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de natura instrumentorum

Welcome to the Fall 2012 project section of ARCH531 (Architectural Intentions from Vitruvius to the Renaissance), a lecture course taught by Prof. Alberto Perez-Gomez at McGill University, School of Architecture, with this project segment conceived and taught by me.

The project theme for this term was de natura instrumentorum, or the nature of instruments, and following books 9 and 10 in Vitruvius’s Ten Books of Architecture, we explored the status of instruments and machines from antiquity through to the Renaissance. In Exercise 1, early forms of computing (e.g. paper machines or volvelles, astrolabes) were investigated, and Exercise 2 involved replicating the map of Rome as well as the ‘horizon’ instrument invented by Leon Battista Alberti (1404-1472). Exercise 3 explored machines, mechanisms, and contrivances from antiquity to the Renaissance, using them as inspiration towards creating original work.

Please click on the links above to see the project and exercise descriptions, and to leave comments.

Yelda Nasifoglu, project instructor for Fall 2012


Exercise 3: Drawing machine – deconstructing a 2-D image


Our initial research focused on several drawing machines from the Renaissance such as Jacopo Barozzi da Vignola’s Bi-Dimensional perspectograph (developed between 1527-1545), Jacques Besson’s Drawing Machine from Theatrum instrumentorum et machinarum (1578), Albrecht Dürer’s perspectograph (Dürer’s Door – 1525) and other machines that have since been reiterated by people such as JH Lambert in 1752 and recently for purposes of abstract art by Eske Rex.

 Inspired in most part by a machine developed by an unknown artist we saw on the internet, we became interested in developing a drawing machine that would generate the cams that could produce a 2-D drawing.

A cam changes the input motion, which is usually rotary, to a reciprocating motion of the follower. Cams can be shaped to change the way the follower moves. The shape of the cam is called the profile. If a cam rotates clockwise, the movement of the follower in 1 would be a gradual rise and fall motion for each rotation; the follower in 2 would rise and fall twice for each rotation; in 3, the follower would be motionless for half the cycle and then rise and fall; and the follower in 4 would rise gradually and then fall suddenly. Note that 1, 2 and 3 would have the same result if the cam was rotating anticlockwise; 3 would jam. These aren’t the only shapes you can use. Anything goes…depending on what sort of motion you want…though there are practical limitations to making these things out of wood.

Our drawing machine, thus, acts to develop a shape (not necessarily circular) that would deconstruct a drawing into horizontal and vertical data. Theoretically, this data could in turn be plugged back into the process to generate the original drawing.

Next we decided to fabricate a machine that would automate step two. For this to occur, a series of gears would navigate two arms vertically and horizontally. These arms are connected to the “master” pen that traces a 2-dimensional drawing. The veritcal and horixzontal arms hold pens to record data on a moving canvas. We decided to deconstruct Henri Matisse’s “Blue Nude” (1952). Matisse’s painting is a rather simple outline of 2-Dimensional figure, one that would be fairly easy to trace with our machine. Furthermore, we were intrigued to see the outcome that would occur by deconstructing an already semi-abstract painting.

Exercise 3: Jack Bian’s Verge and Foliot Escapement Clock


A verge and foliot is a clock reduced to its simplest elements. A suspended mass M provides a torque that drives the verge and foliot escapement (at top) alternately in one direction and then the other.
The verge and foliot escapement mechanism permits a mathematically simple illustration of a dynamical system exhibiting a limit cycle and stability. Similar to the previous exercise in the construction of the astrolabe and Alberti’s map of Rome, the Verge and Foliot Escapement follows a mathematical logic. One can determine time by the dial’s position or the mass’s location above the ground.

The beauty lies in the physical connection to the surrounding world. The clock is gravity-driven and one could sense this connection as the mass drops. Like how the astrolabe relate to the heavenly bodies, the Verge and Foliot relate to the earthly body.

The suspended weight causes the gear wheel to rotate. This rotation brings a peg into contract with one of the pallets. The rotation brings a peg into contact with one of the pallets, causing the verge and foliot escapement to rotate. By the time that the escapement rotation has reached angle P, the right gear wheel has disengaged the right pallet, and now the left gear wheel engages the left pallet, causing the escapement to rotate in the opposite sense. Because of the inertia provided by the foliot masses, the gear wheels’ rotation is interrupted. The result is a regular oscillation of the escapement and a slow rotation of the gear wheels.

Giovanni Dondi’s Astrarium

by Jean-Roch Marion

When first confronted with the task of making a paper astrolabe as our first assignment, I became interested in the history of different types of medieval astronomical computers (clocks, astrolabes, equatoriums) in order to understand the origins of modern technology and modern thought. Contrary to contemporary electronics, the function of mechanical computers can be perceived by our senses and allow us to understand complicated processes with a simple physical action. Just like clocks and watches, the astrolabes were instruments created in order to compute what was perceived as natural cycles (the moon, the sun and the stars). They simulate the movement of the main celestial objects on a small scale map of the skies engraved on discs that you can manipulate to compute various data. Behind the front plate of the astrolabe, you can also find mathematical instruments that allow you to make complicated operations physically (trigonometry, conversion of units, etc.) Geared versions of astrolabes were also created in order to process more complex data. The earliest mechanical computer to be discovered is a calendar computer mechanism found in a shipwreck off the coast of Antikythera in Greece dating back to 80 B.C. (De Solla Price). In 1000 A.D., Muslim scholar al-Biruni describes a geared calendar that features similar mechanisms to the Antikythera and is found in various astrolabes dating from the Medieval Era. These instruments are at the origin of the common mechanical clock (Boudet) that rapidly become common integrations to towers of cathedrals of the 13th century.

dondis astrarium

As a first step to this third project of the semester, I wanted to reproduce an early mechanical computer in order to understand and its mechanism. Researching for detailed drawings and photographs of mechanical calendars and astrolabes lead me to discover Giovanni Dondi’s astrarium: a 14th century astronomical clock which displayed the position of the sun, the moon, five planets as well as the date and time on various dials. On display at the Castello Visconteo at Pavia, the astrarium was a complicated simulacrum of the geocentric universe following the Ptolemaic theory of motion of the planets. It was apparently marvelled upon by Leonardo Da Vinci, who sketched dials of the astrarium in his notebooks (Bedini et Maddison). Dondi also produced a manuscript that describes his masterpiece in detail entitled Tractatus astrarii. Since the original machine disappeared 150 years after its creation, this manuscript allowed replicas of the astronomical clock to be made. A model of the astrarium is on display at the Smithsonian Institute.


Using the Tractatus astrarii manuscript as well as various photographs of the existing replica of the astrarium, I reproduced the mechanism of its moon dial using thick chipboard and wooden dials. This mechanism simulates an epicyclical movement of the moon’s orbit: a eccentric elliptical rotation, as well as a circular oscillation. The complexity of the mechanism is hard to understand, as it integrates both sliding rules and rotating gears. Just like Leonardo in his time, I wanted to draw the movement of to the moon dial in order to understand it. Based on Leonardo’s drawings of the transmission rods found in his Codex Madrid I, I installed a drawing arm to the moon dial in order to trace its movement on paper. I discovered that the machine that I fabricated produced an elaborate oscillating shape with 14 indents that harmonizes every 5th rotation of the dial.

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Bedini, Selvio A. and Francis R. Maddison. “Mechanical Universe: The Astrarium of Giovanni De Dondi.” Transactions of the American Philosophical Society (1996): 1-69.

Boudet, Jean-Patrice. “L’apparition des horloges mécaniques en Occident.” Revue Historique (1998): 145-154.

De Solla Price, Derek. “Automata and the Origins of Mechanism and Mechanistic Philosophy.” Source: Technology and Culture (1964): 9-23.

—. “Gears From The Greeks. The Antikythera Mechanism: A Calendar Computer from ca. 80 B.C.” Transactions of the American Philosophical Society 64.7 (1974): 1-70.

Leonardo’s Machines: Secrets and Inventions in the Da Vinci Codices, Firenze: Giunti, 2005

Sawday, Jonathan, Engines of the Imagination: Renaissance Culture and the Rise of the Machine, New York: Routledge, 2007

Lefevre, Wolfgang, ed., Picturing Machines 1400-1700, Cambridge, MA: The MIT Press, 2004.

Abstract Art by a Gearograph by C.Wong

In Leonardo Da Vinci’s machines, gears are often the main mechanism that initiates rotation and movement. To combine my personal interest and part of Leonardo’s invention, I decided to design a drawing machine from scratch, which can be used to create my own abstract art.



Gearograph, a combination of gear; operates like a pantograph and a spirograph, but with a sense of freedom and unpredictability. Testing on material was the main part of the research, as well as reading different sources of books for inspiration. Two dimensional images contain a capacity for spatial illusion and I think this perspective of abstract art is related to architectural design.

Image    Image


Abstraction comes from the world. The interesting aspect of creating an abstract art is that the author controls the image but not the reaction to it. “Composition, harmony, proportion, light, color, line texture, mass, and motion are all part of the vocabulary of sight, we tap this vocabulary, and the pattern that go with it. When we compose or frame images the commonality that allow us to respond to images, even abstract ones, is rooted in our ability to recognize infinite manifestation of the physical world and the mental constructs to which they correspond.” – Kit White, 101 Things to Learn in Art School

Red, Grey and Black  Red, Grey and Black | pen on 51x 60cm paper

Sandstorm in Pieces Sandstorm in Pieces| Pen on 56x76cm paper



Exercise 3 / project: the contrivances_Omar Alameddine

Vitruvius mentions in Book X of The Ten Books On Architecture, that the astronomical bodies are connected by a mechanism of revolution which rests on the circular geometry as its trajectory. So the items that govern our lives on earth are bound by a geometry which could be imitated and used for human convenience. Vitruvius explains the components of such a contraption emphasizing two main elements: the circle and the line. Combined together, these elements form a machine or an engine which facilitates different tasks such as hoisting materials for construction. Other items stem from this technological advance such as military machines: The Ballistae, The Catapult, and other siege weapons.

The interest on the circular form is not unwarranted. A few experiments with different forms as the backdrops of a rotary machine has led to the realization that the circle is probably the ideal form by which one uniform force can be transformed into another uniform force with the same magnitude but a different direction. The illustrations below show three different geometries which were the subject of this experiment. The result for the square background was an interruption of movement focused around the edges of the square. The result for the elliptical background was an interruption of movement focused around the far ends of the ellipse. The circular background returned an uninterrupted motion.

vitruvius 2 [Converted]

vitruvius 1 [Converted]

vitruvius 3 [Converted]




A prominent figure in the history of Renaissance Art, Leonardo Da Vinci, has displayed an interest in the potential of the machine. Several designs of rotary machines figure in his sketches. The book entitled Leonardo’s Machines: Secrets and Inventions in the Da Vinci Codices reveals many of Da Vinci’s sketches which focused on different functional machinery. His sketches would often describe the different elements of the composition along with the means to assemble them.

An interesting aspect of Da Vinci’s design is that he uses the circle oriented in one direction to manipulate another circle in another direction. For this exercise, I will address the different possibilities generated by this change of orientation to produce a machine not as a tool for production rather a product in itself. Buildings are characterized with having a specific program. By using a mechanical process such as those described in both Vitruvius’ writings and Leonardo Da Vinci’s sketches; is there a way to manipulate the program of a building; or at least its envelope?.

The Facade that has been developed using this mechanical system introduces an architecture that actually announces its activity. If the pattern is opened up, that would mean that the space behind it is active; if the pattern is closed off, that would mean that the space is currently inactive. In this modern age, architects have been aspiring to create architecture that would reveal its function. A government building has a typology that is different from that of a residential building. To push the boundaries of that definition to the point where the architecture would reveal if the space is occupied or not is a breakthrough. Also, when the facade is opened up, it allows light to shine through the openings and creates a patterned shadow on the ground which would move as the day passes.


A Study of the Human Head

Inspired by Leonardo’s drawing of the Vitruvian man, I became interested in the study of human proportion and its evolution over the Renaissance Era. I focused my research by comparing the work of two Renaissance artists, Piero Della Francesca and Albrecht Durer.

Elevations and Horizontal Outlines of the Human Head | Piero della Francesca

Elevations and Horizontal Outlines of the Human Head | Piero della Francesca

In Piero’s manuscript entitled De Prospectiva Pingendi (On the Perspective for Painting) written between 1474 and 1482, he produces a drawing of the human head entitled Elevations and Horizontal Outlines of the Human Head. Through this drawing he attempted to realize the ideal proportions of the human head through significant geometrical and cosmological numbers. His drawing consists of two groups of four sectional plans; radially divided into sixteen sections and two elevations; horizontally segregated into 8 parts. He intersected these two drawings to plot a series of data points on a polar coordinate grid. Through orthographic projection, used these points to reorient and draw the head at any angle. This surveying technique is very similar to that of Alberti in his map of Rome, which Alberti describes by saying,  “The man who possesses them [the numbers] can so record the outlines and position and arrangement of the parts of any given body in accurate and absolutely reliably written forms that not merely a day later, but even after a whole cycle of the heavens, he can again at will situate and arrange the same body.”(Smedley 2001)

In 2001, artist Geoffrey Smedley showcased his work at an exhibition put on at the Canadian Centre for Architecture entitled Meditations on Piero. His work consisted of a series of drawings and sculptural meditations based on Piero’s drawings of the human head. His study engaged concepts of cosmology, perspective, anatomy and surveying, apparent in Piero’s work. In one of his models entitled The Numbers, Smedley constructed a sculpture that consisted of nine horizontal surfaces, which carefully follows Piero’s mapping of the head. This inspired me to construct a model as a way to materialize my own interpretation of Piero’s drawing.

The Numbers | Geoffrey Smedley

The Numbers | Geoffrey Smedley

model based on Piero's drawings

model based on Piero’s drawings

Albrecht Durer’s theory of human proportion derived from the geometrical canon that Leonardo used to construct the Vitruvian man. Durer did not generalize man through idealized proportions, rather recognized the immense diversity of the human form that existed in society. In his book entitled Four Books on Human Proportion he states, “If you wish to make a beautiful figure, it is necessary that you probe the nature and proportions of many people: a head from one; a breast, arm, leg from another.”( Durer 2003) To create the drawings illustrated in this book, he first defines a system of geometrical measure.  He then adjusts this system using relative proportion and fractional relations to conform to different body types. His system divides the profile of the head into eight sections and front of the head into seven.

Human Proportions | Albrecht Durer

Human Proportions | Albrecht Durer

Human Proportions | Albrecht Durer

Through the imposition of Durer’s system on Piero’s head I found that their divisions were essentially the same, excluding the addition of one division line between the top of the head and the top of the hairline in Piero’s drawing. Durer also applied his method of drawing perspective as a way to reorganize the grid to draw different orientations of the head.

Durer's proportions overlaid on model of Piero's head

Durer’s proportions overlaid on model of Piero’s head

A modern exploration by Stephan Marquardt featured in a four-part BBC documentary entitled The Human Face, examines beauty in reference to symmetrical proportions. He used to the golden ratio to create a mask known as the Golden Mast, which he claims encapsulates absolute, universal beauty. He explains his view by stating, “the closer the face conforms to this mask, the more beautiful it is.” (Human Head, BBC One 2001)

Golden Mask | Stephan Marquart

Golden Mask | Stephan Marquart


“Albrecht Durer: 1471-1528.” The Metropolitan Museum of Art. 2000-2012 <http://www.metmuseum.org/toah/hd/durr/hd_durr.htm&gt;

“Collections: Printed Books & Bindings.” The Morgan Library & Museum. New York. <http://www.themorgan.org/collections/collections.asp?id=577&gt;.

Dodds, George., Tavernor, Robert., Rykwert, Joseph. “Body and Building: Essays on the Changing Relation of Body and Architecture.” Massachusetts: Massachusetts Institute of Technology, 2001.

Durer, Albrecht. “De symmetria partium in rectis formis humanorum corporum : Nuremberg.” 1532 Oakland: Octavo Corp, 2003.

Field, J.“Piero Della Francesca: A Mathematician’s Art.”London, 2005

Franscesca, D. Piero. De Prospectiva Pingendi. Firenze: Casa Editrice Le Lettre, 1984.

Gates, H. William. The Art of Drawing The Human Figure. London: Bailliere, Tindall and Cox.

Neher, Allister. “Albrecht Durrer and Nicholas Suanus: the Real, the Ideal, and the Quantification of the Body.” Luxemburg: Jean-Paul Riopelle, 1992. <http://www.uqtr.uquebec.ca/AE/Vol_11/libre/Neher.htm&gt;

Human Head. Dir. Erskine, James., Stewart, David. BBC One, 2001.

Smedley, Geoffrey. Meditations on Piero: Sculptures by Geoffrey Smedley. Montreal: Centre Canadien d’Architecture/Canadian Centre for Architecture, 2001.

Williams, Kim., Lionel, March., Wassel, Stephen., “The Mathematical Works of Leon Battista Alberti.” Washington: Birkhauser, 2010.