What is the source of this geometrical relationship?

October 2, 2021 6:13 PM

In this futilitycloset.com article entitled Court Order, a character in Ken Follett's novel The Pillars of the Earth explains the preferred area relationship between a small courtyard and its surrounding covered arcade, and how to create it. Is there a text or tradition which documents the historical use of this area relationship?
posted by Jackson to Grab Bag (13 answers total)

Likely "A Pattern Language: Towns, Buildings, Construction" by Christopher Alexander.
posted by ajr at 7:36 PM on October 2, 2021

It's almost certainly mentioned/used in Abûl-Wefâ Buzjani (940-998)'s On Those Parts of Geometry Needed by Craftsmen, and possibly Pappus before that. I know that the late prof Keith Critchlow and many of the Sacred Geometry crowd are really into these kind of proportions and would go on about "√2 rhombs" seemingly forever.

The relationship is simple: the ratio of the side of a square to its diagonal is ½√2 (appx 0.7071). Square that, and you have a square that's half the area of the square it's inscribed inside
posted by scruss at 7:39 PM on October 2, 2021 [3 favorites]

FWIW, doubling the area of a square is a key point in Plato's Meno, and it leaves you with a diamond inscribed in a square in the same way as this illustration--except the diamond is the original square, half the area of the larger square. Someone who knows how to do that can probably do the reverse, but I have no idea how obvious the 45 degree rotation would be.
posted by Wobbuffet at 7:48 PM on October 2, 2021 [1 favorite]

There isn't one architectural ancient source that springs to my mind for that specific geometrical construction, but I think the use of it goes way back into antiquity. My first instinct was to look at Hadrian's villa, and yes, there are examples of its use, though not for the square courtyard with surrounding arches described in the Follet text. I don't remember that appears before late Roman / early medieval times, but it is indicated in the St Gallen Plan.
I have read medieval architectural theory, but thirty years ago, and I'm not sure where to look now, but at some point in time, there was a shift in the interpretation of the biblical texts on the temple which seems to have inspired the construction of cloisters based on that particular construction. There was probably also a functional reason.
That all said, back before modern technology, all measuring on construction sites all over the world was basically conducted with a (measuring) stick and a (measuring) string. They had other instruments, including some very sophisticated ones, but for the purpose of building the stick and string concept is very accurate and reliable, if you do the cross measurements right. There might well be Chinese literature on this that predates any Western texts, but I am not familiar with those sources.
posted by mumimor at 2:49 AM on October 3, 2021

Wait, there is exactly such a courtyard in Hadrian's villa, and there is a plan of it in the article I linked to. Stupid me.
Hadrian was very interested in the architecture he saw during his travels, and the villa reflects that. So the origin of that use geometry may be something he picked up on the way.
posted by mumimor at 3:47 AM on October 3, 2021

This states firmly that, pre-renaissance, architecture was an oral tradition/transmission:


C J Dudley's Peterborough Cathedral - The Complete Geometry 1100 - 1500

...

Whatever one may think of the accuracy of the history, the fact remains that the medieval mason clearly understood that the science of Geometry that he used in architectural design and construction had been received from its inventor, Euclid. Unfortunately, for the past two centuries at least, the teaching of Euclid’s geometry in schools and universities throughout the western world has been concerned solely with his system of axioms, theorems and problem-solving though logical argument. From this it has been widely taken for granted that the medieval devotion to Euclid as the founder of architectural design was a pious fantasy, for the simple reason that Euclid’s works were unavailable to any mason. There were a few manuscript copies stored away in a few monastic scriptoria, which were the most secure and inaccessible places in any monastery. They were written in Latin or Greek. Few masons would ever have seen any book except a church Latin Bible, and could not have read it if they had.

But medieval geometer was not concerned with words, but with the manipulation of his drawing instruments -- the compass and the unmarked straightedge.

posted by sebastienbailard at 6:31 AM on October 3, 2021

I took the question to be asking about the preferred dimensions, or ratios of dimensions, that will balance sun, shade, and air movement to achieve the best ambience for whatever the place used for.

And I don't know, but I'm sure builders in Italy, Spain, etc had some ideas on the subject. As a tourist in Puert Rico, I noted that some of the very old buildings were more comfortable than buildings of modern design.
posted by SemiSalt at 6:31 AM on October 3, 2021

Response by poster: Just to clarify my original question: What I want to know is where the specific design guideline presented by the character in the book comes from. I assume Follett did not make this up. Therefore, he had a source for this guideline that was used by builders in this time period. So, is there an historical text or a scholarly paper or some other source where Follett would have found this specific guideline that was used for designing a courtyard/arcade area relationship? I understand the geometry and how it is used here, but it's Follett's source I'm interested in, not the actual geometry. Medieval builders had design and construction practices such as this that were handed down, and it's documentation of these practices - and this design practice specifically - that I'm looking for.

Ajr - Alexander's Pattern Language touches on arcades and courtyards, but does not address this particular topic.

Mumimor - I have not read this article yet in detail, but - interesting as it is - it doesn't appear to present an actual guideline.

Scruss - Asking "the Sacred Geometry crowd" is a good idea. I will pursue that.
posted by Jackson at 7:26 AM on October 3, 2021

This states firmly that, pre-renaissance, architecture was an oral tradition/transmission:

C J Dudley's Peterborough Cathedral - The Complete Geometry 1100 - 1500

It may do so, but if so it is wrong. However, the question is complex.
An architect today does not expect the mason to know about architectural theory, that would be absurd. But the architect should know about theory, and they mostly do, and the medieval architects certainly did. No cathedral investor would pay for something not solidly anchored in the theory (theological interpretation) of the day.
On the other hand, the majority of the workers on site were illiterate, and the master builder had the role of translating the ideas of the architect (mostly a theologist like the legendary Abbé Suger) into the practice on site -- the stick and the string. Most master builders probably had some knowledge of the relevant Latin, just like most building entrepreneurs today have some knowledge of the relevant English regardless of their mother language. They certainly had good drawing skills and understood how to work with scaling.
The idea that cathedrals were built by illiterate builders organized in socialist guilds was launched during the 19th century as part of a anti-elitist romantic understanding of history, and it has had an extraordinary long life, even in academic circles. In spite of the fact that we know the names of several medieval architects.

In religious buildings, from antiquity and up to and including the baroque period, every single element as well as the overall form has a significance that is justified theologically. This would not be possible if the designers didn't have the relevant knowledge, and in the case of medieval cathedrals, that knowledge would be focused on three or four core texts, describing the temple in Jerusalem, and in addition to those, some interpretive texts that may have been abridged for builders. It is not strange that we don't have many of such sources today, since this was before the age of printing, and these documents probably had a rough life, travelling through Europe from site to site.

On the other hand, to this day, a lot of the practical knowledge of construction is what we have chosen to call "tacit knowledge", because our society only partially acknowledges that the knowledge in drawings and models can be as accurate, complex and rich as that in letters and numbers. We don't have tons of drawings and models from the Middle Ages, but we do have some, like the St Gallen Plan linked to above, and they prove that the practice of architecture continued unbroken from antiquity into medieval times.

Since the skills needed to read or create such drawings and models are hard to learn and seem hermetic to outsiders, the myths of the masons grew forward from the late renaissance and onward, contributing to the romantic image of the Bauhütten as well as the creation of Freemasonry.
posted by mumimor at 7:31 AM on October 3, 2021 [1 favorite]

Jackson -- sorry, cross posting.
My medieval source books are not here, so I can't look through them, the main ones everyone refers to are Abbé Sugers writings, mentioned above, and the sketchbook of Villard de Honnecourt.
But thinking a bit more about your question, it seems more likely that Follet has this from a secondary source. Maybe a 19th century or early 20th century medievalist? There are to this day plenty of people who write about the daily practice of building cathedrals, not so much based on the medieval sources as on the knowledge in the Bauhütten which are still extant around Europe.
posted by mumimor at 7:48 AM on October 3, 2021

Best answer: The ultimate source is a diagram in Villard de Honnecourt's sketchbook which you can see here: it's the second diagram in the third row, captioned 'Par chu fait om on clostre autre tant es voies com el prael', 'How to make a cloister whose walkways are equal in area to the central garden'. It's based on the geometrical technique known as quadrature (ad quadratum) or square rotation.

I don't know where Follett got it from, but according to Wikipedia, his sources included two books by Jean Gimpel, The Cathedral Builders (1958), and The Medieval Machine (1975), plus personal conversations with Gimpel himself. The Medieval Machine includes a chapter on 'Villard de Honnecourt: Architect and Engineer', which touches briefly on Villard's knowledge of quadrature.
posted by verstegan at 1:47 PM on October 3, 2021 [4 favorites]

Response by poster: Verstagan - perfect! Thanks to you and everyone else for their suggestions.
posted by Jackson at 3:19 PM on October 3, 2021

I just stumbled across (via a random woodworking blog's outgoing links on drafting) an introduction to the historical French designing and building tradition. So France, not England, and documented from the 13th century onward, instead of 12th century, but the reading list is worth skimming:

http://www.historicalcarpentry.com/l-art-du-trait.html

So what is 'Trait' or 'L'Art du Trait'?


This term, specific to the vocabulary of Compagnonnage, designates a science relative to drawing, or more precisely, a way of representing volumes in depth. An art of stereotomy, the trait is also a kind of applied geometry practiced without the abstract formulas required by descriptive geometry. In more concrete terms, the notion of the trait encompasses a set of graphic techniques facilitating the production - at a reduced scale or full scale, on paper, on parchment, or even on the ground - of working drawings needed to complete structures in wood or stone.

For centuries, youngsters on the Tour de France have been initiated into the mysteries of the trait in courses given by journeyman professors who are past masters of its subtleties. In fact, the working drawings used in realizing great "master pieces" of carpentry are just as remarkable as the artifacts themselves. A visit to the museum in Romaneche-Thorins demonstrates this.

The trait transforms the work as well as the worker. With support and supervision of his professor, the student thinks, reflects, and learns to envision differently. It is in this spirit that we can understand the definition of the trait formulated by the journeyman Aveyronnais la Clef des Coeurs (The Heart Keystone from Aveyron); "The trait makes anyone who has mastered it a visionary in spatial depth. It is the alchemy of solids. Numbers are scientific but lines are initiatory." Finally, we note that the art of the trait is not restricted to journeymen carpenters, joiners, stone cutters,. Like the tinkers, journeymen in many crafts continue to teach this noble skill so dear to the Compagnons.

Here are a few excerpts from the inscription of Le Trait into the UNESCO


'The scribe tradition in French timber framing, or the “trait de charpente” as the carpenters call it, makes it possible to design complex wooden buildings in three dimensions. To understand this system is to master the perception of the volume of buildings and, thereby, possess an essential quality in the art of building. The mastery of the French scribe system is therefore accompanied by powerful symbolic and social practices that play a crucial role in the characteristic representations of the compagnonnages or trade guilds. Knowledge of it enables one, metaphorically speaking, to know exactly how to behave in the universe and society. The art of scribing is a discipline specific to France, with related practices to be found only in Japan and Germany. Since the Middle Ages, it has made possible the construction of the greatest French monuments. It also makes possible a social and professional achievement that is accessible to all, including young people from the underprivileged sections of society. Furthermore, it contributes to intercultural and international dialogue.

...

...

The writer is a France-trained master carpenter who's also a teacher, and along with his courses he sells a reading-list $45 usb stick packed full of ebooks and scans:

Free with every purchase; A lifelong collection of over 700 books, essays, treatises, articles, plates, presentations, and multimedia files. Some of these books and treatise date before the 17th century. For the most part, these files cover the massively wide range topic of carpentry and timber framing from around the world. Also included are books and articles on Stereotomy, Descriptive Geometry, and Art du Trait. Some files cover the topic of the Compagnon and the Compagnonnage system in France and Europe. Because of the nature of these files a lot of them are in French but a large number are in English and a few are in German and Spanish. All files are in the ‘public domain’ which means there is no copyright infringement. As mentioned before, a lot of these books date during and before the 19th century which are found publicly on Google books and random sources of public domain books and articles.

...

He provides the titles of a subset of the books, and people who are interested may want to follow up on google books or archive.org .
posted by sebastienbailard at 11:31 PM on October 12, 2021