da Vitória, Batalha, vulgarmente chamado de grafitos históricos.
É examinada uma variedade de tipologias gravadas, incidindo nas que estão claramente associadas às traçarias, desenhos relacionados com a arquitectura. Esta análise permite uma nova reinterpretação dos espaços, usos e respectivas evoluções.

Lahore is among the major urban centers in Pakistan, with increased energy demands, that has resulted into frequent power blackouts. The growing population, economic disparities and competition, has further fueled the energy requirements and reliance on modern comforts [air-conditioners and heaters]. The standardized design layouts of modern houses suffer regional identity and continuity, that is typical of traditional residential settings rooted in time as they progressively evolved in response to cultural and climatic conditions of the city. This study employs the methodology of the "Deep Beauty Framework" [1] used by Coates that critically looks at architecture [in our case house types] on three levels: functional, typological and archetypal. The comparative analysis between a typical traditional courtyard house and a typical modern detached single-family house allows to understand the diametric conditions in design and planning of the two house-types. It also addresses the paradigm shifts driven by current socio-spatial and cultural conditions, establishing new trends for the future. We argue that the traditional courtyard house offers a valid proposition in the modern design and planning process. The design attributes of the traditional house instill a sense of place rooted in continuity with history, local/indigenous building materials and relies on passive heating and cooling techniques such as cross-ventilation, stack effect and evaporative cooling, that are legitimate propositions for a city that is frequently challenged by energy shortfalls till a permanent solution is achieved. Geometric patterns and arabesque design in the traditional house-types echo cultural continuity with spatial continuity that can be experienced throughout the house with prospect and refuge spaces [2] such as connected courtyards and verandas besides high ceilings for light and cooling. Courtyards in traditional house-types are known to engender social activities within the joint family system in more than one way. Conversely, the present-day detached single-family house in Lahore falls short in meeting the criteria established by the Deep Beauty Framework. The findings offer design opportunities to hybridize the unique spatial and physical characteristics of the traditional house with the spatial proclivities of the modern house for innovative designs that are more responsive to the current need, and to re-establish the continuity between the old and the new.

Scientific Life in Ancient Egyptian Civilization
"Historical and Analytical Study of Geometry, Mathematics, Astrology and Astronomy"
Historical Manifestations of Geometry and Mathematics in Ancient Egypt; Manifestations of Astrological Life in Ancient Egypt; Manifestations of Astronomical Life in Ancient Egypt; Linguistic Identification of Units of Time in Ancient Egyptian Language.
This book presents outlines of History of Sciences in Ancient Egyptian Civilization. The late Egyptian and Greek sources praised the ancient Egyptian cultural centers as a source for knowledge and science, thus ancient Egypt was a source of inspiration for the legislator 'Solon', as well as 'Thales of Miletus' who was a mathematician and astronomer, where He learned and practiced the sciences of ancient Egypt, then taking this knowledge to the Greeks. The First Chapter provides 'The Historical Manifestations of Geometry and Mathematics in Ancient Egypt'. The Second Chapter provides 'The Historical Manifestations of Astrological Life in Ancient Egypt'. The Third Chapter provides 'The Historical Manifestations of Astronomical Life in Ancient Egypt'. The Addendum provides 'The Historical and Linguistic Identification and Treatment of Obsolete Units and Measures of Time in Ancient Egyptian Language and Synonyms'. All of these were divided to several sections and topics to serve the scientific methodologies in ancient Egyptian civilization which was absolutely precious.

Nota e ringraziamenti Avendo da tempo maturato l'interesse per le applicazioni della geometria elementare alla pratica del disegno geometrico, ho potuto svolgere la parte centrale di questo studio nel corso dell'anno accademico 2012-13 durante il quale ho usufruito di un congedo dalle attività didattiche per motivi di studio. La ricerca -che oltre a questo volume ha avuto come esito un saggio sulle interpretazioni settecentesche e ottocentesche della voluta ionica greca, in corso di pubblicazione -potrà proseguire con lo studio di alcuni bizzarri strumenti da disegno di grande interesse, il Volutor ed i cosiddetti "elicografi", costruiti per il disegno a tratto continuo delle spirali.