Abstracts of Research Articles Available Online and in Print

A Generative System for Mamluk Madrasa Form-Making Buthayna H. Eilouti and Amer M.I. Al-Jokhadar. Nexus Network Journal vol. 9 no. 2, pp. 7-30.
In this paper, a parametric shape grammar for the derivation of the floor plans of educational buildings (madrasas) in Mamluk architecture is presented. The grammar is constructed using a corpus of sixteen Mamluk madrasas that were built in Egypt, Syria, and Palestine during the Mamluk period. Based on an epistemological premise of structuralism, the morphology of Mamluk madrasas is analyzed to deduce commonalities of the formal and compositional aspects among them. The set of underlying common lexical and syntactic elements that are shared by the study cases is listed. The shape rule schemata to derive Mamluk madrasa floor plans are formulated. The sets of lexical elements and syntactic rules are systematized to form a linguistic framework. The theoretical framework for the formal language of Mamluk architecture is structured to establish a basis for a computerized model for the automatic derivation of Mamluk madrasa floor plans.

A Computer-Aided Rule-Based Mamluk Madrasa Plan Generator Buthayna H. Eilouti and Amer M.I. Al-Jokhadar. Nexus Network Journal vol. 9 no. 2, pp. 31-58.
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.

Curve Fitting in Architecture Dirk Huylebrouck. Nexus Network Journal vol. 9 no. 2, pp. 59-70.
It used to be popular to draw geometric figures on images of paintings or buildings, and to propose them as an "analysis" of the observed work, but the tradition lost some credit due to exaggerated (golden section) interpretations. So, how sure can an art or mathematics teacher be when he wants to propose the profile of a nuclear plant as an example of a hyperboloid, or proportions in paintings as an illustration of the presence of number series? Or, if Gaudi intended to show chain curves in his work, can the naked eye actually notice the difference between these curves and parabolas? The present paper suggests applying the "least squares method", developed in celestial mechanics and since applied in various fields, to art and architecture, especially since modern software makes computational difficulties nonexistent. Some prefer jumping immediately to modern computer machinery for visual recognition, but such mathematical overkill may turn artistic minds further away from the (beloved!) tradition of geometric interpretations.

Non-Orthogonal Features in the Planning of Four Ancient Towns of Central Italy Giulio Magli. Nexus Network Journal vol. 9 no. 2, pp. 71-92.
Several ancient towns of central Italy are characterized by imposing circuits of walls constructed with the so-called polygonal or "cyclopean" megalithic technique. The date of foundation of these cities is highly uncertain; indeed, although they all became Roman colonies in the early Republican centuries (between the fifth and third centuries B.C.) their first occupation predates the Roman conquest. It is the aim of the present paper to show - using four case-studies - that these towns still show clear traces of an archaic, probably pre-Roman urbanistic design, which was not based on the orthogonal "rule", i.e., the town-planning rule followed by the Greeks, Etruscans and Romans. Rather, the layouts appear to have been originally planned on the basis of a triangular, or even star-like, geometry, which therefore has a center of symmetry and leads to radial, rather than orthogonal, organization of the urban space. Interestingly enough, hints - so far unexplained - pointing to this kind of town planning are present in the works by ancient writers as important as Plato and Aristophanes, as well as in the comment to the Æneid by Marius Servius.

Integrated Function Systems and Organic Architecture from Wright to Mondrian James Harris. Nexus Network Journal vol. 9 no. 2, pp. 93-102.
The development of an architectural form where the individual parts reflect the integrated whole has been a design goal from ancient architecture to the current explorations into self-organizational structures. Organic architecture, with this part-to-whole association as an element of its foundation, has been explored from its incidental use in vernacular structures to its conscious endorsement by Frank Lloyd Wright. Traditionally Piet Mondrian has not been associated with organic architecture but a closer examination of the artistic and philosophical underpinnings of his work reveals a conceptual connection with organic architecture.

Traditional Patterns in Pyrgi of Chios: Mathematics and Community Charoula Stathpoulou. Nexus Network Journal vol. 9 no. 2, pp. 103-118.
Ethnomathematical research has revealed interesting artifacts in several cultures all around the world. Although the majority of them come from Africa, some interesting ones exist in Western cultures too. Xysta of Pyrgi are a designing tradition that concerns the construction of mainly geometrical patterns on building façades by scratching plaster. The history and the culture of the community, the way that this tradition is connected with them, as well as the informal mathematical ideas that are incorporated in this tradition are some of the issues that are explored here.

The Arch: Born in the Sewer, Raised to the Heavens Matthys P. Levy. Nexus Network Journal vol. 8 no. 2, pp. 7-12.
The great ancient civilizations all knew about the arch yet the Greeks relegated its use to underground sewers and never raised an arch above ground: Why? The Egyptians, the Babylonians, the Assyrians and of course the Romans all exploited the arch as a means of spanning and enclosing space. Yet, curiously, Greece, one of the most cultured of the ancient civilizations and the builder of magnificent temples used stone in a most unnatural way, as a beam. The arch as a construction technique is intimately connected to the material of which it is constructed, namely masonry. Stone, when used as a beam is limited by size, scale and proportion. The Greeks certainly understood this as they closely spaced columns to support stone lintels. They also meekly tried to use stone in an A-frame configuration using a corbelled arch in Mycenea (1325 BC) but they never made the transition to the true arch using stone voussoirs. WHY? Perhaps the answer lies in a lack of understanding of the mechanics of the materials and the nature of compression and tension.

As Hangs the Flexible Line: Equilibrium of Masonry Arches Philippe Block, Matt Dejong, John Ochsendorf. Nexus Network Journal vol. 8 no. 2, pp. 13-24.
In 1675, English scientist Robert Hooke discovered "the true… …manner of arches for building," which he summarized with a single phrase: "As hangs the flexible line, so but inverted will stand the rigid arch." In the centuries that followed, Hooke's simple idea has been used to understand and design numerous important works. Recent research at MIT on the interactive analysis of structural forces provides new graphical tools for the understanding of arch behavior, which are useful for relating the forces and geometry of masonry structures. The key mathematical principle is the use of graphical analysis to determine possible equilibrium states.

Galileo Was Wrong! The Geometrical Design of Masonry Arches Santiago Huerta. Nexus Network Journal vol. 8 no. 2, pp. 25-52.
Since antiquity master builders have always used simple geometrical rules for designing arches. Typically, for a certain form, the thickness is a fraction of the span. This is a proportional design independent of the scale: the same ratio thickness/span applies for spans of 10m or 100m. Rules of the same kind were also used for more complex problems, such as the design of a buttress for a cross-vault. Galileo attacked this kind of proportional design in his Dialogues. He stated the so-called square-cube law: internal stresses grow linearly with scale and therefore the elements of the structures must become thicker in proportion. This law has been accepted many times uncritically by historians of engineering, who have considered the traditional geometrical design as unscientific and incorrect. In fact, Galileo's law applies only to strength problems. Stability problems, such as the masonry arch problem, are governed by geometry. Therefore, Galileo was wrong in applying his reasoning to masonry buildings.

Gateway to Mathematics. Equations of the St. Louis Arch Paul Calter. Nexus Network Journal vol. 8 no. 2, pp. 53-66.
Eero Saarinen's Gateway Arch in St. Louis has the form of a catenary, that is, the form taken by a suspended chain. The catenary can be reproduced empirically, but it can also be precisely formulated mathematically. The catenary is similar to the paraboloid in shape, but differs mathematically. Catalan architect Antoni Gaudi used the catenary to great effect in his Church of the Sagrada Familia in Barcelona, but he also used the paraboloid as well.

Arches and Culture. Donald L. Hanlon. Nexus Network Journal vol. 8 no. 2, pp. 67-72.
Technological innovation is the driving force of our civilization. Therefore, we assume all other civilizations would exploit a technological advantage to the same degree that we would. We forget, however, that technology is an aspect of culture, and as with any other aspect of culture, it may be more or less important to any given civilization. The history of the arch is an interesting case in point. The arch is a structural device in architecture that has distinct advantages over post-and-beam construction. People have known how to build the arch and how to use it since the third millennium B.C., but did not use it because its form and meaning did not fit with other dimensions of their respective cultures.

Solving Ertha Diggs's Ancient Stone Arch Mystery Michael Serra. Nexus Network Journal vol. 8 no. 2, pp. 73-78.
According to legend, when the Romans made an arch, they would make the architect stand under it while the wooden support was removed. That was one way to be sure that architects carefully designed arches that wouldn't fall! Educator Michael Serra led AAAS symposium participants in a surprising and fun hands-on arch construction project using familiar objects-Chinese take-out cartons-in an unfamiliar way: "these are stone voussoirs from an ancient miniature bridge uncovered by my friend, archaeologist Ertha Diggs. She has asked us to determine the number of stones in the original bridge." This makes it possible to understand both arch mechanics and the mathematics behind the arch through actually constructing them.

Mathematical Elements in Historic and Contemporary Architecture Elena Marchetti and Luisa Rossi Costa. Nexus Network Journal vol. 8 no. 2, pp. 79-92.
Starting from the idea that Mathematics plays an important role in planning any aesthetically attractive and functional construction, this work focuses on curves and surfaces easily recognisable in buildings. Many contemporary examples, but also some intriguing forms connected by classical geometrical questions are illustrated. Nowadays as well in the past, architects often give a splendid interpretation of the beauty of Mathematics; at the same time they introduce modern aspects of this important subject, related to the social and environmental field.

On Division in Extreme and Mean Ratio and its Connection to a Particular Re-Expression of the Golden Quadratic Equation x²-x-1 = 0. J. Iñiguez, A. Hansen, I. Pérez, C. Langham, J. Rivera, J. Sánchez and J. Acuña. Nexus Network Journal vol. 8 no. 2, pp. 93-100.
The golden quadratic x²-x-1=0, when re-expressed as (x)(1)=1/(x-1), x=1.618, can be interpreted as the algebraic expression of division in extreme and mean ratio (DEMR) of a line of length into a longer section of length 1 and a smaller of length x-1. It can, however, also be interpreted as the formulation of the area of a golden rectangle of sides x=1.618 and 1, and as the system of equations constituted by y=x, and y=1/(x-1). Based on the well-known connection existing between the first two of these interpretations, the authors address the problem of finding out the thread connecting the golden rectangle with the system of equations referred to above. The results obtained indicate first that this system, like the golden rectangle, also carries in its geometry the essential traits of DEMR; and, second, that it implicitly subsumes the simpler rectangular geometry of its alternative interpretation. The process of developing these connections brought forward a heretofore apparently unreported golden trapezoid of sides PHI, 1, phi, and root-2.

Structure of Phenomenological Forms: Morphologic Rhythm Luisa Consiglieri and Victor ConsiglieriNexus Network Journal vol. 8 no. 2, pp. 101-110.
The images in architecture are handed down through mathematical forms. The meaning of the plastic value of the forms and the conflict between their visual boundaries are a result of the geometrical composition of the object. Since Stonehenge in Britain, the Egyptian pyramids, the Greek Parthenon or the Roman Pantheon, architecture has been a reflex of simple boundaries without accidental confrontations. Nowadays materials are organised through movement/change in order to represent the required profile. This developed structure emerges in the artistic manifestations according to the theory of continuity. As an expression of the formal quality in opposition to the ancient characteristics of quantity, a new conception of rhythm appears. The concept of a cell as an architectural element that can have any biological form and can be grouped itself according to different ways and functions (such as repetition of floors) is introduced. This concept of cell permits eurhythmy (harmony in the proportion of a building) through the notion of rhythm once all the elements of a building are situated among themselves

The Acropolis of Alatri: Architecture and Astronomy Giulio Magli. Nexus Network Journal vol. 8 no. 1, pp. 5-16.
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.

The Stylistic Characteristics of the Shampay House of 1919: A Formal Analysis Jin-Ho Park. Nexus Network Journal vol. 8 no. 1, pp. 17-32.
This paper analyzes the stylistic characteristics of the Shampay House with a series of formal methodologies. It focuses on three parts: spatial arrangement, symmetry and proportion. For the thorough analysis, archival drawings are enhanced through reconstructing new drawings and through the building of a quarter-inch scale model.

Timely Timelessness: Traditional Proportions and Modern Practice in Kahn's Kimbell Museum Steven Fleming and Mark Reynolds. Nexus Network Journal vol. 8 no. 1, pp. 33-52.
The twentieth century witnessed declining interest in architectural proportioning systems, which were virtually eclipsed by technical, social and fiscal agendas. Louis Kahn is a seminal architect, whose most acclaimed building, the Kimbell Art Museum (1966-72), represents a compelling case-study in the use proportions by twentieth-century architects. In spite of a raft of peculiarly modern restrictions (both technological and programmatic), Kahn appears - despite his espoused ambivalence concerning proportion - to have intentionally produced a building with an array of approximate geometrical as well as precise harmonic proportions.
This two-part paper presents the findings of a multifaceted research project that examined the Kimbell's proportions from numerous standpoints. Part 1 presents a textural analysis of Kahn's statements regarding proportion, as well as the findings of an archival study of correspondence between the architect and his client and consultants. Part 2 presents a prima facie geometrical analysis of the construction drawings for the project. The division into parts reflects an apparent discrepancy between Kahn's buildings and what he had to say about them.

Origins of an Obsession DJP Marshall. Nexus Network Journal vol. 8 no. 1, pp. 53-64.
Though many geometric shapes can be constructed from circles, this paper is about the geometric square. It will be demonstrated that while the square is not the easiest of the polygons to construct initially, it is both easily enlarged and easily subdivided. Step-by-step manipulations of the square provide an explanation for the architectural design of the Forum of Augustus.

Nexus Architecture and Mathematics. Aims -- Methods -- Criteria David Speiser. Nexus Network Journal vol. 7 no. 2, pp. 7-9.
David Speiser, frequent Nexus conference participant and presenter, discusses what the aims, methods, and criteria for research into the relationships between architecture and mathematics should be

A Tale of Bridges: Topology and Architecture Jean-Michel Kantor. Nexus Network Journal vol. 7 no. 2, pp. 13-21.
In modern times geometry has had a new development : topology, a field with more freedom and new dreams for the mathematician and the architect.We describe some of its successes and problems, from Euler to Poincaré, from Riemann to strings.

Mathematics, Astronomy and Sacred Landscape in the Inka Heartland Giulio Magli. Nexus Network Journal vol. 7 no. 2, pp. 22-32.
It is very well known that the "Inca space" was a sacred space in which directions, places, monuments, springs and so on all had a sacred content. In recent years, new insights into this complex cosmographic view have been obtained with the study of the so-called Cusco ceque system. Further, new insights have been obtained in the field of Incan astronomical lore, with the identification of Incan dark cloud constellations of the Milky Way. Giulio Magli proposes possible new connections between the Inka view of the sky, the Inka system of notation of numbers and dates called Khipus, and the sacred landscape of the capital of the empire.

How Should We Study Architecture and Mathematics? Sandro Caparrini and David Speiser. Nexus Network Journal vol. 6 no. 2, pp. 7-12.
The 1996 paper by John Clagett on "Transformational Geometry and the Central European Baroque Church," presented at the first Nexus conference on architecture and mathematics is taken as a starting point in a discussion that intends to shed light on how to study the Nexus of Architecture and Mathematics.

Andrea Palladio's Villa Cornaro in Piombino Dese Branko Mitrovic. Nexus Network Journal vol. 6 no. 2, pp. 15-30.
Villa Cornaro in Piombino Dese is one of Andrea Palladio's most influential works. As for many of Palladio's buildings, modern surveys do not exist, are incomplete, omit information about important aspects such as the use of the classical orders, or have been published without dimensions indicated in the plans. The analysis presented here is based on a June 2003 survey of the villa made by Steve Wassell, Tim Ross, Melanie Burke and author Branko Mitrovic. In his treatise, Palladio listed his preferred room types: circular, square or rectangular with length-to-width ratios 2/1, 3/2, 4/3, 5/3 or Ö2/1. Half a century ago, this kind of speculative search for the comprehensive interpretation of Palladio's proportional system received great impetus from Rudolf Wittkower's Architectural Principles in the Age of Humanism. It is, however, important to differentiate between the derivation of certain proportional rules and their explanation.Wittkower asserted thatthe use of ornamentation -- and especially the orders -- did not matter in Palladio's design process. Refuting this theory, Mitrovic argues that Palladio, in the early 1550, formulated a very different approach to the use of the orders, combining the principle of preferred room proportions and the use of a columnar system to determine the placement of walls. The proportions of the main sala and porticos are derived on the basis of the proportional rules for the order used; the proportions of the side rooms on the basis of preferred ratios. Ultimately, the result is that the mathematics of the orders became decisive for Palladio's design principles and the use of proportions from the early 1550s.

Geometric Methods of the 1500s for Laying Out the Ionic Volute Denise Andrey and Mirko Galli. Nexus Network Journal vol. 6 no. 2, pp. 31-48.
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.

Musical Symbolism in the Works of Leon Battista Alberti: From De re aedificatoria to the Rucellai Sepulchre Angela Pintore. Nexus Network Journal vol. 6 no. 2, pp. 49-70.
On the basis of a new survey, Angela Pintore analyzes the micro-architecture of the Rucellai Sepulchre in Florence, because the sepulchre is 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 summit 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.

Guarino Guarini, Mathematics and Architecture: The Restoration of the Chapel of the Shroud in Turin
Mirella Macera, Paolo Napoli, Fernando Delmastro. Interview by Kim Williams, edited by Sandro Caparrini
(English version) Nexus Network Journal vol. 6 no. 2, pp. 73-90.
Guarini'sChapel of the Shroud in Turin, a major monument of the Italian Baroque, was davastated by fire in 1997. Plans are now underway for its restoration. An important intial phase of the restoration project is to understand Guarini's original design process and the construction techniques used. In this interview with Mirella Macera, Paolo Napoli and Fernando Delmastro, coordinators of the restoration project, the nature of the damage caused by the fire, the steps taken thus far to stabilize the structure, and new discoveries about the Chapel as a result of the fire are examined. The interview is by Kim Williams, edited by Sandro Caparrini.

Guarino Guarini, matematica e architettura: Il restauro della Cappella della SS. Sindone a Torino
Mirella Macera, Paolo Napoli, Fernando Delmastro. Intervista di Kim Williams, curato da Sandro Caparrini
(versione italiana) (Autumn 2004)
I problemi del restauro della Cappella della SS. Sindone di Guarino Guarini a Torino, destrutto dall'incendio nel 1997, sono stati esaminati in un intervista con i responsabile per il restauro arch. Mirella Macera, ing. Paolo Napoli e arch. Fernando Delmastro in un intervista di Kim Williams curato da Sandro Caparrini.

The Use of the Golden Section in the Great Mosque of Kairouan Kenza Boussora and Said Mazouz. Nexus Network Journal vol. 6 no. 1, pp. 7-16.
The geometrical analysis conducted reveals very clearly a consistent application of the golden section. The geometric technique of construction of the golden section seems to have determined the major decisions of the spatial organisation. The golden section appears repeatedly in some part of the building measurements. It is found in the overall proportion of the plan and in the dimensioning of the prayer space, the court and the minaret. The existence of the golden section in some part of Kairouan mosque indicates that the elements designed and generated with this principle may have been realised at the same period. This suggests and opens the possibility for further inquiry into the dating of the transformations that took place in this mosque. Authors Kenza Boussora and Said Mazouz examine earlier archaeological theories about the mosque, demonstrate the geometric constructions for the golden section, and apply these constructions to the plan of the mosque to test their hypothesis.

Philosophy and Science of Music in Ancient Greece: Predecessors of Pythagoras and their Contribution Graham Pont Nexus Network Journal vol. 6 no. 1, pp. 17-29.
Although the writings of the classical Greeks and their Roman and Arabic successors remain the foundation of western philosophy and science of music, as well as their sometimes problematic applications to architecture and other constructive arts, there has been a steady renewal of interest in the old science of harmonics, and it is recognized that much of the Greek theory and practice of harmonics was unquestionably derived from earlier cultures, the still shadowy predecessors of Pythagoras. Though hardly any modern writers would describe themselves as Pythagoreans, some of their ideas have important connections with the old tradition and all are symptomatic of a new era in the history of thought when mechanistic and reductionist paradigms are giving way to a holistic and organic world-view. Modern scholarship has established that most of the doctrines traditionally ascribed to Pythagoras were really the contributions of the older high civilisations, particularly of Mespotamia and Egypt. The rise and dissemination of these perennially influential doctrines remains one of the most formidable problems for the historian of ideas. Graham Pont.

The Fractal Nature of the Architectural Orders Daniele Capo Nexus Network Journal vol. 6 no. 1, pp. 30-40.
Daniele Capo tests with regards to architectural elements certain concepts that are proper to fractal geometry. The purpose is not to show that the architectural orders are true fractal objects, but rather that how fractal "instruments" can be used to approach certain objects and what kinds of information can be gleaned by such an approach. Understanding the orders, which for centuries have provided the basis for Western architecture, in light of the analysis presented above, allows us to observe, through the analysis of numerical data, how small elements are inserted in a continuous and coherent whole. If we interpret this structure fractally we do not distinguish between the essential and the inessential; everything is essential and so creates in this way a greater (fractal) coherence. It could be said, in this light, that the general form is not what counts the most, but rather, what is really important is the way in which parts hold together.

An American Vision of Harmony: Geometric Proportions in Thomas Jefferson's Rotunda at the University of Virginia Rachel Fletcher. Nexus Network Journal vol. 5 no. 2, pp. 7-47.
Thomas Jefferson dedicated his later years to establishing the University of Virginia, believing that the availability of a public liberal education was essential to national prosperity and individual happiness. His design for the University stands as one of his greatest accomplishments and has been called "the proudest achievement of American architecture." Taking Jefferson's design drawings as a basis for study, this paper explores the possibility that he incorporated incommensurable geometric proportions in his designs for the Rotunda. Without actual drawings to illustrate specific geometric constructions, it cannot be said definitively that Jefferson utilized such proportions. But a comparative analysis between Jefferson's plans and Palladio's renderings of the Pantheon (Jefferson's primary design source) suggests that both designs developed from similar geometric techniques.

Rudolph M. Schinder: Space Reference Frame, Modular Coordination and the "Row" Lionel March. Nexus Network Journal vol. 5 no. 2, pp. 48-59.
While Rudolph Schindler's "space reference frame" is becoming better known, its relationship to the "row" has only been recently investigated. The theory of the "row" counters traditional proportional notions, many of which are derived from the principle of geometric similitude: a principle which is mostly represented in architectural drawings by regulating lines and triangulation. Here, Lionel March presents the simple mathematics of row theory. A short background note concludes the paper.

Rudolph M. Schindler: Proportion, Scale and the "Row" Jin-Ho Park. Nexus Network Journal vol. 5 no. 2, pp. 60-72.
Jin-Ho Park interprets Rudolph M. Schindler's 'reference frames in space' as set forth in his 1916 lecture note on mathematics, proportion and architecture, in the context of John Beverley Robinson's1898-99 articles in the Architectural Reconrd. Schindler's unpublished, handwritten notes provide a source for his concern for "rhythmic" dimensioning in architecture. He uses a system in which rectangular dimensions are arranged in "rows". Architectural examples of Schindler's Shampay, Braxton-Shore and How Houses illustrate the principles.

The Cinema as Secular Temple: Ethos, Form and Symbolism of the Capitol Theatre Graham Pont. Nexus Network Journal vol. 5 no. 2, pp. 73-99.
Since the Tetraktys is the most important symbol of the Pythagorean school and system, its prominence in the form and decoration of the Capitol Theatre, Melbourne (Australia), suggests that this building was designed in the Pythagorean spirit. The Tetraktys encodes the fundamental proportions or harmonies of the musical scale (1:2, 2:3, 3:4) and so one would expect to find these same proportions used in the form and decoration of the Capitol as well as evidence of relevant musical thought and inspiration in the writings of its designers, Walter Burley and Marion Mahony Griffin. Graham Pont explains the significance of the Tetraktys in the Pythagorean tradition, identifies the Capitol as a "secular temple" in form and ethos, and indicates possible contemporary influences on the theatre's remarkable motif of the "Crystal Tetraktys" and other symbolism.

Perspective as a Symmetry Transformation György Darvas. Nexus Network Journal vol. 5 no. 1, pp. 7-21.
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. Nexus Network Journal vol. 5 no. 1, pp. 22-48.
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. Nexus Network Journal vol. 5 no. 1, pp. 49-63.
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. Nexus Network Journal vol. 5 no. 1, pp. 64-98.
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. Nexus Network Journal vol. 5 no. 1, pp. 99-124.
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.

Mathematics and Design: Yes, But Will it Fly? Martin Davis and Matt Insall. Nexus Network Journal vol. 4 no. 2, pp. 9-13.
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.

How Should We Measure an Ancient Structure? Harrison Eiteljorg, II. Nexus Network Journal vol. 4 no. 2, pp. 14-20.
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.

A Light, Six-Sided, Paradoxical Fight Marco Frascari. Nexus Network Journal vol. 4 no. 2, pp. 21-37.
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. Nexus Network Journal vol. 4 no. 2, pp. 38-53.
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.

The Double Möbius Strip Studies Vesna Petresin and Laurent-Paul Robert. Nexus Network Journal vol. 4 no. 2, pp. 54-64.
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. Nexus Network Journal vol. 4 no. 2, pp. 65-78.
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.

Nexus 2002 Round Table Discussion: Mathematics and Architecture Education Moderated by Judith Moran, with the participation of Mario Kruger, Arselio Martins, Maurizio Vianello, Stephen Wassell, and Jean Brangé. Nexus Network Journal vol. 4 no. 2, pp. 81-100.
The round table discussion on mathematics in the architecture curriculum took place at the Nexus 2002 conference, 17 June 2001. Moderated by Judith Flagg Moran, the questions were discussed: Is mathematics a necessary part of the education of an architect? What effect does the increasing use of computers have on the mathematics that an architect needs to know? What kind of skills are required at the secondary level to prepare students adequately to prepare students to do architectural work at university?

The Golden Section in Architectural Theory Marcus Frings. Nexus Network Journal vol. 4 no. 1, pp. 9-32.
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. Nexus Network Journal vol. 4 no. 1, pp. 33-43.
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. Nexus Network Journal vol. 4 no. 1, pp. 45-58.
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. Nexus Network Journal vol. 4 no. 1, pp. 59-82.
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.

Comments on the Nexus 2000 Round Table Discussion Leonard K. Eaton. Nexus Network Journal, vol. 3, no. 2, pp. 7-12.
Leonard K. Eaton reflects on the Nexus 2000 Round Table Discussion and on the relationships of architecture and mathematics through history.

The Engineering Achievements of Hardy Cross Leonard K. Eaton. Nexus Network Journal vol. 3 no. 2, pp. 15-24.
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.

Palladio's Villa Emo: The Golden Proportion Theory Rebutted Lionel March. Nexus Network Journal vol. 3 no. 2, pp. 85-104.
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. Nexus Network Journal vol. 3 no. 2, pp. 105-112.
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. Nexus Network Journal vol. 3 no. 2, pp. 113-126.
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. Nexus Network Journal vol. 3 no. 2, pp. 127-136.
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 Michael Leyton. Nexus Network Journal vol. 3 no. 2, pp. 39-58.
This is a tutorial on the mathematical structure of architecture. The purpose of these tutorial 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. Nexus Network Journal vol. 3 no. 2, pp. 59-84.
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.

Euclidism and Theory of Architecture Michele Sbacchi. Nexus Network Journal vol. 3 no. 3, pp. 25-38.
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.

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. Nexus Network Journal vol. 3 no. 1, pp. 43-72.
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. Nexus Network Journal vol. 3 no. 1, pp. 99-109.
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. Nexus Network Journal vol. 3 no. 1, pp. 11-15.
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.

Iannis Xenakis - Architect of Light and Sound Alessandra Capanna. Nexus Network Journal vol. 3 no. 1, pp. 19-26.
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.

"Fractal Architecture": Late Twentieth Century Connections Between Architecture and Fractal Geometry Michael J. Ostwald. Nexus Network Journal vol. 3 no. 1, pp. 73-84.
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. Nexus Network Journal vol. 3 no. 1, pp. 85-98.
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. Nexus Network Journal vol. 3 no. 1, pp. 27-42.
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.

The Arithmetic of Nicomachus of Gerasa and its Applications to Systems of Proportion Jay Kappraff. Nexus Network Journal 2 (2000): 41-55.
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.

Introduction to Slavik Jablan's Modular Games Donald W. Crowe. Nexus Network Journal 2 (2000): 15-16.
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.

Modular Games Slavik Jablan (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. Nexus Network Journal 2 (2000): 93-102.
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.

Methodology in Architecture and Mathematics: Nexus 2000 Round Table Discussion Carol Martin Watts, Moderator. Nexus Network Journal 2 (2000): 105-130.
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 Relationship Between Architecture and Mathematics in the Pantheon Giangiacomo Martines. Nexus Network Journal 2 (2000): 57-61.
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?

How to Construct a Logarithmic Rosette (Without Even Knowing It) Paul Calter. Nexus Network Journal 2 (2000): 25-31.
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.

Under Siege: The Golden Mean in Architecture Michael Ostwald. Nexus Network Journal 2 (2000): 75-81.
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.

Pythagorean Triangles and the Musical Proportions Martin Euser. Nexus Network Journal 2 (2000): 33-40.
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.

Pavements as Embodiments of Meaning for a Fractal Mind Terry M. Mikiten, Nikos A. Salingaros, Hing-Sing Yu. Nexus Network Journal 2 (2000): 63-74.
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.

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. Nexus Network Journal 2 (2000): 19-25.
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.

Environmental Patterns: Paving Designs by Tess Jaray Kim Williams. Nexus Network Journal 2 (2000): 87-92.
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.

A Geometrical Ensemble to Generate the Squaring of the Circle Helmut Sander. Nexus Network Journal 2 (2000): 83-85.
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.

In the Footsteps of the Prince: A Look at Renaissance Ferrara Charles M. Rosenberg. Nexus Network Journal 1 (1999): 43-63.
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).

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. Nexus Network Journal 1 (1999): 23-42.
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.

Study the Works of Peter Eisenman? Why?! Adriana Rossi. Nexus Network Journal 1 (1999): 65-74.
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.

Architectural Traces of an Admirable Cipher: Eleven in the Opus of Carlo Scarpa Marco Frascari. Nexus Network Journal 1 (1999): 7-21.
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.

Architecture and Mathematics in the Gothic of the Mendicants Marcello Spigaroli. Nexus Network Journal 1 (1999): 105-115.
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.

The Sky Within: Mathematical Aesthetics of Persian Dome Interiors Reza Sarhangi. Nexus Network Journal 1 (1999): 87-97.
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.

Architecture, Patterns and Mathematics Nikos Salingaros. Nexus Network Journal 1 (1999): 75-85
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.

Cosmati Pavements at Westminster Abbey John Sharp. Nexus Network Journal 1 (1999): 99-104.
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.

Spirals and Rosettes in Architectural Ornament Kim Williams Nexus Network Journal 1 (1999): 129-138.
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.

The Mathematics of Palladio's Villas: Workshop '98 Stephen R. Wassell. Nexus Network Journal 1 (1999): 121-128.
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.

"Triangulature" in Andrea Palladio Vera W. de Spinadel Nexus Network Journal 1 (1999): 117-119.
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.

Copyright ©2007 Kim Williams Books