Kim Williams

• B. Architectural Studies
• Editor at Birkhauser Publishers, Basel

David Speiser and I first met through correspondence in 1995, after I had published an article in The Mathematical Intelligencer on the geometry and the proportions of the pavement layout of the Baptistery of Florence. At that time, those of us who worked on theories and applications of proportion, geometry, symmetry and similar mathematical concep...

The aim of this research is to evaluate not so much the quantity and quality of mathematical notions that writings by Vitruvius in antiquity and Villard de Honnecourt in the late middle ages conveyed to their readers and to posterity, but rather the quantity and quality of the theoretical mathematical background that the authors appear to have lear...

A sixteen-folio treatise by fifteenth-century polymath Leon Battista Alberti, entitled Ludi matematici, contains an explanation of how to aim a bombard using a device that Alberti calls an “equilibra”. The aims of this present paper are: to examine Alberti’s exposition of and solution to the bombard problem, at last providing what we believe to be...

Daniele Barbaro examined ‘geometric ratio’ at length in his 1567 commentary on Vitruvius, explaining the properties of and operations with ratios involving continuous quantities such as lengths. While some of this is lost on us today, his explanations were certainly clear to those of his contemporaries who were studying Euclid thanks to new transla...

This is the first-ever English translation of Daniele Barbaro’s 1567 Italian translation of and commentary on Vitruvius’s Ten Books of Architecture, an encyclopaedic treatment of science and technology whose influence extended far beyond its day. Intended to both interpret and expand upon the Vitruvian text, Barbaro’s erudite commentary reflects hi...

This is the English translation of the treatise on ratio and proportion published by the Vicentine engineer Silvio Belli in 1573. In it the author aims to present the material, which was the object of much study and application in the Renaissance on the part of architects and artists, in a way that facilitates learning. The work consists of three “...

Editor-in-Chief Kim Williams discusses the versatility and range of tools used in exploring questions of form-finding, both ancient and modern, and introduces the articles in Nexus Network Journal vol. 24 no. 4 (2022). She also makes an important announcement about the future of the NNJ.

Editor-in-Chief Kim Williams discusses the versatility and range of tools used in exploring questions of form-finding both ancient and modern, and introduces the articles in Nexus Network Journal vol. 24no. 1 (2022).

I will treat that part of perspective that the Greeks call scenographia, that is, the description of the scenes, which, with admirable rationale (ragione) of lines regulated by a point according to the distances, shows the surfaces of bodies, the reliefs, the moving away and coming back, the projections of the fabrics of all the bodies, and even mo...

Before I come to describe plans, it is necessary to practice the way of describing the figures called ‘polygons’, that is, those having many equal angles and sides circumscribed by a circle. So, the description of the triangle is conjoined to the definition of the figure of 6 equal sides and angles.

The half-sphere is used to make sun clocks in various planes, but that serves us for only one elevation. We have imagined an instrument that is not only a clock in itself, but is also an instrument for making clocks in every plane, at every elevation, and with every sort of hour.

When Barbaro alludes to his forthcoming treatise on perspective in his commentary on Vitruvius, it is to make two promises: to explain the ways in which perspective interweaves two distinct sciences – geometry and natural vision – into one,and to show how an understanding of certain geometric figures are a great help in many things, ‘and especially...

There is no doubt at all that the power of perspective depends on the nature of the angles, and that the appearance of things changes when the centre and the eye change, and that it is not sufficient for the erudite and expert to see and consider the bare lineaments of bodies and shapes put into perspective in order to judge things.

Many times, with no less delight than marvel, are seen some panels or sheets of perspective in which, when the viewer’s eye is not placed in a determined point, what appears is something other than that which is depicted, but then, when viewed from its point, there appears that which was actually made according to the intention of the painter, be i...

I hope that the difficulty of the things before this will make scenographia appear easy, for all that I have said in the three previous parts will be of use for it.

When Daniele Barbaro published his treatise in 1568 there had not yet appeared a text on perspective that was without errors and ambiguities.

Magnificent and excellent brother, our studies having been shared for many of our years under the care of good preceptors, and the knot of friendship between us having been tied by the virtue of honest and thus firm connections, it comes to me to publish our closeness under part of those titles (whatever they may be) that with worthy efforts we hav...

A fine, ingenious and useful invention was that of the ancients for casting the points and circles of the sphere onto planes with proportion and correspondence of ratios, because its demonstrations were of the greatest benefit to the investigators of celestial things.

There are three ways that are most used to raise bodies from their plans, of which the first is the following.

The human body has by various persons been measured variously. Some have regarded the larger parts; others the smaller parts.

Editor-in-Chief Kim Williams examines the sometimes very sophisticated use of fundamental mathematical elements—curves, grids, simple polygons and polyhedra—in ancient and contemporary architecture and introduces the articles in Nexus Network Journal vol. 23 no. 3 (2021).

Three years ago, at Nexus 2018, we could never have imagined how much our conferences – indeed, how much our whole world – would change as a result of the COVID-19 pandemic that began in early 2020. The conference that we had planned to take place in the Kaiserslautern, Germany, in July 2020 was suspended in hopes that 2021 would allow us to meet i...

Editor-in-Chief Kim Williams examines the way intersections of architecture and mathematics in ancient architecture are being brought to light thanks to increased information and new technologies, and introduces the articles in Nexus Network Journal vol. 23 no. 2 (2021).

A year after the second edition of his famous translation and commentary on Vitruvius, Daniele Barbaro published The Practice of Perspective, a text he had begun working on many years before. Barbaro was the first to publish a formal treatise entirely dedicated to the science of geometric perspective. In an informal style especially addressed to pr...

Editor-in-Chief Kim Williams examines the role of vision and interpretation in establishing relationships between architecture and mathematics, and introduces the articles in Nexus Network Journal vol. 22 no. 3 (2020).

Editor-in-Chief Kim Williams examines the role of the search for pattern in establishing relationships between architecture and mathematics, and introduces the articles in Nexus Network Journal vol. 22 no. 2 (2020).

Nexus 2018, the 12th international, interdisciplinary conference for architecture and mathematics, took place in Pisa in June 2018. From among the 43 presentations, 12 papers were selected for publication in the Nexus Network Journal. In what follows the conference directors the papers that make up vol. 22, no. 1 (2020).

Nexus 2018, the 12th international, interdisciplinary conference for architecture and mathematics, took place in Pisa in June 2018. From among the 43 presentations, 13 papers were selected for publication in the Nexus Network Journal. In what follows the conference directors the the papers that make up vol. 22, no. 1 (2020).

Geometry plays a number of fundamental roles in architecture: as support for structure, as an enrichment of experience, as a powerful design tool, as an endower of beauty. Such additional factors for the examination of geometry—including the impact of perception, representation, construction and reception—are crucial for understanding architectural...

Cover, Silvio Belli, On Ratio and Proportion

The entirety of fine works, most illustrious and reverend Sir, the more they are looked at and contemplated by men, the more they reveal their beauty and the skilfulness of the master, though often at first glance one feels none of the appreciation that comes later when they have been carefully examined and considered. I need not exert myself to pr...

Prudently and usefully our elders decided to leave to posterity by means of reports in the commentaries the thoughts of their minds, so that they would not be lost, but increasing in every age, and brought to light with the volumes, little by little over a great age they would arrive to the highest subtlety of the doctrines. Thus it is not with lit...

In the name of God Glorious, I, Daniele Barbaro, a Venetian nobleman, committed myself to expose and interpret the ten books of architecture by M. Vitruvius. My intention with this honest endeavour was to be of use to the scholars of artful inventions, and to provide an opportunity for others to write more clearly about those things (since many of...

Having completed the part that was dedicated to religion, there follows that devoted to what is convenient and opportune for citizens, in which will be shown the arrangement of the forum, the basilica, the aerarium, the curia, the prison, the theatre, and the things that appertain to the theatre such as stage sets, porticoes, and tiers, the baths,...

Delphic Apollo, in the reply given to Pythia, affirmed Socrates to be the wisest of men. This one (it is said) with prudence and great learning said that the breasts of men should be as windows, and open, so that their feelings were not hidden, but were open for consideration. Would God that nature, following the opinion of Socrates, had made men’s...

I having, O Emperor, observed that many have left precepts of architecture and volumes of commentary that are not ordered, but begin with dismembered particles, I thought it a worthy and most useful thing to first reduce the corpus of this discipline to perfect order, and then explicate in each volume the prescribed and certain qualities of the man...

The elders of the Greeks conferred such great honours on those noble athletes who had won the Olympic, Pythian, Isthmian, and Nemean games that not only were they lauded before the multitude of men with the palm and crown but also, when they returned victorious to their own countries, they were carried triumphantly inside the walls in chariots and...

It is said that in Ephesus, noble and large city of the Greeks, there was established by their elders, with hard condition but reasoning that was not unjust, an ancient law whereby the architect, when he sets out to make a public work, first promises how much expense must go into it. Having given the estimate to the magistrates, he pledges his poss...

Thales of Miletus, one of the seven sages, said that water was the principle of all things; Heraclites, fire; the priests of the Magi, water and fire; Euripides, auditor of Anaxagoras, which philosopher the Athenians named Scaenicus, said air and earth, and that the earth, having been impregnated by the celestial rains, generated in the world the k...

Architect Dinocrates, confiding in his thoughts and his astuteness, Alexander being the ruler of the world, departed from Macedonia to go to the army, desirous of being commended by the ruling majesty. Departing from his country, his relatives, and his friends, he obtained letters of recommendation addressed to the principals and powerful ones at c...

Aristippus, Socratic philosopher, thrown by shipwreck onto the shores of the Rhodians, having noticed some figures of geometry in the sand, is said to have exclaimed thus, ‘Let us have hope, O companions, since here I see the traces of men’. Having said this, he hastened without delay to the land of Rhodes, going straight to the gymnasium where, di...

This “Letter from the Editors” begins with reflections on frameworks for research and development of architectural knowledge. Drawing on the both old and new ways of thinking, the editors show how two forms of knowledge—observational and propositional—have figured in the identification of two types of architects: architectus ingenio, the observer a...

This ‘Letter from the Editors’ introduces two theories about the role of play in culture and society. Drawing on the work of Roger Caillois, the editors identify two types of play, broadly aligning each to the concerns and approaches taken in the two parts of this issue of the Nexus Network Journal. Thereafter, they introduce the papers that make u...

Founder Kim Williams relates the history and development of the international conference series "Nexus: Architecture and Mathematics" and the journal that grew out of it, "Nexus Network Journal"

The Co-Editors-in-Chief of the Nexus Network Journal introduce the contents of vol. 19, no. 1 (2017).

The Co-Editors-in-Chief of the Nexus Network Journal introduce the contents of vol. 18, no. 2 (2016).

This is a book review of Giulio Magli’s Archaeoastronomy, a welcome addition to the literature regarding this relatively new science.

The book aims to provide an overview of the state of the art on the mechanics of arches and masonry structures. It is addressed to an international audience, arising from the international context in which the Associazione Edoardo Benvenuto has carried out its activities in recent years, under the honorary presidency of Jacques Heyman. As is well k...

Co-Editor-in-Chief of the Nexus Network Journal, Kim Williams, introduces sixteen papers in vol. 17, no. 1 (2015).

The frontispiece of the thirteenth century Bible Moralisee conserved in Vienna portrays a Christ-like figure leaning over a primordial world and using a pair of compasses to measure and inscribe its limits (Fig. 3.1). Titled ‘God as architect of the world’, it depicts the use of a mathematical instrument to determine the functional, symbolic and ae...

Every age and every culture has relied on the incorporation of mathematics in their works of architecture to imbue the built environment with meaning and order. Mathematics is also central to the production of architecture, to its methods of measurement, fabrication and analysis. This two-volume edited collection presents a detailed portrait of the...

Three ways of thinking about the history of ideas“the revolutionary, the reactionary and the revivalist”are useful when examining connections and relationships between architecture and mathematics from the sixteenth century to the present, and projecting them into the future. Here the authors provide an overview of historical developments in the pr...

This paper reports on a survey of an architectural element in Michelangelo’s Laurentian Library. The element is the portal that exits the reading room and leads into the vestibule. The information obtained is significant as the information in terms of what it reveals about Michelangelo’s use of a proportional system. The dimensions of the portal ma...

The tombslab by Verrocchio commemorating Cosimo de’ Medici, patriarch of the wealthiest of Florentine families, is a relatively small memorial marker laid in the floor of the crossing in the basilica of San Lorenzo in Florence (Fig. 45.1). In spite of its size, it contains interesting lessons on the rich relationships between mathematics and design...

What is the nature of the relationship between architects and mathematicians between or architecture and mathematics? As they are commonly understood these two groups seem to have few obvious connections. The word ‘architecture’ is used to describe either the practice of creating buildings or a particular class of constructed—architectural—objects....

Every age and every culture has relied on the incorporation of mathematics in their works of architecture to imbue the built environment with meaning and order. Mathematics is also central to the production of architecture, to its methods of measurement, fabrication and analysis. This two-volume edited collection presents a detailed portrait of the...

Archaeoastronomer Giulio Magli is no stranger to readers of the Nexus Network Journal. He began sharing his research with us in 2005, and has authored or co-authored several papers (2005, 2006, 2007, 2009a), and most importantly, was guest editor of a special issue of the NNJ devoted to architecture, mathematics and archaeoastronomy (NNJ 15, 3, 201...

Alberti's contribution to the nascent science of fortifications in the 1450s is often ignored, but a careful reading of his descriptions of fortifications show that he was the first to describe the elements of fortification formally in the precise mathematical terms of shape, measurements, relationships, proportions. In forming his ideas, Alberti e...

Editor-in-Chief Kim Williams opens volume 16 number 1 (Winter 2014) of the Nexus Network Journal by presenting new developments for this journal, including the introduction of Michael J. Ostwald as co-Editor-in-Chief.

A three-day workshop on the design and construction of reciprocal structures allowed a group of international scholars to experiment with and exchange ideas about dome structures inspired by the designs of Leonardo da Vinci.

A foreword to Rachel Fletcher's "Infinite Measure"

NNJ editor-in-chief Kim Williams introduces the papers in NNJ vol. 15, no. 2 (Summer 2013).

NNJ editor-in-chief Kim Williams introduces the papers in NNJ vol. 15, no. 1 (Spring 2013).

NNJ editor-in-chief Kim Williams introduces the papers in NNJ vol. 14, no. 3 (Winter 2012).

NNJ editor-in-chief Kim Williams introduces the papers in NNJ vol. 14, no. 2 (Autumn 2012).

NNJ editor-in-chief Kim Williams introduces the papers in NNJ vol. 14, no. 1 (Winter 2012).

The ninth edition of the biennial conference series “Nexus: Relationships Between Architecture and Mathematics” took place at the Politecnico di Milan in June 2012. There are many connections between architecture and mathematics: mathematic principles may be used as a basis for an architectural design, or as a tool for analyzing an existing monumen...

This collection of essays by physicist and historian of science David Speiser on relationships between science, history of science, history of art and philosophy shows how the limits of individual disciplines can be pushed and sometimes completely overcome as the result of input from and interactions with other fields, and how progress may even be...

The eighth edition of the international, interdisciplinary Nexus conference on architecture and mathematics took place from Sunday 13 June through Tuesday 15 June 2010, in Porto, Portugal.

Have you ever seen a blind man teach the way to someone who sees? Here with these briefest of notes, which we call Elements, you will see that someone who perhaps does not himself know how to draw can show the true and sure reasoning and the way to become a perfect draughtsman, as long as you don’t shy away from learning that which you judge to be...

The way of measuring a two-cusped2 figure composed of two curved lines as shown in the figure3 Contrary to the opinion of the many who say that figures composed of lines that are curved and circular cannot be squared perfectly, most of all of those that are portions of circles, they say this in my opinion by the authority of Aristotle, who says tha...

Those who are charged with the highest affairs know by experience how important it is to have a very trustworthy person with whom to reveal projects and decisions of the most secret nature, without ever having reasons to regret it.

Se volete col veder[e] sendo in capo d’una piaza misurare quanto sia alta quella torre quale sia a pié d[e]lla piaza fate in questo modo fichate uno1 dardo in terra et fichatelo chegli stia a piombo fermo et poi scostatevi da questo dardo quanto pare a voi òsei òocto piedi et indi mirata alla cima d[e]lla torre dirizando il vostro vedere a mira p[e...

Leon Battista Alberti (1404-1472) was a highly prolific polymath of the fifteenth century. Although his contributions to architecture and the visual arts are well known and available in good English editions, as are many of his literary and social writings, his mathematical works are not well represented. This present volume was planned to fill tha...

With the new architecture still in constant evolution, and its forms fragmented, bent, twisted and folded out of any immediately recognizable shape, a comparison with older, more canonical architecture also helps us to see the new, and to understand what we are seeing. KeywordsMichael Ostwald-Baroque architecture-architecture of complexity-archite...