Author Archives: xproulx

The “Replication Box”

Le projet de recherche est issu d’une expérimentation à l’aide d’instruments de relevés inventés à la Renaissance. Il s’agit d’une suite logique des exercices sur la mesure et la cartographie effectuées dans le cadre de ce cours.

La plupart des instruments de mesures des arpenteurs de l’époque dépendent d’artifices mathématiques dont certains ont été résumés dans l’œuvre de Leon Battista Alberti, Ludi Mathematica. Dans cet ouvrage, il explique diverses astuces mathématiques relatives à l’art de la mesure. Suite à la reproduction de la carte de Rome d’Alberti en deuxième exercice, nous avons défini deux points d’intérêts pour le projet final : la numérisation des points du relevés d’Alberti est indépendante du support sur lequel le relevé a été fait; il est possible pour n’importe qui de reproduire cette carte peu importe sa taille, pourvu que l’instrument qu’il décrit soit reproduit consciencieusement. L’élément fascinant est que sans le savoir, Alberti utilisait un système d’analyse computationnel peu différent que le langage d’un ordinateur moderne. Deuxièmement, l’instrument utilisé par Alberti pour construire sa carte démontre à notre avis l’introduction d’un système de coordonnées polaires appliquées dans l’art de la mesure. Ce système allait définir le point de départ du projet.

 Le projet est né d’une recherche approfondie sur les divers instruments de mesure inventés par Alberti. Nous avons consulté tout d’abord les Ludi Mathematica afin de comprendre comment Alberti avait transposé les éléments remarquables de Rome en nuage de points mathématiques. Il s’avère qu’Alberti utilisait un instrument circulaire de grande dimension déposé sur le sol permettant d’établir à vue les angles des points d’intérêts par rapport au lieu de mesure. Au cours de nos recherches, nous avons lu attentivement le traité De Statua, où Alberti décrit un deuxième instrument de mesure inspiré du premier permettant de mesurer une statue. Alberti donne un système de proportion à son instrument dans la mesure où la hauteur de la statue est égale à 6 pieds. Il présente ensuite l’instrument de mesure, le finitorium. Il s’agit d’un instrument semblable à celui utilisé pour la carte de Rome, excepté qu’il implique maintenant une composante tridimensionnelle à son fonctionnement en la qualité d’un fil à plomb suspendu à son rayon. Ce dernier est de 3 pieds, chacun divisés en 10 uncia et subdivisées en minuta.

En utilisant l’instrument décrit par Alberti, de manière très imprécise disons-le, on comprend qu’il serait possible de diviser le contour de n’importe quel objet en coordonnées spatiales, qu’il serait ensuite possible de retracer sans la présence de l’objet de référence. Nous avons fait d’autres recherches à comprendre exactement le fonctionnement de cet instrument. Nous avons établi que sa reproduction exacte n’avait à notre connaissance jamais été tentée. Mathématiquement, ce sytème repose sur les coordonnées cylindriques où un point dans l’espace est défini par le rayon, l’angle dans le plan horizontal et z, la distance verticale. D’autre part, il s’avère que le principe d’Alberti a été effectivement utilisé par les sculpteurs afin de reproduire des statues. Le principe est le suivant : le sculpteur positionne deux ou trois points de référence majeurs (le coude, le genou, etc) par rapport au centre de l’axe vertical et sait ainsi la position de chaque point par rapport à l’épaisseur du bloc de matériel à tailler.

Le projet final se veut une hybridation des différents principes de mesure inventés par Alberti. Nous avons voulu modifier l’instrument afin de le rendre capable de mesurer n’importe quel objet. Le résultat final est une « boîte de duplication », où un objet peut être inséré, mesuré, transformé en une série de coordonnées tridimensionnelles, et par la suite reproduit. Plus le nombre de points mesurés au départ a été grand, plus la reproduction de l’objet sera précise.

 Le définisseur mesure 30 cm de diamètre et la hauteur de la boîte mesure 24 cm, soit 80% du rayon de mesure. On pourrait étendre ce système de proportion à volonté, en multipliant ces mesures du facteurs de notre choix, afin de mesurer un objet plus gros.

 La première étape est donc d’insérer un objet dans la boîte et d’utiliser la plaque de mesure afin de noter une série de points judicieux en terme de rayon, d’angle et de distance z. Par la suite, si on imagine ne posséder que cette série de coordonnées, il est possible de reproduire le contour de l’objet à l’aide d’une seconde plaque contenant un tableau de coordonnées polaires. On suspend alors chaque point sur cette plaque à l’aide d’un fil à plomb. Le nuage de points qui en résulte reprend alors le contour de l’objet mesuré au départ.

 Finalement, nous avons souhaité expérimenter avec la notion de projection, de vision et de parallaxe en tentant d’introduire une grille de mesure sur les faces latérales de la boîte. Nous espérions pouvoir obtenir une projection parallèle du contour de l’objet une fois les points suspendus dans l’espace. Cette méthode aurait alors pu fournir l’élévation de des contours de l’objet mesuré.

***

The research project is the result of an experiment using surveying instruments invented in the Renaissance. It is a logical step forward from previous exercises on measuring and mapping.

 Most instruments surveyors used at the time depend on mathematical artifices. Some of them were summarized in the work of Leon Battista Alberti, Ludi Mathematica. In this treatise, he explains various mathematical tricks on the art of measurement. Following the reproduction of the map of Rome as a second exercise for this class, we identified tree points of interest for the final project:

 –        Alberti’s surveys coordinates are independent of the medium on which the map has to be drawn. It is possible for anyone to reproduce this map in any size, assuming Alberti’s instrument is conscientiously reproduced.

–        The other fascinating element is that without knowing it, Alberti used a computational analysis system which is in our opinion not much different than the language of a modern computer.

–        Finally, the instrument used by Alberti to build his map introduces the notion of polar coordinate system applied in the art of measurement. This system defined the starting point of the project.

Alberti's Finitorium, as pictured in De Statua treatise.

Alberti’s Finitorium, as pictured in De Statua treatise.

The project originated from extensive research on various measuring instruments invented by Alberti. We first consulted the Ludi Mathematica treatise to understand how Alberti had transposed the remarkable features of his Rome map into a discrete mathematical array of points. It turns out that Alberti used a large circular instrument placed on the ground in order to establish the angles of points of interest in relation to his location on site (For the Map of Rome, the zero coordinate was the summit of Capitola). During our research, we then examined Alberti’s De Statua, where Alberti describes a second measuring instrument used to survey a standing statue. Alberti confers a proportion system into his instrument by defining that the height of the statue to be measured is to be “6 feet” tall. It then presents the measuring instrument itself, which is called the finitorium. It is an instrument similar to the one used for the map of Rome, except that it now involves a component for its operation in a three-dimensional space. The added component is a plumb line suspended from the radius of the instrument. The radius is 3 feet, each divided into 10 uncia and subdivided into minuta. By using this defined proportional system, one’s can enlarge or reduce the instrument according to the height of the statue to be measured.

 By replicating the instrument described by Alberti, we can understand that it would be possible to divide the outline of any object into spatial coordinates. Having these coordinates at hand, it would also be possible to replicate its outline without the presence of the reference object. Some other research was done in order to understand exactly how this instrument could work in a practical way. To our knowledge, an exact replication of the finitorium instrument has never been attempted. Mathematically, this point-by-point survey of a body is based on cylindrical coordinates, where any point in space is defined by it’s radius, it’s angle in the horizontal plane, and z it’s vertical distance from the horizontal plane where the radius sits. On the other hand, it turns out that the measurement principle invented by Alberti was actually picked up by sculptors to reproduce statues. The principle is as follow: the sculptor positioned two or three major reference points (elbow, knee, etc.) from the center of the vertical axis and thus could know the position of each point relative to the thickness of the material to be cut.

The final project is intended to be an hybridization of these different measurement principles invented by Alberti. We wanted to change the above instrument to make it able to measure any object. The end result is a “replication box”, where an object may be inserted, measured, converted into a series of three-dimensional coordinates, and then reproduced. The greater the number of points is measured, the greater the resolution of the reproduced object will be.

 The Finitorium measures 30 cm in diameter and the height of the box is 24 cm, or 80% of the radius span. In order to measure a larger object, this proportion system can be easily extended. One’s only need to multiply these dimensions by a given factor of choice.

 The first step is to insert an object into the box and use the measuring plate to compute a succession of wisely chosen points in terms of radius, angle and z distance. Subsequently, it is possible to reproduce the contour of the object with a second plate containing a polar array. Each point is to be suspended from the plate with a plumb line, with correct z distance. The resulting points cloud will approximate the contour of the object.

 Finally, I wanted to experiment with the concept of projection and vision, by trying to introduce a measurement grid on the sides of the box. That way, I was hoping to catch a parallel projection of the object’s contour points, once being suspended in space. This method could then provide key elevation contour points of the measured object.

Bibliography

 Académie d’Art de Meudon et des Hauts De Seine, Histoire d’une commande imago pietatis Les procédés de reproduction, [online], http://www.academieart-meudon.fr/wdp/wp-content/uploads/2011/10/Texte-rencontre-Agn%C3%A8s-Bracquemond1.pdf

 Alberti, Leon Battista. Ex ludis rerum mathematicarum, in Williams, Kim, The Mathematical Works of Leon Battista Alberti, Birkhäuser, Berlin, 2010.

 Alberti, Leon Battista, De La Statue et de La Peinture, traduit du Latin en Français par Claudius Popelin, A. Lévy Éditeur, Paris, 1868.

 Carpo, Mario, The Alphabet And The Algorithm, MIT Press, Cambridge, 2011, 169p.

 Grafton, Anthony, Leon Battista Alberti Master Builder Of The Renaissance, Hill and Wang, New-York, 2000, 417p.

 Miller, Naomi, Mapping the City, Continuum, London, 2003, 270p.

 Scaglia, Gustina, Instruments Perfected for measurements of man and statues illustrated in Leon

Battista Alberti’s De Statua, Nuncius, Volume 8, Number 2, 1993, pp.555-596.

 Williams, Kim, The Mathematical Works of Leon Battista Alberti, Birkhäuser, Berlin, 2010.

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On Alberti’s surveying instruments

Xavier Proulx

Preliminary research and starting point

The starting point of our project is the history of cartography in the Renaissance, or more precisely, surveying instruments. As a first exercise for this class, we have already investigated the Mariner’s Astrolabe, an ancient surveying instrument (see the previous post on the blog). The logical step forward in our research has taken place in the second exercise, where we have produced Alberti’s map of Rome. That said, this exercise is linked to our area of interest for the final project.

 ***

The way Alberti’s map is constructed is pretty interesting because of the fact that he mistrusted the manuscript copyists and thus provided only a set of polar coordinates in order for the reader to redraw the map. In that spirit, Alberti has been the first computational geometry practitioner in history, suggesting the construction of a city plan on a mathematical basis. By interpreting a set of points in an array, the reader becomes the equivalent of a modern plotter. Alberti’s work is not turned toward drawings – actually there are few drawings in his treaties – he instead relies on words to describe most of his instruments and findings:

 “He simply chose not to use illustrations that would not have be reproducible, replacing them with textual descriptions. In his map of Rome, he instead replaced pictures with computational instructions designed to transform the image in a digital file and then recreate a copy of the original picture when needed.” [3, p.54].

On the other hand, the spirit of a place has been traditionally linked with the sense of vision. You know a place because you see it. With the introduction of surveying instruments, we are beginning to witness a “progressive shift away from the individual human body as a reliable agent for recording spatial information, towards dependence upon instrumentation as the guarantor of accuracy and objectivity in survey data”. [4, p.159]

We shall put maps at the center of attention for this project. At the most basic level, a map, or chorography, is “a mode of description in which truth to the individuality, personality and uniqueness of a place or region was the goal.” [4, p.7] Simply put, mapping was seen as much an art as a science. Alberti’s work is only the beginning of the science of cartography, which later evolved into the art of orthographic projections and picturing machines (cf. Mariano Taccola, Albrecht Dürer, and Gerardus Mercator). However, “Alberti’s images were meant to be carriers of precise quantitative information, and to record measurable data – data that could be used and acted upon.” [3, p.68]

***

Our preliminary research about surveying, cartography, and data collection throughout history attracted our curiosity about the way Alberti conducted his survey of Rome, as reproduced in the second exercise. In particular, we are interested in the way Alberti used mathematics to “order spatial relationships”, a quality that can be shared by both mapmaking and painting during the Renaissance. [7, p.177] Consequently, as a project goal, we want to keep our focus on the genesis of surveying instruments that were used to render these maps.

In particular, the starting point of our project is a desire to explore the instrumentation that lead to Alberti’s map of Rome and the process in which he transformed his survey data into the basics of a digital file. Alberti certainly took great care to keep a sense of mystery on these in his survey of Rome treatise. However he later explained the surveying instruments he used in his Mathematical Games (Ex Ludis rerum mathematicarum) manuscript. Reading this small treatise let us understand Alberti’s interest in mathematics. Several “games” are depicted and many surveying instruments are drawn. We shall therefore narrow down our interest to the surveying instruments developed by Alberti, as seen in this manuscript. Several examples are pictured here such as the equilibra for leveling surfaces and regulating the course of water, weighing objects or aiming a bombard (a true multipurpose instrument!). He also constructed a device for measuring lengths of distances along a road, and addressed the construction of an instrument for measuring the speed of a ship using wind. Finally, he described the instrument he used for his survey of Rome: a circular disc used to measure angles and estimate distances. These instruments will be the starting point of our investigation.

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We believe that the choice of Alberti’s work as a starting point is relevant in the process of conducting this project. His unique work sits right between the humanism of the Renaissance, and the influence of medieval craftsmanship: he mastered all the traditional arts of a medieval courtier and all new ones of the Renaissance intellectual [7]. He lived at a time right at the beginning of the Renaissance, when the influences of the Middle Ages – notably in craftsmanship – were still present. But at the same time he retained the influence of engineers whose knowledge had started to spread, notably in the science of war: “The engineers were augurs as well as mathematicians and artists.”[5] These explain Alberti’s choice for celebrating these disciplines. In his way, he connected the liberal arts of geometry and astrology with the crafts of painting and sculpture [5]. As we began to discover in our preliminary research, Alberti’s time was a period when engineers’ work blended with the architectural practice in the art of building – but also perhaps in the art of surveying?

In order to nourish this project, we are aiming to pursue research on the historical context that pushed Alberti to undertake his surveys and his work on instruments in his Mathematical Games. At the same time, we will soon choose a precise surveying instrument among the ones identified above and begin the project by replicating it, in the hope that it contributes to further innovation for the final step.

Alberti used instruments to translate the vision of a place on paper. However he added an extra step, which was to transform the measurement into a mathematical array of points. Maybe can we replicate and use one of his instruments taken from his Mathematical Games and then apply the same computational data structure on the resulting measurements? Maybe this instrument’s usage can be actualized or modified in regards to modern architecture practice? As such, we believe nowadays architecture projects could benefit from this basic surveying language, especially in the matter of feeling the sense of a place by surveying it first on a more organic approach basis, rather than the purely computerized methods of today.

Preliminary research bibliography

[1] Alberti, Leon Battista. Ex ludis rerum mathematicarum

[2] Brown, Llyod A., The Story Of Maps, Bonaza Books, New-York, 1949, 393p.

[3] Carpo, Mario, The Alphabet And The Algorithm, MIT Press, Cambridge, 2011, 169p.

[4] Cosgrove, Denis. Geography and Vision: Seeing, Imagining and Representing the World. : I.B. Tauris, London, 2008, 256p.

[5] Grafton, Anthony, Leon Battista Alberti Master Builder Of The Renaissance, Hill and Wang, New-York, 2000, 417p.

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

[7] Miller, Naomi, Mapping the City, Continuum, London, 2003, 270p.

[8] Woodward, David, Cartography in the European Renaissance, The University of Chicago Press, Chicago , 2007, 2272p.

[9] Williams, Kim, The Mathematical Works of Leon Battista Alberti, Birkhäuser, Berlin, 2010.

Xavier’s Descriptio Urbis Romae

Synopsis

In Descriptio Urbis Romae, Alberti describes a very simple way to record information from his survey of Rome, thus allowing anybody to redraw the resulting map of the city, based on his observations. The original point of observation of Alterti’s survey– that is point (0,0) in the tables – is the summit of the Capitol. The process used by Alberti is very simple. Strings would be stretched in the air (!), between the survey’s point of origin and the point of interest1. The distance and angle of measurement was then recorded in tables we still have nowadays in Alberti’s book. This surveying method was quite efficient for the time since the steep city of Rome made very difficult a direct measurement on foot (according to Pr. Bruno Queysanne, Queysanne, Bruno, Alberti et Raphaël, Descriptio Urbis Romae ou comment faire le portrait de Rome, Plan Fixe Édition, École d’architecture de Grenoble, Lyon, 87p, p.14.)

Alberti transcribed his measurements in a form that is basically polar coordinates (r,θ). The survey’s map can then be transcribed by anyone, in any chosen format. First, the “horizon” is divided into 48 equal parts (which are called degrees) around a circle (any chosen radius). The north is thus given by the 48th division, south by its 24th, west is number 36 and east number 12. Each of these 48 parts is then divided again in 4 equal parts, which are called “minutes”. This numeric value corresponds to the angle (θ) of the polar coordinates in Alberti’s survey.

The radius coordinate (r) is given by the “spoke”, which is basically a ruler that spins around the center of the map – that is the center of the horizon’s circle. The length of the ruler is the radius of the previously traced circle. It is again divided into 50 equal parts called degrees. These 50 parts are subdivided into 4 “minutes”.

To map of Rome can then be traced point by point by finding the corresponding points in the given tables of Alberti’s book; angle on the horizon scale, then radius on the “spoke” scale. There should be a certain “magic” feeling to see this map, surveyed by Alberti centuries ago, appear before your eyes, points after points. The final step is to connect these points.

It should be noted that there’s two different kind of point in Alberti’s tables. “Corners” point must be connected with a straight line. “Apex” points should be connected with a curved line. It should be the summits of an arc on the final map (see figure 1).

Construction of the instrument 

The instrument was constructed in thick white cardboard (Peterborough wall mounting cardboard). The “spoke” will be attached to the center of the map with a pin so it can be rotated around the horizon. For now we consider a 20 cm long “spoke”. This should give a bigger map of around 40 cm tall, this allowing us to see better details.

Legend

The letters on the map correspond to the differents gates surveyed by Alberti.

  • A: Porta del Popolo
  • B: Porta Pinciana
  • C: Porta Salaria
  • D: Porta La Donna
  • E: Porta San Lorenzo
  • F: Morta Magiore
  • G: Porta del Laterno
  • H: Porta Latina
  • I: Porta Appia
  • J: Porta San Paolo
  • K: Porta Portuense
  • L: Porta San Pancrazio
  • M: Porta Gianicolense
  • O: Porta Santo Spirito Gianicolo
  • P: Rear Gate over the valley
  • Q: Gate on the hill
  • R: Porta Palatina
  • S: Porta Castello

Download the complete paper here. [pdf].

Xavier’s Mariner’s Astrolabe

The mariner’s astrolabe was an instrument used to determine the latitude of a ship at sea by measuring the altitude of the sun at noon or a star of known declination at night (in other words any celestial body above the horizon, measured at its maximum altitude – that is on the meridian through your location). The star or sun declination was then looked up in an almanac. The latitude was given by the formula : (90º- measured altitude) + declination (that is the angle between the equator and sun rays. At the equinox this angle is zero, but it goes up to 23,45º during solstices. It is to find in an almanac for the corresponding altitude depending of the date).

Because it was difficult to know for sure if the sun or star was at it’s maximum altitude at the time of measurement, sailors where usually taking several measurements in order to estimate correct altitude. 

Mariner’s astrolabe are thus very inaccurate instruments, and errors between 4 or 5 degrees where common at the time. The base plate is simply a graduated ring in degrees with an alidade on top of it. Holes are pierced on the plate to allow wind to come through and stabilize the instrument while sailing. It is also for stabilization issues that the original astrolabes were made of heavy thick brass.

The upper-left and bottom-right quadrants labels already include the (90º-measured altitude) calculation for simplicity (a dimension called Zenith Distance). The upper-left and bottom-right quadrants are labeled in regular degrees (from 0 to 90) in order to easily estimate the altitude of a standing object on ground.

Moreover, measuring longitude was not possible at the time. Therefore the ship was sailed to a known measured latitude, then moved east or west toward the desired location. The instrument could also be used to estimate the height of a tall object.

Instructions

To determine current latitude:

  • Using the horizon as a point of reference, align the base of the astrolabe by firmly holding the instrument’s handle with your hand.
  • If measuring the sun’s altitude at noon, rotate the alidade until its shadow appear between the alidade’s notches. The sun will then be correctly aligned and it’s altitude can be measured. DO NOT look directly at the sun, use its shadow !
  • If at night, look toward the point of interest by aiming through the alidade’s notches.
  • Perform the (90º- measured altitude) operation, if required.
  • Correct this angle by adding the declination value by looking in an almanac.

To determine the height of an object

  • Walk from the base of the object until you can aim to its top while maintaining a 45º angle on the astrolabe. Measure this distance (or approximate it by estimating your pace-length).

The height of the object is then given by the distance from it’s base plus the height of your eye from the ground while aiming with the astrolabe.

  • If it is not possible to aim to the object with a forced 45º angle, the distance will then be given by the expression: D•Tan(measured angle) + the height of your eyes from the ground, where D is the distance from the base of the object of interest.

Download the full paper here [pdf]. Astrolabe