Perspective as a Symmetry Transformation György Darvas (Spring 2003)
From the quattrocento to the end of the nineteenth century, perspective has been the main tool of artists aiming to paint a naturalistic representation of our environment. In painters' perspective we find a combination of affine projection and similitude. We recognise the original object in the painting because perspective is a symmetry transformation preserving certain features. The subject of the transformation, in the case of perspectival representation, is visible reality, and the transformed object is the artwork. The application of symmetry transformations developed from the origin of perspective through the centuries to the present day. The single vanishing point could be moved (translated), and even doubled, developments that made it possible to represent an object from different points of view. In the twentieth cenutyr, the application of topological symmetry combined with similitude resulted in new ways of seeing, new tools for artists such as cubists and futurists.

Distance to the Perspective Plane Tomás García-Salgado (Spring 2003)
Distance is an integral concept in perspective, both ancient and modern. Tomás García-Salgado provides a historical survey of the concept of distance, then goes on to draw some geometric conclusions that relate distance to theories of vision, representation, and techniques of observation in the field. This paper clarifies the principles behind methods of dealing with the perspective of space, in contrast to those dealing with the perspective of objects, and examines the perspective method of Pomponius Gauricus, contrasting it with the method of Alberti. Finally the symmetry of the perspective plane is discussed.

From the Vaults of Heaven Marco Jaff (Spring 2003)
Many clues lead Marco Jaff to conjecture that Brunelleschi knew about the use of the astrolabe, an instrument very often used in his times; among his friendships we find the astronomer Paolo Dal Pozzo and engineer Mariano di Jacopo da Siena, who certainly knew how to use the astrolabe accurately. Because this instrument is based on the principle of stereographic projection, a particular kind of central projection, it is quite possible that Filippo applied this principle either for the perspective construction outline for Masaccio's Trinità in S. Maria Novella, as well as for the two lost panels of the Baptistery of Florence.

Speculations on the Origins of Linear Perspective Richard Talbot (Spring 2003)
Richard Talbot demonstrates an approach and method for constructing perspectival space that may account for many of the distinguishing spatial and compositional features of key Renaissance paintings. The aim of the paper is also to show that this approach would not necessarily require, as a prerequisite, any understanding of the geometric basis and definitions of linear perspective as established by Alberti. The author discusses paintings in which the spatial/geometric structure has often defied conventional reconstruction when the strict logic of linear perspective is applied.

Visual sensibility in Antiquity and the Renaissance: The Diminution of the Classical Column David A. Vila Domini (Spring 2003)
David Vila Domini looks at the recommendations regarding optical adjustment of the columnar diminution in the architectural treatises of Vitruvius, Alberti, and Palladio. He examines the variation in diminution of column thickness according to the height of the column, and its implications for our understanding of the various practices with regard both to columnar proportion and visual sensibility in Antiquity and the Renaissance. He also examines possible sources for the methods by which the ratios of column height to diameter were derived.

How Should We Measure an Ancient Structure? Harrison Eiteljorg, II (Autumn 2002)
Harrison Eiteljorg, II, examines the questions of precision and accuracy in the measurement of ancient buildings, taking into account the separate requirements of both scholarship and preservation. Modern technology has changed matters significantly and promises to continue to bring change. Whereas the problem was once measuring as precisely as possible or as precisely as a scaled drawing could display, the problem is now to measure and record as precisely as required for the particular project. For each survey project, the answer must be unique, but it must be well and carefully argued with respect to the tools at hand and the subject. It is no longer appropriate to assume that the most precise measurements are necessary. Technology has advanced; now the decisions are ours.

The Double Möbius Strip Studies Vesna Petresin and Laurent-Paul Robert (Autumn 2002)
The curious single continuous surface named after astronomer and mathematician August Ferdinand Möbius has only one side and one edge. But it was only in the past century that attention in mathematics was drawn to studies of hyper- and fractal dimensions. As Vesna Petresin and Laurent-Paul Robert show, the Möbius strip has a great potential as an architectural form, but we can also use its dynamics to reveal the mechanisms of our perception (or rather, its deceptions as in the case of optical illusions) in an augmented space-time.

Villard de Honnecourt and Euclidian Geometry Marie-Thérèse Zenner (Autumn 2002)
In this reprint from a popular science journal, Marie-Thérèse Zenner presents a brief overview of the survival of Latin Euclid within the practical geometry tradition of builders, taking examples from an eleventh-century French Romanesque church, Saint-Etienne in Nevers, and a thirteenth-century Picard manuscript of drawings (Paris, Bibliothèque nationale, MS fr. 19093), known as the portfolio of Villard de Honnecourt.

Mathematics and Design: Yes, But Will it Fly? Martin Davis and Matt Insall (Autumn 2002)
Martin Davis and Matt Insall discuss a quote by Richard W. Hamming about the physical effect of Lebesgue and Riemann integrals and whether it made a difference whether one or the other was used, for example, in the design of an airplane. The gist of Hamming's quote was that the fine points of mathematical analysis are not relevant to engineering considerations.

A Light, Six-Sided, Paradoxical Fight Marco Frascari (Spring 2002)
Built structures and their architectural representations are places where geometry, mathematics and construction discover their common nature, that is, the capability of human imagination to merge architectural objects with the telling of enjoyable tales. In this ppaer Marco Frascari takes aim at the forces that have shaped a system of critical thoughts on how to fight gravity with a happy architecture based on light structures combined with the dilettante's approaches to hexagonal design, interweaving the thoughts of Alberti, Kahn and Le Ricolais with those of master story-tellers Calvino and Rebelais.

The Fire Tower Elena Marchetti and Luisa Rossi Costa (English version) (Spring 2002)
The Fire Tower was a project by Johannes Itten, one of the most important exponents of the Bauhaus movement. The aim of this paper by Elena Marchetti and Luisa Rossi Costa is to describe the shape of The Fire Tower with the language of linear algebra and give a virtual reconstruction, in order to understand how Itten managed to concretise his strong mathematical intuition in an artistic form, even though he was unable to formalise it entirely with adequate instruments.

La Torre di Fuoco Elena Marchetti and Luisa Rossi Costa (versione italiana) (Spring 2002)
La Torre del Fuoco è un progetto di Johannes Itten, uno dei più importanti esponenti del Bauhaus. Scopo del presente lavoro di Elena Marchetti e Luisa Rossi Costa è quello di descrivere la forma della Torre del Fuoco attraverso il linguaggio dell'algebra lineare e di darne una ricostruzione virtuale, nella consapevolezza di quanto Itten fosse capace di concretizzare nell'arte il suo forte intuito matematico, pur non potendo formalizzarlo fino in fondo con adeguati strumenti.

The Golden Section in Architectural Theory Marcus Frings (Winter 2002)
In the never-ending - but always young - discussion about the Golden Section in architecture never lacks the hint at Luca Pacioli and the architectural theory. But what always lacks is a thorough study of this topic, the Golden Section in architectural theory. The paper aims to present this analysis. Marcus Frings traces Golden Section from the mathematical and rather theoretical character of Pacioli's concept, examines Alberti, Serlio, Palladio and other architectural treatises, to arrive to Adolf Zeising in the nineteenth century and to theorist Matila Ghyka and the practitioners Ernst Neufert and Le Corbusier in the twentieth. In the latter's writings and designs the Golden Section seems to play the role of a scholarly element which shows the master's theoretical erudition, leading to contemporary architects such as Ricardo Bofill.

The Pythagopod Christopher Glass (Winter 2002)
In 1967 lecture at Yale Architecture School Anne Tyng discussed integrating of the five Pythagorean solids into a single shape and suggested the shape as an architectural solid. Christopher Glass aim is to sphere the cube in the manner of Buckminster Fuller, but with reference not only to the engineering models he uses but to the cultural models of the Pythagorean proportions as well. The author has developed computer models of the resulting plan at least two scales: the original glass house and a smaller hermitage pod.

More True Applications of the Golden Number Dirk Huylebrouck and Patrick Labarque (Winter 2002)
Dirk Huylebrouck and Patrick Labarque try to provide a positive answer to the question that the golden section corresponds to an optimal solution. It is but a college-level rephrasing exercise, but it could reboot the mathematical career of the golden section. An extension to the related silver section is given as well. The authors betin their examination with the definition of the golden number, then proceed to its applications to architecture, grey-tone mixing, colour mixing and bicycle gears.

Spirals and the Golden Section John Sharp (Winter 2002)
The Golden Section is a fascinating topic that continually generates new ideas. It also has a status that leads many people to assume its presence when it has no relation to a problem. It often forces a blindness to other alternatives when intuition is followed rather than logic. Mathematical inexperience may also be a cause of some of these problems. In the following, my aim is to fill in some gaps, so that correct value judgements may be made and to show how new ideas can be developed on the rich subject area of spirals and the Golden section. The paper is divided into four parts: Introduction; Types of spirals; Spirals from the Golden rectangle, Triangles and the pentagon by approximation; Mathematics of true Golden Section spirals; The myth of the nautilus shell.

Palladio's Villa Emo: The Golden Proportion Theory Rebutted Lionel March (Autumn 2001)
In a most thoughtful and persuasive paper [Fletcher 2000], Rachel Fletcher comes close to convincing that Palladio may well have made use of the 'golden section', or extreme and mean ratio, in the design of the Villa Emo at Fanzolo. What is surprising is that a visually gratifying result is so very wrong when tested by the numbers. Lionel March provides an arithmetic analysis of the dimensions provided by Palladio in the Quattro libri to reach new conclusions about Palladio's design process.

Palladio's Villa Emo: The Golden Proportion Theory Defended Rachel Fletcher (Autumn 2001)
At Nexus 2000, Rachel Fletcher argued that Palladio may well have made use of the 'golden section', or extreme and mean ratio, in the design of the Villa Emo at Fanzolo. In the Autumn, 2001 issue of Nexus Network Journal, Lionel March argued that the Golden Section is nowhere to be found in the Villa Emo as described in I quattro libri dell'archittetura. In the present paper, Rachel Fletcher defends her original thesis, comparing the Villa Emo as actually built to the project for it that Palladio published in his book.

Rosettes and Other Arrangements of Circles Paul L. Rosin (Autumn 2001)
The process of design in art and architecture generally involves the combination and manipulation of a relatively small number of geometric elements to create both the underlying structures as well as the overlaid decorative details. In this paper we concentrate on patterns created by copies of just a single geometric form - the circle. The circle is an extremely significant shape. By virtue of its simplicity and its topology it has been highly esteemed by many different cultures for millennia, symbolising God, unity, perfection, eternity, stability, etc. For instance, Ralph Waldo Emerson considered the circle to be "the highest emblem in the cipher of the world"

Violins and Volutes: Visual Parallels between Music and Architecture Åke Ekwall (Autumn 2001)
In early Greek architecture, above all in the Ionic order, the volute was developed with particular perfection and grace. From 1957 to 1965, I carried out an extensive investigation into how the violin acquired its singular shape. One aspect of violins that I studied was the strong spiral line of the f-holes and scroll. The present paper compares the constructions of Vitruvius, Alberti and Palladio for the volute to my own analyses performed on the scrolls of historic violins. It also seeks a parallel for constructions of volutes with arcs of different degrees in the volutes of the Medici Chapel by Michelangelo.

Group Theory and Architecture II: Why Symmetry/Asymmetry?  Michael Leyton (Autumn 2001)
This is the second in a sequence of tutorials on the mathematical structure of architecture. The first was Group Theory and Architecture 1 (NNJ vol. 3 no. 3 Summer 2001). The purpose of these tutorials is to present, in an easy form, the technical theory developed in Leyton's book, A Generative Theory of Shape [Springer-Verlag, 2001], on the mathematical structure of design. In this second tutorial Michael Leyton looks at the functional role of symmetry and asymmetry in architecture.

Gothic Flemish Town Halls In and Around Flanders, 1350-1550: A Geometric Analysis Han Vandevyvere (Summer 2001)
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. Among the most famous of them, we can mention Brussels, Louvain, Oudenaarde and Bruges, all of them built from the late 14th till the early 16th 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.

The Engineering Achievements of Hardy Cross Leonard K. Eaton (Summer 2001)
Leonard K. Eaton resurrects the reputation of Hardy Cross, developer of the "moment distribution method" and one of America's most brilliant engineers. The structural calculation of a large reinforced concrete building in the nineteen fifties was a complicated affair. It is a tribute to the engineering profession, and to Hardy Cross, that them were so few failures. When architects and engineers had to figure out what was happening in a statically indeterminate frame, they inevitably turned to what was generally known as the "moment distribution" or "Hardy Cross" method. Although the Cross method has been superseded by more powerful procedures such as the Finite Element Method, the "moment distribution method" made possible the efficient and safe design of many reinforced concrete buildings during an entire generation.

Euclidism and Theory of Architecture Michele Sbacchi (Summer 2001)
Michele Sbacchi examines the impact of the discipline of Euclidean geometry upon architecture and, more specifically, upon theory of architecture. Special attention is given to the work of Guarino Guarini, the 17th century Italian architect and mathematician who, more than any other architect, was involved in Euclidean geometry. Furthermore, the analysis shows how, within the realm of architecture, a complementary opposition can be traced between what is called "Pythagorean numerology" and "Euclidean geometry." These two disciplines epitomized two overlapping ways of conceiving architectural design.

Group Theory and Architecture I: Nested Symmetry  Michael Leyton (Summer 2001)
The present series of articles by Michael Leyton, of which this is the first, will give an introduction to a comprehensive theory of design based on group theory in an intuitive form, and build up any needed group theory through tutorial passages. The articles will begin by assuming that the reader has no knowledge of group theory, and we will progressively add more and more group theory in an easy form, until we finally are able to get to quite difficult topics in tensor algebras, and give a group-theoretic analysis of complex buildings such as those of Peter Eisenman, Zaha Hadid, Frank Gehry, Coop Himmelblau, Rem Koolhaas, Daniel Libeskind, Greg Lynn, and Bernard Tschumi. This first article is on a subject of considerable psychological relevance: nested symmetries.

Applications of a New Property of Conics to Architecture: An Alternative Design Project for Rio de Janeiro Metropolitan Cathedral Juan V. Martín Zorraquino, Francisco Granero Rodríguez and José Luis Cano Martín (Spring 2001)
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 of the the integration of a ellipical base and a cross in the top plane. Two alternative designs are presented for the cathedral, based on the choice of either the Latin or Greek cross.

Modularity and the Number of Design Choices Nikos Salingaros and Débora M. Tejada (Spring 2001)
Nikos Salingaros and Débora Tejada analyze one aspect of what is commonly understood as "modularity" in the architectural literature. There are arguments to be made in favor of modularity, but the authors argue against empty modularity, using mathematics to prove their point. If we have a large quantity of structural information, then modular design can organize this information to prevent randomness and sensory overload. In that case, the module is not an empty module, but a rich, complex module containing a considerable amount of substructure. Empty modules, on the other hand, eliminate internal information, and their repetition eliminates information from the entire region that they cover. Modularity works in a positive sense only when there is substructure to organize.

On Precision in Architecture Costantino Caciagli (English version) (Spring 2001)
In architecture, the term precision, in the sense of "respect for order and exactness", says everything and nothing. In fact, "precision in architecture" can be used in reference to diverse aspects such as the carrying out of program functions, to execution, to forms, to distribution of forces, to dimensions, but we could never arrive at a conclusion if the characteristics taken into consideration were not commensurable to a reference sample.

A proposito della precisione in architettura Costantino Caciagli (versione italiana)(Spring 2001)
Precisione, nel senso di "rispetto dell'ordine e dell'esattezza", in architettura dice tutto e non dice nulla, infatti ci si può riferire allo svolgimento delle funzioni alla esecuzione, alle forme,alla distribuzione dei pesi, delle dimensioni, ma non potremmo arrivare a nessuna conclusione, se i caratteri presi in considerazione non sono commensurati ad un campione di riferimento.

Iannis Xenakis - Architect of Light and Sound Alessandra Capanna (English version) (Spring 2001)
Alessandra Capanna summarizes the life and work of Iannis Xenakis, who passed away on 4 February 2001.He was a musician, but above all he was a theorist and pure researcher who used mathematical thought as a basis for of his compositions. Because of this, his way of working more closely resembles that of a philosopher of science than that of an artist, whose instinctive creations are sometimes controlled only by aesthetical aims. he was also an architect. In 1956 Le Corbusier entrusted his sketches for the Philips Pavilion for the Brussels World's Fair to Xenakis, who was charged to translate them through mathematics.

Iannis Xenakis -- Architetto della luce e dei suoni Alessandra Capanna (versione italiana) (Spring 2001)
Alessandra Capanna ripercorre il carierra di Iannis Xenakis, uno dei musicisti contemporanei più celebri, scomparso il 4 febbraio 2001. Un musicista, ma anche un teorico e un ricercatore puro che, ponendo alla base di tutte le sue articolazioni compositive il pensiero matematico. Era anche un'architetto. In ottobre del 1956 che egli ricevette da Le Corbusier l'incarico di tradurre attraverso la matematicai suoi schizzi per la Padiglione Philips per l'Expo di Bruxelles.

"Fractal Architecture": Late Twentieth Century Connections Between Architecture and Fractal Geometry Michael J. Ostwald (Winter 2001)
For more than two decades an intricate and contradictory relationship has existed between architecture and the sciences of complexity. While the nature of this relationship has shifted and changed throughout that time a common point of connection has been fractal geometry. Both architects and mathematicians have each offered definitions of what might, or might not, constitute fractal architecture. Curiously, there are few similarities between architects' and mathematicians' definitions of "fractal architecture". There are also very few signs of recognition that the other side's opinion exists at all. Practising architects have largely ignored the views of mathematicians concerning the built environment and conversely mathematicians have failed to recognise the quite lengthy history of architects appropriating and using fractal geometry in their designs. Even scholars working on concepts derived from both architecture and mathematics seem unaware of the large number of contemporary designs produced in response to fractal geometry or the extensive record of contemporary writings on the topic. The present paper begins to address this lacuna.

Analysis and Synthesis in Architectural Designs:A Study in Symmetry Jin-Ho Park (Winter 2001)
Ordered designs are frequently encountered in art and architecture. The underlying structure of their spatial logic may be discussed with regard to the use of symmetry principles in mathematics. In architectural designs, the use of symmetry may sometimes be apparent immediately by just looking at designs, although the final design is seemingly asymmetrical; or various symmetries are manifested in the parts of the designs, yet not immediately recognizable despite an almost obsessive concern for symmetry. At this point, it is crucial to develop a formal methodology that may clearly elucidate different hierarchical levels of the use of symmetry in architectural designs.In an effort to do this, before proceeding to analytic and synthetic applications, we discuss a methodology founded on the algebraic structure of the symmetry group of a regular polygon in mathematics. The approach shows how various types of symmetry are superimposed in individual designs, and illustrates how symmetry may be employed strategically in the design process. Analytically, by viewing architectural designs in this way, symmetry, which is superimposed in several layers in a design, becomes transparent. Synthetically, architects can benefit from being conscious of using group operations and spatial transformations associated with symmetry in compositional and thematic development. The advantage of operating with symmetry concepts in this way is to provide architects with an explicit method not only for the understanding of symmetrical structures of sophisticated designs, but also to give architects insights for the construction of new designs by using symmetry operations.

The Squaring of the Circle in two Early Norwegian Cathedrals? Dag Nilsen (Winter 2001)
The squaring of the circle is impossible, but it can be represented geometrically, as demonstrated by Dr.-Ing. Helmut Sander in "A geometrical ensemble to generate the squaring of the circle". I immediately recognized his diagram as being very close to a diagram that I have found by analyzing two early Norwegian basilican- plan cathedrals, and which, at first glance, I believed might have been used in determining the ratios between some important dimensions. This spurred me to make further investigations, revealing that it was not quite that simple. However, this pursuit revealed some alternative, but related possibilities, including a way of combining Ö2 and Ö5 -- albeit approximately, but close enough to fool a non-mathematician working by small-scale geometry into make a false assumption similar to Le Corbusier's when he was developing the Modulor.

UPDATED! Architecture, Patterns and Mathematics Nikos Salingaros (April 1999, republished in an updated version Winter 2001)
One of the roles served by architecture is that of offering professionals and laymen alike the possibility to experience mathematical pattern. Nikos Salingaros examines how the revolution in architectural style at the end of the nineteenth century and the beginning of the twentieth, aimed at banishing an irrevelant architectural ornamentation, also banished pattern from architecture, much to the detriment of man's experience of the built environment. Using the architecture of Mies van der Rohe and Le Corbusier and the theories of Christopher Alexander as a base, the author explains the malady and the cure for twentieth century architecture. This paper appears in print in the Nexus Network Journal 1 (1999): 75-85.

The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion Jay Kappraff (October 2000)
Nicomachus of Gerasa has gained a position of importance in the history of ancient mathematics due in great measure to his Introduction to Arithmetic, one of the only surviving documentations of Greek number theory. Prof. Kappraff discusses a pair of tables of integers found in the Arithmetic and shows how they lead to a general theory of proportion, including the system of musical proportions developed by the neo-Platonic Renaissance architects Leon Battista Alberti and Andrea Palladio, the Roman system of proportions described by Theon of Smyrna, and the Modulor of Le Corbusier. This paper appears in print in Nexus Network Journal 2 (2000): 41-55.

Introduction to Slavik Jablan's Modular Games Donald W. Crowe (October 2000)
Donald Crowe, Professor emeritus of mathematics at the University of Wisconsin and known for his collaboration with Dorothy Washburn on the book Symmetries of Culture : Theory and Practice of Plane Pattern Analysis, introduces a new interactive tiling program by Slavik Jablan called Modular Games, also published in the issue of the NNJ. Prof. Crowe provides an overview of the program's function as well as a brief background to the concepts of tiling and combinatorials. This paper appears in print in Nexus Network Journal 2 (2000): 15-16.

Modular Games Slavik Jablan (October 2000)
Slavik Jablan, editor of the e-journal VisMath, has created an interactive tiling program for the NNJ. Jablan presents four sets of prototiles called “OpTiles”, “SpaceTiles”, “Orn(amental)Tiles” and “KnotTiles”. Each involves a small set of square tiles which can be combined by the reader in various orientations and reversals to make a bewildering array of designs and patterns. The reader may contemplate his or her constructions at leisure, and with a simple inkjet printer they can be printed out to use in any way you like. This program appears as a Supplementary CD to the Nexus Network Journal 2 (2000).

Hugues Libergier and His Instruments Nancy Wu (October 2000).
One of the most frequently illustrated images of a medieval architect is the tomb slab of Hugues Libergier, architect of the Abbey of Saint-Nicaise in Reims. Hugues (d. 1263) is immortalized by a famous effigy now found in the Cathedral of Reims. As might be expected from the effigy of an architect, it is accompanied by several instruments of his profession: a square, a compass, and a measuring rod. These instruments are frequently found in conjunction with the representation of architects, on tomb slabs, sculpture, in construction scenes on manuscript pages or stained glass panels, the subject of study by scholars in search of the secrets of medieval construction. This paper appears in print in Nexus Network Journal 2 (2000): 93-102.

Methodology in Architecture and Mathematics: Nexus 2000 Round Table Discussion Carol Martin Watts, Moderator (October 2000).
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, the panelists were Rachel Fletcher, Paul Calter, William D. (Bill) Sapp and Mark Reynolds. This report is a transcript of the audio tapes made during the discussion, which covered three areas:

The workshop discussion appears in print in Nexus Network Journal 2 (2000): 105-130.

The Relationship Between Architecture and Mathematics in the Pantheon Giangiacomo Martines (July 2000).
An examination of the latest Pantheon studies illustrates the newest theories of relationships between architecture and mathematics in Rome's most celebrated building. This paper was presented at the Nexus 2000 conference on architecture and mathematics, 4-7 June 2000, Ferrara, Italy. Many studies on the Pantheon are carried out far from Rome and so ideas on the monument cannot be checked easily or frequently. For this reason, a group of architects and archaeologists are working in Rome , trying to resolve some seemingly banal but still unanswered questions. For instance, one question that is often asked is: Could the inside of the Pantheon have been an astronomical observatory? This paper appears in print in Nexus Network Journal 2 (2000): 57-61.

How to Construct a Logarithmic Rosette (Without Even Knowing It) Paul Calter (April 2000)
Paul Calter explains what a logarithmic rosette is and gives some examples of their occurrence in pavements. Then he gives a simple construction method which is totally geometric and requires no calculation. He then proves that it gives a logarithmic rosette, with the exception that the spirals are made up of straight-line segments rather than curved ones. This paper appears in print in Nexus Network Journal 2 (2000): 25-31.

Under Siege: The Golden Mean in Architecture Michael Ostwald (April 2000)
Michael Ostwald briefly describes the Golden Mean and its history before examining the stance taken by a number of recent authors investigating the Golden Mean in architecture. He addresses the theories of Husserl, Derrida and Ingraham, who separately affirm that tacit assumptions about the relationship between geometric forms and other forms - say geometry and architecture - must be constantly questioned if they are to retain any validity. This paper appears in print in Nexus Network Journal 2 (2000): 75-81.

Pythagorean Triangles and the Musical Proportions Martin Euser (April 2000)
Martin Euser researches the factor root-(2N - 1) and its interesting relations between musical proportions and Pythagorean triangles. The simple scheme N +/- root-N is also interesting as a generative set of pairs of numbers. This set looks like a prototype for the generative set of pairs of numbers discussed in a previous article by the author. The findings are presented summarily and it is left to the reader to elaborate upon them. This paper appears in print in Nexus Network Journal 2 (2000): 33-40.

Pavements as Embodiments of Meaning for a Fractal Mind Terry M. Mikiten, Nikos A. Salingaros, Hing-Sing Yu (April 2000)
This paper puts forward a fractal theory of the human mind that explains one aspect of how we interact with our environment. Some interesting analogies are developed for storing ideas and information within a fractal scheme. The mind establishes a connection with the environment by processing information, this being an important theme seen during the evolution of the brain. The authors assert that pavements play a role in connecting human beings to surrounding structures by acting as a vehicle for conveying meaning, and argue that the design on pavements transfers meaning from our surroundings to our awareness. This paper appears in print in Nexus Network Journal 2 (2000): 63-74.

Pisa baptistry is giant musical instrument, computers show Rory Carroll (April 2000)
A music professor at the University of Pisa and a Catholic priest have joined forces to show that the extraordinary acoustics of the Baptistery in Pisa are intentional and that it is a large musical instrument.

The Architecture of Curved Shapes Kazimierz Butelski (January 2000)
In the 20th century, architecture remains the part of art where formal principles are very important for creators and spectators. Because form in architecture is so important, two questions arise: How can architects nowadays create forms? How can forms be described and classified? When we work only with formal analysis, we can point to an important criterion of innovation, that is, that certain forms have never before been seen in the history of architecture. In the present day, CAD/CAM technology permits us to realize any form our imaginations can create. This paper appears in print in Nexus Network Journal 2 (2000): 19-25.

Environmental Patterns: Paving Designs by Tess Jaray Kim Williams (January 2000)
There is no greater opportunity for mathematics and architecture to interact than in paving designs. Where walls are often broken by windows, doors and pilasters, or are covered by paintings, and ceilings (especially modern ceilings) are occupied by lighting fixtures, air vents and smoke alarms (once called "ceiling acne" by architect Robert Stern), floors are usually large unbroken surfaces. For this reason, pavement design has flourished from ancient times. Kim Williams discusses the pavements for urban centers and public spaces designed by British Artist Tess Jaray. Jaray's patterns are derived from the proportional properties of the bricks she uses, and are inspired by the centuries' old masonry tradition. Jaray's designs are a geometric link between architecture and mathematics. This paper appears in print in Nexus Network Journal 2 (2000): 87-92.

A Geometrical Ensemble to Generate the Squaring of the Circle Helmut Sander (English version) (January 2000)
The purely geometrical squaring of the circle with straightedge and compass is possible only within the tolerance of an approximation. But knowing the value of the irrational number pi of the circle (p = 3,14159265 ...), it is possible to transform it as a line or rather as a shape of a circle or a square. This paper appears in print in Nexus Network Journal 2 (2000): 83-85.

Ein Schaubild zur Kreisquadratur Helmut Sander (Deutsche Version) (January 2000)
Die rein geometrische Kreisquadratur init Zirkel und Lineal hat sich längst als nur annähernd möglich erwiesen. Weil aber die Kreiszahl Pi mit p = 3,14159265... bekannt ist, lässt sie sich trotzdem als Strecke und sogar als Kreis- oder Quadratfläche darstellen.

In the Footsteps of the Prince: A Look at Renaissance Ferrara Charles M. Rosenberg. (October 1999)
The narrow cobblestone streets of Ferrara, some scarcely wider than a footpath, give a real sense of what the city was like in the middle ages and early Renaissance: the Via Chiodaiuoli, street of the ironmongers, crossed by a file of slim, brick buttresses; the Via Ragno, lined by typical red-brick houses with protruding sporti; the dramatic Via Volte, bridged by a succession of enormous pointed vaults supporting the second and third stories of buildings which actually span the roadway; the still vibrant arcaded commercial Via Romano, as well as the more twisting paths in the district of the castrum. The history of Ferrara and its princes has left a clear and readable imprint on the city's streets, palaces and churches. Written in their stones is the memory of what has gone before. (Ferrara was the site of the Nexus 2000 conference on architecture and mathematics). No longer online, this paper appears in print in the Nexus Network Journal 1 (1999): 43-63.

A Comparative Geometric Analysis of the Heights and Bases of the Great Pyramid of Khufu and the Pyramid of the Sun at Teotihuacan Mark Reynolds. (October 1999)
Looking back into the murky mysteries of ancient times, there are reminders of past glories in the art, architecture, and design of our ancestors, and, in the number systems they employed in those designs. These number systems were clearly expressed in the geometry they used. Among these works are the mammoth pyramids that dot the Earth's surface. Accurate in their placement as geodetic markers and mechanically sophisticated as astronomical observatories, these wonders of ancient science stand as reminders that our brethren of antiquity may well have known more than we think. This paper appears in print in Nexus Network Journal 1 (1999): 23-42.

Study the Works of Peter Eisenman? Why?! Adriana Rossi (English version). (October 1999)
In architecture it is possible to demonstrate, as Peter Eisenman states, "...all the changes can in some way refer to cultural changes... the most tangible changes... were determinated by technological progress, by the development of new conditions of use and by the change in meaning of certain rituals and their field of representation" [Eisenman, 1989]. Thus in the simple use of geometric solids, he limits himself to the promotion of a language orientated with a correspondent systematic order. In the spatial manipulations of plans and sections, Eisenman experiments with the "laws of thought" (1854) put in place in the nineteenth century by George Boole and Augustus De Morgan. In the same way that the two English logicians brought to extreme consequences the Aristotelian syllogisms which prelude to mechanised reasoning, Peter Eisenman manipulates an idea, submitting it to a sort of propositional calculation. Through probings and attempts which follow each other in a sequence of approximations made possible by a new conception of notation and representation, and beginning with elementary solids or simple internal relations, architectural space takes shape. This paper appears in print in Nexus Network Journal 1 (1999): 65-74.

Studiare le opere di Peter Eisenman? Perché?! Adriana Rossi (versione italiana). (October 1999).
In architettura, si può dimostrare -come afferma Peter Eisenman- che: "...tutti i cambiamenti possono in qualche modo far riferimento a cambiamenti culturali... i mutamenti più tangibili... sono stati determinati dal progresso tecnologico, dallo sviluppo di nuove condizioni d'uso e dal cambiamento del significato di certi rituali e del loro campo di rappresentazione" [Eisenman, 1989]. Cosicchè nel fare semplice uso di solidi geometrici, si limita a promuovere un linguaggio orientato insieme a un corrispondente ordine sistematico. Nelle manipolazioni spaziali di piante e sezioni Eisenman sperimenta le "Leggi del pensiero" (1854), messe a punto nell'ottocento da George Boole e Augustus De Morgan. Come i due logici inglesi portavano alle estreme conseguenze i sillogismi aristoteliani che preludono ai ragionamenti meccanizzati, così Peter Eisenman manipola l'idea, sottoponendola ad una sorta di calcolo proposizionale.

Architectural Traces of an Admirable Cipher: Eleven in the Opus of Carlo Scarpa Marco Frascari. (July 1999)
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. No longer available online, this paper appears in print in Nexus Network Journal 1 (1999): 7-21.

Architecture and Mathematics in the Gothic of the Mendicants Marcello Spigaroli (English version) (July 1999)
The universal essence of beauty consists of the resplendence of form on the material parts in proportion. This luminous statement by Albertus Magnus could be chosen as the synthesis of the esthetic thought of the thirteenth century, and more generally, of the entire late medieval period. The whole range of philosophy and science of this period centers on the theme of proportional relationships as the origin of unity, coherence and the intelligibility of the universe and its infinite parts. From the mendicant orders would come the major exponents of the scientific philosophy, the assumptions of which hinged on the principle of proportions. The city is the theatre where beauty and truth coincide in celebration of political power founded on a mercantile economy, justifying at once an ideology and a way of life. No longer available online, this paper appears in print in Nexus Network Journal 1 (1999): 105-115.

The Sky Within: Mathematical Aesthetics of Persian Dome Interiors Reza Sarhangi. (July 1999)
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. No longer available online, This paper appears in print in Nexus Network Journal 1 (1999): 87-97.

Cosmati Pavements at Westminster Abbey John Sharp (April 1999).
Architecture in thirteenth century England was as much of a textbook as it was a shelter. John Sharp examines one of the most beautiful "texts": the decorated pavements created by Cosmati artists for Henry III. Besides explaining technical details of the panels such as materials and workmanship, Sharp reveals the number symbolism of the inscription that surrounds the Great Pavement, showing how sacred meaning was encrypted in a mathematical symbol system. No longer available online, this paper appears in print in Nexus Network Journal 1 (1999): 99-104.

Spirals and Rosettes in Architectural Ornament Kim Williams (April 1999).
By now noted for both its frequency and its many variations in nature, the spiral has inspired architectural forms for many centuries. The logarithmic spiral was adapted by the Greeks for the ionic volute; many generations of architects developed geometrical constructions to approximate the curves of the spiral. A development on the theme of the spiral is the fan pattern, in which spiral segments are translated about the center of a circle. The superimposition of opposing fan patterns results in the rosette. The easily-constructed circular rosette is an ancient and beautiful pavement pattern, and can be varied to lay the base for many other motives. No longer available online, This paper appears in print in Nexus Network Journal 1 (1999): 129-138.

Re-issued! The Mathematics of Palladio's Villas: Workshop '98 Stephen R. Wassell (April 1999)
Stephen Wassell describes the aims and results of the 1998 and 1999 workshop tours of the villas of Renaissance architect Andrea Palladio. An interdisciplinary group of scholars took advantage of visits to nine villas in Italy's Veneto region to examine Palladio's use of proportions, geometry and symmetry. A review of the literature purtaining to Palladio's use of these mathematical principles sets the stage for new work to be produced by workshop participants. This paper appears in print in Nexus Network Journal 1 (1999): 121-128.

Re-issued! "Triangulature" in Andrea Palladio Vera W. de Spinadel (January 1999).
At the June 1998 workshop on the architecture of Andrea Palladio, the dimensions of the rooms were much remarked. Vera Spinadel convincingly argues that Palladio used precise mathematical relationships as a basis for selecting the numerical dimensions for the rooms in this villas. The integer dimensions are demonstrated to be approximants linked to continued equations, and a particular way of deriving these integers through the use of a continued fraction expansion that approximates by excess is introduced. This paper appears in print in Nexus Network Journal 1 (1999): 117-119.

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