Nexus Network Journal

Published by Springer Verlag
Online ISSN: 1522-4600
Publications
Triangular, oblong and square Pythagorean numbers. For the Pythagoreans, number was not represented by such symbolic forms as Arabic numerals (the number “3” for example) but rather as clusters of points. “Certain numbers were triangular,” notes Dr. Tyng, “such as 1, 3 and 6. Oblong numbers were even, being generated by the summation of even numbers as the areas of rectangles, such as 2, 6, 12, 20... Some numbers, like 6, were both triangular and oblong, making them magical in the eyes of the Pythagoreans.” As such, numbers evoked qualitative as well as quantitative associations. Drawing Robert Kirkbride. 
Image of measuring instruments depicted in the Gubbio studiolo. In the Gubbio studiolo (1476-83), one of two small meditation chambers created for the Duke of Urbino and his son, there are several emblems that point to the significance of architecture and geometry for education. The set-square/level and divider, instruments of the architect and geometer, were believed to conduct natural talent toward well-tempered, ethical action. The term “Ingenioq,” located above, represented innate genius or ability, as compared with experience and training. Likewise, the musical proportions embodied in the cittern, an instrument strung with metal wire to better sustain pitch for impromptu performances, assist the grace of the body through dance, as recommended by Plato. Photo Robert Kirkbride. 
Pascal Triangle and its diagonal summations. “Blaise Pascal discovered (c. 1650) his triangular table of probability, some 450 years after the similar ‘Chinese Triangle’ was conjectured to have been known to Fibonacci, but there is apparently no evidence that either Pascal or Fibonacci found the Fibonacci series in the sums of its diagonals.” (A. Tyng, “The Universe Plays Dice...” unpublished manuscript). Drawing Anne Tyng 
The Super Pythagorean Theorem. “A ‘Super Pythagorean Theorem’ is found in a Fibonacci-divine proportion overlapping additive sequence of circled sides of right-angled triangles. Circled sides have the same relative areas to each other as squared sides. As in numbers or quantity, the right-angled triangle’s Pythagorean theorem is seen as a two- and three-dimensional overlapping containment of process when sides are circled (or sphered) in a ‘Super Pythagorean Theorem,’ rather than the traditional squaring of sides. The first Fibonacci triangle—half a square cut on its diagonal—the ‘ 1, 1, 2 ‘ triangle with sides circled can all be encompassed by a ‘ 3’ circle that also encompasses the hypotenuse of the ‘ 1, 2, 3 ‘ Fibonacci triangle, which, in turn, with all its circled sides is encompassed by a ‘ 5 ‘ circle. Again the ‘ 5 ‘ circle encompasses the hypotenuse of the ‘ 2, 3, 5 ‘ triangle, which, with its circled sides, is encompassed by an ‘8’ circle—an overlapping sequence of Fibonacci triangles toward the precise divine proportioned ‘1 , φ , φ 2 ‘ triangle with sides encompassed by a ‘ φ 3 ’ circle. The two-dimensional overlapping triangulated Fibonacci relationships also define three-dimensional close-packed spheres within spheres, with Fibonacci to divine proportion fit of spherical surfaces within spherical surfaces. The Super Pythagorean Theorem of circled or sphered sides plays a significant role in atomic and molecular structures, offering variations of size and quantity of bonded and non-bonded contact radii of atoms for molecular building blocks” (A. Tyng, “The Universe Plays Dice...” unpublished manuscript). Drawing Anne Tyng 
Article
On November 11 and 13, 2003, Robert Kirkbride interviewed Anne Tyng, Fellow of the American Institute of Architects and a member of the National Academy of the Arts on the potentials of geometry and number in architectural practice. Through such examples as Pascal’s Triangle and her “Super Pythagorean Theorem,” Dr. Tyng asserts that geometry is not only a metaphor for thought and the creative process, it is a spatial demonstration of how the mind generates associations by the combination, or layering, of pattern and chance.
 
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In Hebrew, Christian and Islamic revelation, the world is created in six days of physical activity, followed by a single day of stillness and rest. Geometry provides a fitting metaphor, for the radii of six equal circles mark out the circumference of an identical circle placed in the center
 
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Han Vandevyvere undertakes an investigation into some geometrical schemes that can be supposed to underlie the plans and facades of a number of Flemish Gothic town halls, all of them built from the late fourteenth till the early sixteenth century. To govern his study he founded a set of basic ordering rules: a search for simple series of integer numbers, so as to obtain simple ratios between the dimensions; a check to see that what is found to set up a plan is also found in the elevations; the preferential use of geometrical constructions that can easily be constructed with the compass and the carpenter’s square; checking the design in the measurement units that were in use at the moment and place of construction; a check for the use of construction based on a circle, its inscribed square and equilateral triangle.
 
Article
Volutes, a distinguishing feature of the Ionic order, are the double curls in the form of spirals on either side of the Ionic capital. In the Renaissance, the Ionic volute was the object of study for architects who were concerned with the development of the new theories of architectural forms. In addition to studies of its proportions, research focused on the search for a sure and elegant method for laying out the volute. The point of departure for the elaborate theories were the ruins of buildings from the classical era and the treatise by Vitruvius. Authors Denise Andrey and Mirko Galli compare and contrast three methods by Sebastiano Serlio, Giuseppe Salviati and by Philandrier for laying out the Ionic volute.
 
Article
A brief description of Palladio’s life and works. The focus is on the evolution of his design methodology, including the growing importance of proportion to his approach. Selected mathematical details are cited in the endnotes, and the list of references includes many publications focused on the relationships between architecture and mathematics in Palladio’s designs.
 
– Area distribution of the Blocks showing the power law dependency.  
Church area distribution evidencing the exponential decay of the areas.
Area decay for the permanence areas.
– Diagrams that represent the area of space dedicated to public space in different periods of time. Source:(KRIER, 1999, p.147)  
Article
The present study, based on a comparative analysis of several plans for Lisbon’s Baixa district, with an emphasis on that area’s public space, contributes to an understanding of the urban design process and presents a fresh perspective on dealing with historical data by conducting a posteriori analysis using mathematical tools to uncover relations in the historical data. The nine plans used were quantified and evaluated in a comparative manner. While CAD was used to quantify the urban morphology of the different plans, comparative tables make it possible to register the data, which was further evaluated through two interrelated processes: mathematical analysis and the urban analysis. The results show the existence of power law relations for the areas of each of the city’s different elements (e.g., blocks, churches, largos and adros). We discuss how this contributes to the understanding of the plans’ elements. Keywordsurban design–design analysis–design theory–CAD–computer technology–morphology–patterns–proportion–measurement–mappings–geometry analysis–fractals–graph theory–statistical analysis
 
Chapter
In recent years the importance of mathematics as a basic discipline not only for the sciences but also for the humanities has been rediscovered and become the subject of interdisciplinary investigations. Two examples among many are the publication Die mathematischen Wurzeln der Kultur. Mathematische Innovationen und ihre kulturellen Folgen (The mathematical roots of culture — mathematical innovations and their cultural consequences) edited by Jochen Brüning and Eberhard Knobloch, Munich 2005, and the conference “Was zählt. Präsenz und Ordnungsangebote von Zahlen im Mittelalter” (What counts. The presence and medial function of number in the Middle Ages) held 2006 in Berlin. The specific role of geometry in the field of landscape architecture has been pointed out lately and rather pertinently by Volker Remmert in his article “‘Il faut etre un peu geometre’- Die mathematischen Wissenschaften in der Gartenkunst der Frühen Neuzeit” (One has to be a bit of a geometer — the mathematical sciences in the art of the garden in early modern age), published in the catalogue of the exhibition “Wunder und Wissenschaft — Salomon de Caus und die Automatenkunst in Gärten um 1600”, Düsseldorf 2008, (Miracle and science — Salomon de Caus and the art of automatons in gardens of the 1600s).
 
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The four answers to the prize question of the Brussels’ Academy of 1783 on the development of a theory of beams demonstrates that the modeling and mathematical mastering of the problem remained for a long time limited to a very small circle of men. With savants such as the Viscount de Nieuport (1746–1827), who worked on the calculus of vaults and who formulated this question on beams, the Academy appears such a privileged milieu. In Belgium, it would at take least until the creation of the polytechnic school at the State University of Ghent in 1835 for this approach to be diffused in a systematic and institutionally way among (an elite of) construction professionals. Keywordsstructural mechanics-Belgian Royal Academy-De Nieuport-arches-vaults-beams-structural analysis
 
Article
John Hatch examines the friendship between Theo Van Doesburg and El Lissitzky, which was fuelled by a shared interest in scientific theories. Both moved from painting to architecture in seeking out a form best suited to conveying the spatiotemporal experiences phrased by Relativity, resulting in some remarkably innovative architectural designs and theories. KeywordsEl Lissitzky-Theo Van Doesburg-De Stijl-Relativity-relationships between art and science
 
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Guest Editor Lionel March introduces Nexus Network Journal volume 13 number 1 (Winter 2010) dedicated to Shape and Shape Grammar. Keywordsshape grammar–design theory
 
Article
The Justified Plan Graph (JPG) technique was developed in the late 1970s and refined in the following two decades as a means of undertaking qualitative and quantitative research into the spatial structure or permeability of buildings. Famously used by Space Syntax researchers to uncover the social logic of architectural types, the technique remains an important, if not widely understood, approach to the analysis of the built environment. This paper uses the JPG method to undertake a three-stage analysis of the early houses of Pritzker Prize winning architect Glenn Murcutt; the stages are visual analysis, mathematical analysis and theoretical analysis. Through this process the paper offers a rare application of the JPG method to multiple works by the same architect and demonstrates the construction of a series of “inequality genotypes”, a partial “statistical genotype” and a partial “statistical archetype” for these houses. Instead of seeking to uncover the social structure of Murcutt’s housing, the paper analyses the architect’s distinctive approach to ordering space within otherwise simple volumes or forms. The ultimate purpose of this analysis is to offer an alternative space-based, rather than form-based, insight into this architect’s work. KeywordsGlenn Murcutt–Justified plan graph–graph theory–Space Syntax–mathematical analysis–plan analysis
 
Article
Andrea Pagano and Laura Tedeschini Lalli present a brief note on what has to be considered a teaching success story. They tell of ten years of teaching mathematics at the Faculty of Architecture of the Università di Roma Tre, describing some new course content that has been introduced, the methods used and, above all, the spirit that has driven the ideas about teaching mathematics to future architects. The experience covers the full curriculum of the student: from first year courses to individual projects.
 
Article
Geometer Mark Reynolds examines a ratio that is related to the golden section but is relatively unknown to many architects and designers. Equal to phi times its square root, ϕ√ϕ, the ratio mu,μ=1:2.058, offers its user an opportunity to work with the golden section but with a different “look.”
 
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Article
The Nexus 2000 round table discussion on methodology in architecture and mathematics took place on Tuesday 6 June during the course of the Nexus 2000 conference in Ferrara, Italy. Moderated by Carol Martin Watts (far left in the photo), the panelists were (from left to right) Rachel Fletcher, Paul A. Calter, William D. (Bill) Sapp and Mark Reynolds. The following is a transcript of the audio tapes made during the discussion.
 
Article
The author reflects on the Nexus 2000 Round Table Discussion “Methodology in Architecture and Mathematics” and on the relationships of architecture and mathematics through history.
 
Article
The round table discussion on mathematics in the architecture curriculum took place at the Nexus 2002 conference, 17 June 2002. Moderated by Judith Flagg Moran, panel members discussed issues pertinent to teaching mathematics to architecture students. This paper is the transcription of the audio tapes made of the discussion.
 
Article
B. Lynn Bodner reports on the Bridges 2006 conference.
 
Article
The eighth edition of the international, interdisciplinary Nexus conference on architecture and mathematics took place from Sunday 13 June through Tuesday 15 June 2010, in Porto, Portugal.
 
The participants of the 1998 workshop on the steps of the Villa Bisani, Bagnolo: first row (lr), Rocco Leonardis, Michael Ostwald, Stephen Wassell; second row (l-r), David Speiser, Ruth Speiser, Countess Cornelia Ferri di Lazara, Susan Knight, Kim Williams, Vera Spinadel.
The Villa Rotonda, showing the juxtaposition of geometric elements in the form of classical motifs, ordered by strict rules of symmetry.
Article
Much has been written about the mathematical qualities of Andrea Palladio’s architecture, including his own I quattro libri dellarchitettura. Often this has been analyzed within the context of a larger collection of architectural treatises, underscoring the importance of proportion, symmetry and geometry in Renaissance Italy. This essay provides a review of the mathematical aspects of Palladio’s work as it has been discussed in the literature and offers a novel perspective on his mathematical approach to architectural design. The author argues that, given the amount of discussion already focused on the role that harmonic proportions played in the Palladio’s architecture, it is now time to search further for other mathematical facets of his design philosophy. The analysis is arranged in three sections: geometry, proportion and symmetry.
 
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John Sharp reports on the history, meaning and execution techniques of the Cosmatesque pavements of Westminster Abby.
 
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The astronomical alignments of the Acropolis of Alatri, Italy, are investigated. The results strongly support a dating of the magnificent polygonal walls of the site to a pre-Roman period.
 
Article
It is commonly believed that the longitudinal axes of churches extend exactly in an east-west direction. However, thorough investigations have shown that this is not always correct; rather, both southern and northern deviations of up to about 25° can occur. The angular deviation between the church axis and true east is called the Holy Alignment. This present study presents the possibility that the nave is oriented towards the direction of the sun-rising point on the name day of the patron saint of the church. If several saints share the patronage, the Holy Alignment equals the algebraic sum of the angular distances for each saint. The orientation of a nave can be analyzed by means of common mathematical relations used in geodesy, astronomy and gnomonics. In order to perform such an analysis, it is necessary to know the history of the patronage of the church; the Gauss-Krüger coordinates of the ground plan; and characteristic astronomical quantities at the time the church was built. A calculated example with the saints Andrew, James and Philip for the year 980 illustrates the analysis. An appendix deals with the influence of atmospheric refraction on the apparent altitude of the sun near the horizon.
 
Article
Consciously or unconsciously, part of the apparatus that architects use in their daily fabrications of the built environment grows out of their understanding of numbers and numerals. Embodied in tectonic events and parts, numbers hinge the past and the future of buildings and their inhabitants into a search for a way of life with no impairment caused by psychic activity. Whether sensible or intelligible, tectonic numbers articulate the vigor of human mind’s eye, and ultimately they refer to psychic regimes immersed in the vital ocean of imagination and wonder. The essential influence on Scarpa’s numerical thinking is the combinatorial procedures devised by Raymond Roussel for writing his books, the upturned geometry of Rene A. Schwaller De Lubicz and Surrealistic processes of invention. Scarpa’s architecture is a prudent and playful project that relates to the traces of numbers embodied in a tradition. In Scarpa’s opus, it is true that One and One Equals Two, but it is also wonderfully true that A Pair of Ones Makes an Eleven.
 
Outputs of a computer program developed after Vitruvius's Ten Books of Architecture using Autolisp, the scripting language of Autocad. (Tiago Sousa and Duarte Pape, Programming and Fabrication for Architecture course, TU Lisbon, 2002. Instructor: J. P. Duarte.) From left to right and from top to bottom: digital models of theatres for 5,000, 10,000 and 15,000 spectators; and a 3D print model of the latter. 
Article
The research described in this paper is part of a project aimed at decoding Alberti’s De re aedificatoria by inferring the corresponding shape grammar using the computational framework provided by description and shape grammars to determine the extent of such an influence in the Counter-Reformation period in Portugal. Here we concentrate on the theoretical foundations that enable the translation of Alberti’s text into a shape grammar and then use it in determining its influence on Portuguese Renaissance architecture. Keywordstheory of architecture–shape grammars–Leon Battista Alberti–De re aedificatoria–design automation–rapid prototyping
 
Article
Until recently, environmental control systems have been more often suppressed than expressed, hidden from casual observers and building users, rarely featured as architectural design elements, or considered aesthetically. While the impact of the overall form of a building on its thermal environmental performance may not always be apparent, the mutual influences of shape, form and orientation should be evident to – if not a basic activity of – a wellinformed professional. A primary aim of this paper is to encourage a new aesthetic sensibility for the 21st century; one that conceives architectural form with respect to environmental context and ecological efficiency. Toward this end, I propose a method of comparative analysis, using several recently completed and speculative architectural projects. KeywordsEnvironmental control systems-architectural design elements-architectural form-geometry and aesthetics
 
Article
In the absence of metal beams, domes had been an essential part of the architecture of official and religious buildings around the world for several centuries. Domes were used to bring the brick structure of the building to conclusion. Based on their spherical constructions, they provided strength to the building foundations and also made the structure more resistant against snow and wind. Besides bringing a sense of strength and protection, the interior designs and decorations resemble sky, heaven, and what a person may expect to see beyond ”seven skies.” Some contemporary religious buildings or memorials still incorporate domes, no longer out of necessity, but rather based on tradition or for esthetical purposes. Yet the quality of the interior decoration of these new domes is diminishing. The aim of this article is to study the spatial effects created by dome interior designs and to provide information about construction of such a design. Decorations in dome interiors demonstrate art forms such as stucco, tessellated work, ceramics, paintings, mirror work, and brick pattern construction, as well as combinations of these forms.
 
Article
Keywordschurch-baptistery-basilica-centrally-planned building-chapter house-Middle Ages-meaning-symbolism-geometry-harmonic proportion-measurement-number-Platonic solids-sacred geometry-shapes
 
Article
A computer-aided rule-based framework that restructures the unstructured information embedded in precedent designs is introduced. Based on a deductive analysis of a corpus of sixteen case studies from Mamluk architecture, the framework is represented as a generative system that establishes systematic links between the form of a case study, its visual properties, its composition syntax and the processes underlying its design. The system thus formulated contributes to the areas of design research and practice with a theoretical construct about design logic, an interactive computerized plan generator and a combination of a top-down approach for case study analysis and a bottom-up methodology for the derivation of artifacts.
 
Article
The architectural complexes composed by the two main pyramids of Giza together with their temples are investigated from an inter-disciplinary point of view, taking into account their astronomical alignments as well as their relationships with the visible landscape. Combining already known facts together with new clues, the work strongly supports the idea that the two complexes were conceived as parts of a common project. Comment: Archaeoastronomy/History of Astronomy
 
Article
On the basis of a new survey, Angela Pintore analyzes the microarchitecture of the Rucellai Sepulchre in Florence, the only object designed ex novo by Leon Battista Alberti. Attention is also given to the relationship established between the sepulchre and the chapel that houses it, and to the modifications made to the chapel by Alberti himself. Alberti studied carefully the combinations between the number of the elements of the front elevation and that of lateral elevation and of the apse so that the relationship between them would recall the harmonic musical ratios that he set forth in De re aedificatoria, in which he outlines the correspondence between architectural proportions and harmonic musical ratios that will become the element that characterizes Renaissance architectural theory, inaugurating a tradition that will begin to see a decline only in the eighteenth century. In spite of the myriad difficulties of establishing if these speculations had indeed any concrete effect on architecture, it is clear that Alberti’s theory is not the result of individual reflection, based solely on the classical sources that Alberti himself explicitly cites in his treatise, but rather is the culmination of an age-old tradition of thought that, during the whole arc of the Middle Ages, had deepened the study of the symbolic and expressive value of harmonic ratios.
 
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KeywordsLeon Battista Alberti–Ludi matematici–cipher codes–measuring–history of mathematics
 
Article
KeywordsKirsti Andersen-perspective-perspective geometry-Leon Battista Alberti-Gaspard Monge-Guidobaldo del Monte
 
Chapter
Kay Bea Jones reviews the exhibit of the work of Franco Albini in Milan.
 
Berliner Mietskasernen. Exterior and interior views of tenement building development in the early years of the twentieth century
Method of the successive increments. Example of comparison and evaluation of several plan diagrams reduced to the same scale (1 living room, 1 bedroom for parents, 1 bedroom for children)
The Gross-Siedlung of Bad Dürrenberg in Leipzig. View of the area
Plans of typological lodgings for the Gross-Siedlung of Bad Dürrenberg
Article
The urgent need for housing for the working class following World War I led to a search to determine new standards for housing to ensure that minimum requirements for living, or Existenzminimum, would be respected while costs were kept low. Overshadowed by more famous architects such as Gropius, Taut, and Le Corbusier, Alexander Klein’s important role in this search is often neglected. Klein’s remarkably innovative and mathematically rigorous design methodology began with the comparison of various types of dwellings, aimed at the determination of objective terms for the valuation of the design quality. Klein studied the problem of dwelling in its complexity, even considering the effects induced by conditions of living to the human psyche. His points of reference were the needs of the family and the individual, rather than impersonal hygienicsanitary parameters. Klein’s methodological research may still be considered an important point of reference. Keywordshousing–design analysis–design theory–Alexander Klein–minimal surfaces
 
Article
Lines and surfaces are boundary elements of objects and buildings: it is very important to give the students a mathematical approach to them. Elena Marchetti and Luisa Rossi Costa present linear algebra (by vectors and matrices) as an elegant and synthetic method, not only for the description but also for the virtual reconstruction of shapes.The aim of our activity is to facilitate—and, at the same time, to develop—the comprehension of crucial mathematical tools involved in the realization of forms and shapes in arts, architecture and industrial design and in computer graphics. Another important aspect of linear algebra to be pointed out to the students is its application in graphics software packages, which work with transformations that change the position, orientation and size of objects in a drawing
 
Article
This paper describes the mathematical discovery of a new property of conics which allows the development of numerous geometric projects for use in architectural and engineering applications. Illustrated is an architectural application in the form of an alternative project for Río de Janeiro Metropolitan Cathedral featuring the integration of a circular base and a cross in the top plane. Two alternative designs are presented for the cathedral, based on the choice of either the Latin Immisa or Greek cross.
 
Chapter
Leonardo da Vinci used geometry to give his design concepts both structural and visual balance. The paper examines aesthetic order in Leonardo’s structural design, and reflects on his belief in analogy between structure and anatomy. Leonardo’s drawings of grids and roof systems are generated from processes best known from ornamentation and can be developed into spatial structures assembled from loose elements with no need for binding elements. His architectural plans are patterns based on principles of tessellation, tiling and recursion, also characteristic of the reversible, ambiguous structures which led to Leonardo’s further inventions in structural and mechanical design as well as dynamic representations of space in his painting. In recent times, the ambiguous structures in the art of Joseph Albers, the reversible and impossible structures of M. C. Escher, the recurring patterns and spherical geometry of Buckminster Fuller and the reciprocal grids in structural design of Cecil Balmond display a similar interest. Computer models and animations have been used to simulate processes of perceiving and creating ambiguity in structures.
 
Article
It is a great pleasure for me to review this book, because I was present at its birth and watched it mature. Prof. Eaton’s aim, first with an article published in the NNJ and then in detail with this fine new book, was to show how American engineer Hardy Cross developed a method for analyzing indeterminate structures that minimized the inconveniences and risks involved in the use and development of reinforced concrete. Along the way, however, Eaton also provides us with a tapestry of other considerations as to the interactions of engineering and architecture, the relationship of engineering and mathematics, and the contrasts between American engineers and their peers overseas. This is as culturally rich a book on a single technical argument as you could wish to find.
 
Article
KeywordsJavier Navarro de Zuvillaga-form-representation-perspective-geometry
 
Top-cited authors
Nicoletta Sala
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Ricardo David Valdez-Cepeda
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Sara Eloy
  • ISCTE-Instituto Universitário de Lisboa
Jose Duarte
  • Pennsylvania State University
Jiang Liang
  • Dalian University of Technology