Chartres Cathedral Analysis of the Floor Plan

In the previous post I searched for possible patterns in the floor plan using a rather detailed  floor plan published by Jean Villette and a plan available on the web that is based on the work of Dehio and Bezold. The post ended with the finding that in the measures of certain distances in the plan the Fibonacci numbers showed up. In this post I am carrying on my search by analyzing the floor plan in more detail.

 

CIRCULAR ALIGNMENTS IN THE CHEVET OF CHARTRES CATHEDRAL.

In the previous post of July 9^th 2015 we have seen that the Cathedral on the East side ends in the rounded shape of the Chevet. Within the Chevet we can notice the Choir ending with the 6 piers of the Apse arranged in a semi-circle around the High Altar Point.  Beyond the piers of the Apse lies the double Ambulatory with its 6 rounded columns.  The Ambulatory can be thought to be limited by the ‘inner wall’ of the large buttresses positioned around the Ambulatory on either side of the three major Chapels. These buttresses can be seen to be arranged along a semi-circle (see previous  Figure 6 and next Figure 7.)

Are there more semi-circular alignments in the plan of the Chevet that can be identified? Apart from the three obvious one, just discussed, there are at least four more!.The first less obvious one I would like to point out is the circle that can be drawn around the outer buttresses that support each of the three major chapels of the Chevet. I have added this circle to the previous Figure. It is the largest circle shown in the next Figure 7 (click on the image to enlarge):

Figure 7     Showing Circular alignments in the Chevet II

Fig 7 Circular Alignments in the Chevet of Chartres Cathedral

Note that a circle drawn through the round Ambulatory columns has been omitted from   the above figure 7.

I would like to leave the discussion of the other three less obvious circles for the time being. I will come back to these later, but for now I like start paying attention to the question of the straight line alignments in Choir and Nave and the subsequent question of if and how these linear alignments relate to the circular alignments of the Chevet.

 

JOINING OF ALIGNMENTS OF THE CHEVET WITH THOSE IN THE CHOIR AND NAVE. 

The piers of the Nave and Choir, including the four large piers of the Crossing, line up very well as Villette and James have pointed out and as even a cursory look at the floor plan can show. Having found the circular alignments in the Chevet as shown above then the question came up: do these straight line alignments of Nave and Choir piers have any bearing on the circular alignments in the Chevet?  How do these two types of alignments relate? Are they interconnected?   Do these different alignments ‘intersect’  and if so, how do they intersect?

To find out lets take the last Figure 7 and draw straight lines through the centers of the piers of Nave and Choir running from West to East. They indeed all do line up very well, alright. When following these straight lines towards the East end, I noticed these lines abut tangentially to the Apse circle as you can see for yourselves in the next Figure 8 where these straight lines have been shown in blue.

Now what about the Ambulatory circle? Are there any similar straight lines that can be drawn from West to East that abut this circle tangentially in a similar fashion? And surprisingly yes ! These lines run along the outside surface of the Nave’s major buttresses as can be seen in the same Figure 8.  I found this to be a very interesting observation! At the first impression of seeing the floor plan one would not immediately suspect there to be such joining of the linear and circular alignments!  See next Figure 8 (click on image to enlarge):

Figure 8.  Joining Nave and Choir alignments with those of the Chevet.Analysis Floor plan

Fig 8 Alignments in Nave and Choir joining semi – circular alignments in the Chevet

Next, if I now focus on the large circle around the outer buttresses of the three major Chapels can I find something similar?  Yes Indeed !. When taking the straight lines formed by the North- and South inner walls of the Transept and when I extend them towards the East they meet this large circle in the same manner: they abut tangentially to that large circle!  And we find here again an alignment one would not immediately suspect!

 

SUMMING UP FOUND ALIGNMENTS AND MEASUREMENTS IN ONE FIGURE

In the following Figure I have added to the last Figure 8 the measurements I have found and discussed in the previous post for the divisions along the North – South line forming the end of the Choir running through the High Altar Point ( that is the right hand side of the ‘Grand Square’ ). This Figure 9 sums up some of the interesting finds in the analysis of the Chartres floor plan. Note the measurement numbers are all members of the Fibonacci series whose ratio’s are an approximation of the Golden Mean  or Golden Ratio! (click on the image to enlarge):

Figure 9.  Nave  and Choir plus Chevet alignments including measurement numbers

Fig 9 Chartres Cathedral Floor Plan with Linear and Circular Alignments plus measured Fibonacci numbers

In the next post I will carry the analysis further.  For now I want to share some thoughts and musings about the findings so far.

 

SOME MUSINGS ABOUT THE LINEAR AND CIRCULAR ALIGNMENTS IN NAVE AND CHEVET. 

I want to start with the last finding first, that of the large circle and the straight lines along the North and South inner walls of the Transepts. Is the joining of these alignments sheer coincidence? The position of  the the inner walls of the Transept has clearly to do with the sizing, the dimensioning of the Transept arms and, I believe, must be part of a larger pattern in the design. I will come back to that later too.  One thing that crossed my mind is the blue straight lines run along an inner wall and connect with the large circle encompassing the outer Chapels’ buttresses. This is a thought similar to the one I had when I first ‘saw’  the Grand Square. One of its vertical sides runs for the most part through the inside of the Cathedral, to be specific, through the piers ending the Choir and the High Altar Point and the opposite vertical side also runs through the inside of the Nave and Labyrinth Centre, whereas the horizontal sides are running along the outside of the major buttresses of the Transept’s North and South portals.  Should those horizontal sides not also have ran along the inside of these major buttresses? It would have increased the total length of the Transept and have it stand out more. At my first look at the floor plan my impression was that Transept looked a bit short in relation to the total length of Nave plus Choir! Is the length of the Transept the result of some geometrical pattern in the design or is it the result of practical considerations by the builder/designer to limit the extent to which new foundations would have to be dug out and laid ? After all for the construction of new Nave and Choir the existing foundations of Fulbert’s previous church could be used .

Next, I want to pay attention to the straight lines at 55 mm from the main Nave axis running along the outside of the Nave buttresses and joining up with the Ambulatory circle along the ‘inner wall’ buttresses of the Chevet.  Is this alignment a sheer coincidence thing or did this have had some meaning in the original design? There are unfortunately no records left! Nevertheless, I could for example observe that these straight alignments delimit how far these buttresses extend from the outer walls of the side aisles of the Nave. In other words, could these lines have been taken as guidelines directing the size , how far the buttresses would extend from the Nave walls? Often commentators have remarked on the formidable heaviness of these buttresses. Compare these for example with the flying buttresses of the Paris cathedral that was finished earlier. It is true that the vaulting of the Nave with its pointed arches in the cathedral at Chartres has the largest span (16.4 meter) of any completed cathedral built in France, including the one in Amiens[1] which was intended to be a bigger and higher edifice! When we look at the heaviness of these buttresses from a construction point of view the wider the ‘open space’ that has to be ‘bridged’ by the pointed arch vaulting, the larger the weight of roof and vault, the greater  are the lateral stresses working on the piers. The more heavy the roof and the vaulting, the more the lateral stresses on the  piers try to push them away from the central axis of the Nave sideways.  Thus the greater the counter force that is needed to be provided by the buttresses and their weight.

In general how did the designers and builders of these cathedrals determine what the size and heaviness of the flying buttresses needed to be to safely support the vault and roof? Trial and error I would say. I also believe there is an incremental- learning and making of improvements taking place with each finished church or cathedral. Take for example the mature Romanesque church, the Madeleine, at Vezelay built by the Cluniac monks in ca 1118 , a time well before Suger started to re-built his abbey church into what is now recognized as the first early Gothic church building[2]. In the Madeleine, to support its clerestory and vault plus roof above it, extensive use was made of flying buttresses along the Nave and around the Chevet[3].

Did geometry play a role too in helping to arrive at positioning and sizing of the buttresses ? When I speculated above that the 55 mm lines could have been used as guidelines, as directing lines for sizing of the buttresses I am speculating about the builders taking cue from geometry of the plan.  Perhaps theses alignments I have discussed here are hinting at that?   This question brings us back to the general question of what role geometry played in helping to determine what shape and what proportion the various elements in the construction of a cathedral or church needed to have (see of course Jean Villette’s little booklet’s title referred to above).

[1]  The Cathedral of Paris begun 1163 and finished in 1250. Its Nave vaulting is 33 m high and spans 12     meter. The cathedral of Chartres begun 1194 and finished 1250 has a Nave vaulting of 37 m high and spans 16.4 meter. The cathedral of Amiens begun 1220 and finished in 1270. Its Nave vaulting is 42.3 m high and spans 14.6 meter.

[2] Suger started to rebuild his abbey church in 1135 – 1140 and is often portrayed as the ‘inventor’ of the Gothic building style. It is fair to say that the major elements characteristic of the Gothic building style like the pointed arch and the flying buttress were already used in the abbey- and priory churches built in Roman style by the Cluniac and Cistercienzer monks all over France.  K.J. Conant says that the first pointed arches in West Europe were constructed in the third abbey church at Cluny (Cluny III ) in 1086-….. In this church pointed arches were built over the bays of the Nave arcade. See “Carolingian and Romanesque architecture, 800 – 1200, by Kenneth John Conant, published by Yale University Press, 6^th edition, 1978, page 207.

[3] Built during 1096 to 1137, see Carolingian and Romanesque architecture, 800 – 1200 , by Kenneth Joan Conant, Yale University Press, 6th edition,  page 193.

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In Search of the Golden Mean in Chartres Floor Plan

Keywords: Chartres’ floor plan , cruciform, orientation, design concept , symbolism, proportions, Golden Mean, Phidias, geometry, ad quadratum, ad triangulum, basic geometric figures , ratio’s, proportions, conceptual plan, implementation,  green field site, constraints, dimensioning,   design notes, design process,  maitre d’oeuvre,  maitre macon.

Visiting Chartres’ Cathedral It is one of the most beautiful cathedrals dedicated to Our Lady , “Notre Dame” in France. Visiting end October 2003 gave an indelible experience. Like  G. Bomans, a famous Dutch author, once wrote, the emotion, the devotion, the prayers uttered over the centuries, have penetrated the stone. Once inside, the outer world falls silent, the dim light, almost darkness, enveloping, you are quieted, taken up in its embrace. When the eyes are accustomed, there is the colored light coming through – the mostly all original – stained glass windows, drawing your eyes to their patterns, figures, and medallions. Everyday of our 12 day visit to Chartres we went inside the Cathedral. It was well outside the season and not a lot of touristy visitors or groups were around. Everyday when setting out to visit, I needed to think whether to take my camera with me to capture the images or would I leave it behind and just immerse myself, let the Cathedral’s space envelop me and experience her embrace. It was an extraordinary experience, and not in the least because it still is a living, functioning church where people come to pray and attend services. And we attended and witnessed nearly all different kinds of services, from regular mass to funeral service to the ordination of three new priests, to a marriage.

We bought tickets for a guided tour through the crypts, that right of the bat led us into a niche near the entrance with murals of saints painted on it, and my wife halted and went to the wall and got strong impressions about something “special”, something “holy” beyond that wall. I wondered what is that? (made a mental note of it). Even the deep excavations around the crypt of St Lubin underneath the choir did not evoke such reactions on her part.

We did not get a chance to walk the Labyrinth, as it was covered with chairs at the time. One question remained with me: why is that Labyrinth so far away from the crossing and the choir? It seemed odd ! One of the last days of our visit I bought in the little shop inside, next to the North Tower, a little book with a floor plan of the Cathedral in it. It was titled “Le Plan de la Cathedrale de Chartres , Hasard ou stricte geometrie? ” by Jean Villette,  troisieme edition , published by Editions Jean-Michel Garnier, 3 Cloitre Notre Dame in Chartres. Perhaps it could answer the questions that had come up in my mind about what is it that makes the experience of this cathedral special?  Is it in the layout, in the arrangement of the main elements of this cathedral?  Perhaps I could learn something from that floor plan in the little book?

Once home and the experience had settled down, then the desire to know more about the cathedral came up. Before our trip we had done no previous reading about the background and history of the Cathedral. We went to seek some solace from the loss of our young daughter. It allowed us to have all these impressions and let the space and energy of the place soak in without our minds interfering with prejudices, without expectations and without having pre-formed images. But now, I was curious, wanted to know. I read  Charpentier’s “Mysteries of Chartres Cathedral”  an interesting read, although it seemed  to me to include speculations and interpretations that were just that. More importantly it had diagrams with actual measurements based on – I believe- Lassus’ measurements. It had a full page diagram in it with the cathedral’s floor plan. The diagram had squares and circles superimposed on it to ‘explain’ how the builders had arrived at the shape and proportions of the cathedral’s plan.  Although these basic geometric figures did hit some of the significant (physical) points in the plan, it did not convince me as there seemed to be a certain arbitrariness, why this particular sequence of circles and squares and why of this size and so on.

I wanted to take a closer look at the plan for myself. So I made a 1:1 copy of Jean Villette’s plan on 11×17 inch paper (no magnification) and with graduated ruler and compass tried to analyze it. Yes the Apse at the end of the choir has seven openings but to proclaim that the cathedral’s floor plan hence is based on a seven pointed star seemed farfetched. With so many piers and buttresses it is easy to take a basic geometric figure and make it overlay on the ‘as built’ floor plan and see if it ‘hit’ some existing pier or buttress or other significant points in the underlying plan.  J.V. ‘s  drawing of the Chartres’ cathedral floor plan does not have any of ‘drawn to scale’ indication shown on it. Also there are no dimensions shown nor is there a measurement key given, no scale indication. It is however drawn to rather fine detail suggesting measurements must have been used to be able to make this drawing. I will get back to this question later on.

Having no further information I assumed that this floor plan has been drawn to scale, at least a reasonable assumption as Jean Villette attempts to investigate whether geometry played a role in establishing this plan or whether it was just haphazardly arrived at as his subtitle says. That latter point of view one could possibly argue fore as the present Chartres’ cathedral had to be erected on the vast, remaining foundation and crypt of the earlier cathedral built under bishop Fulbert after the fire of 10 June 1094.

How do you analyze such plan? What am I looking for?  How do you establish that there is a pattern underlying the design?

 

Orientation of the cathedral.

Putting the floor plan drawing before me with the towers on the left- and the round end with the chapels on the right hand side, I imagined myself standing at the crossing facing North. The main entrance, the  Western Royal Portal, lying on my left hand side and the Eastern Chevet‘s chapels on the right hand side. A closer look into the orientation question shows that the main axis of the cathedral actually is pointed in the North by North East direction. The orientation of the building with its Choir and High Altar directed towards the point on the horizon where the first Sun light in the morning arises, chasing away the darkness of the night. As I found later this to be the first hint on the importance of metaphor and symbols incorporated into the conceptual floor plan of the cathedral.

 

Familiarization with the floor plan.

 The cathedral is laid out in the shape of a crucifix. In the bird’s eye view of aerial photographs of the cathedral this shape is easily recognizable. When looking at the floor plan, immediately your eyes are drawn to the massive towers on the West side, to the semi-circular Apse with its seven chapels on the East side and to the crossing of the nave and the transept midway. The ‘arms’ of the transept, the north side-  and the south side transept seem rather short. The spaces between these three area’s in the plan are filled with piers, windows and buttresses placed at more or less regular distances from each other. There are three portals, the ‘Royal’ portal in the West, the portal on the North side of the transept and one at the South side of the transept. With each portal having three doors there is a total of nine major doors that are giving access and entry into the cathedral. How many “openings” (bays) are there between the towers and the beginning of the transept?  Six. How many across the transept? There are three (doors). How many between the eastern wall of the transept and the end of the choir ? Three.  Looking at the size of the openings the one across the crossing looks like double the size of the other “openings”.  Taking this in account the numbers are 6 – 4 –  3 .  The number of piers across the same space is 12, which include the two heavy piers at the crossing of nave and transept that flank the double opening of the crossing.  Is there any significance to these numbers? We can use these counts as a rough measure of the relative length of these sections of the cathedral. The choir is ended with a semi-circular space – the apse or ‘rondpoint’  –  with six smaller piers allowing seven windows to be positioned around the semi circular apse. Is there a symbolical meaning to these numbers? I will leave that for later.  One feature of the Chartres cathedral that is important to note, although I did not at first grasp its significance, is that the crossing with its four major piers is in the form of a rectangle and not in the form of a square as is – as I found out later – more common in most other Gothic churches (Amiens- Cologne –   etc). The following Diagram shows  the Chartres Floor Plan (click on the image to enlarge):

Floor Plan of Chartres Cathedral
Source : Library.Pitt.edu

The Nave has a total of 7 bays up to the crossing rectangle and 4 bays for the Choir that ends with the semi-circular apse.  Such simple counting of features is a first approach to look at the cathedral with an ‘analytical’ rather than an experiential eye.

 

Ratio’s and Proportions.

The cathedral impresses you not only through its sheer size, but also through the sense of unity and proportion that it evokes, the well balanced relative placement and positioning of the piers, the bays, walls, apse, like the width and length of the nave, the width of the bays and their overarching high vaults. How was this accomplished? The sizing of a bay along the nave in comparison with the width of the flanking piers or the relative sizing would give me a ratio and would lead me to asking is this ratio applied to the other bays? And is this ratio also found in other parts? If so that would be one step in the direction of trying to find a pattern that possibly was used in the design of the cathedral’s plan.  However you quickly find out with so many features, so may significant points to choose from, this leads quickly to an overwhelming amount of ratios to determine.  Add to this, the realization that strictly speaking a proportion and a ratio are technically speaking different things. A proportion is like a ratio of ratio’s. Just like a multiplication is a summation of a number of repeated additions. A comparison of two ratio’s and finding them to be equal points to a proportion.  Thus we need to find if and what proportions have been used in the design. Our visual brains already have a built in sense of proportion and symmetry. To ascertain such observations we need to analyze, to count and measure in order to make those observations explicit and tangible.

How has proportioning been used in the design of this cathedral? What was the thinking behind the cathedrals’ plan?

 

Proportioning and the Golden Mean.

The Golden Mean was known to the ancient Greeks, Euclid in his “Elements” explains the Golden Mean in detail.  Phidias, the famous sculptor of Apollo’s statue, used it to help bring a realistic proportion to the sculpture. Euclid describes the Golden Mean as:  “A straight line is said to have been cut in extreme and mean ratio when the whole line is to the greater segment, so is the greater to the less”.  In other words a given, whole line piece  is cut in two parts, but in such a way that the ratio of the length of the largest part (a) over the shorter part (b) is equal to the ratio of the original line length (a+b) to the larger cut part.  A Golden Mean, or Golden Ratio cut is obtained when   a / b = (a+b) / a.  Calling PHI the ratio of a / b  then the quadratic equation in PHI is solved and the resulting value for PHI is 1.618034….. If we take the total length (a+b) equal to 1 then it follows that the length of the largest segment  a  is equal to 0.618034…..  and the length of the smallest section  b  equals 0.38197……

Division of a Line in two parts a and b. And their repeated division of these parts in their turn

The Golden Mean has many remarkable mathematical properties, for example the inverse of PHI equals the decimal part of PHI: 1/1.618034  =  0.618034 . and many more which for the moment we let rest

 

Symbolic interpretation of the Golden Mean. 

We can take the process of dividing a given line into two parts according to the Golden Mean and ask ourselves what is it that we are doing? What is special about this “Golden cut” above a random cutting in two?  And the answer is that after the cut we have created two pieces whose lengths still have a proportional relationship with the original whole line before the cut. From One (piece) we have created Two (pieces), where the ratio between the Two (lengths) are in relation to the ratio of the largest of the Two to the Original One (piece). There is a fixed relationship between these ratio’s.  And we can carry on with this process of division by dividing each of the two pieces again using the “Golden cut” and those (four) new pieces in their turn stand into a (golden) relationship with their originals.  And those ‘second generation’ of pieces can also be given a “Golden cut” and so on ad infinitum.  We can see that in this fashion of repeated division according to the Golden Mean, we have created a whole array of parts whose lengths all stand into a fixed Golden Mean relationship to each other.

After ending the process of division we have created a whole array of large and small parts that can be used to start the process of assembling and building up new ‘wholes’ using those parts and thereby finishing the creation cycle. These parts are “stamped” as it where with their creator’s mark: the Golden Mean. It is the golden thread that binds them all together.

The above interpretation is like a Cosmology in which the Creator creates by dividing and separating not randomly but with an underlying pattern in Mind. From the Original Primordial Matter (the Ur-Waters) by dividing same He created the Firmament, the Heavens and the Earth. And so it was (Good).  (paraphrasing Genesis)

Creation encompasses the two polar opposites of Spirit and Matter, the Heavenly Realm and the Earthly Realm. With Man as the Divine creation carrying the invisible stamp of their Creator.

 

What does the “Golden Mean”  mean for the practice of designing?

There are geometric methods to construct the Golden Mean. For example a square is divided in two equal parts by a line through the middle of two opposite sides of the square. Next, a line is drawn from a corner point, say C, to the opposite middle point M. This diagonal is rotated around that midpoint M and aligned with the side A D. The newly created line A E is now divided into two parts according to the Golden Cut. The ratio of the length of A E over A D equals PHI  = 1.618034 ……..  See the Figure below:( Click to enlarge):

Geometric Construction of a Line with length of PHI ( = 1.618034 ……….)

That this construction indeed yields a PHI division we can verify by using Pythagoras to calculate their respective lengths. If we assign a length of 1 to the side of the square then the length of the diagonal MC we have drawn follows from MC ^2 = 1^2 +(1/2)^2  which gives MC =  ½ time square root of 5  and this is also equal to ME. The total length AE then is ½ + ½ * SQRT 5  or 1.618034 ……….!   And hence the Ratio of AE /AD =  1.618034…. / 1.00000… = PHI .

And with this verification through calculation we have demonstrated another way to practically create a Golden Cut division. That is through measuring out each of the two lengths such that there is  a Golden Ratio between them  in terms of their numerical relationship. Setting out each length through measurements.

In order to measure we need a standard, a fixed unit of measurement to be established that we are going to use throughout. our construction work and in such a way to lay the basis for a unified edifice.

However, there remains a practical problem if we want to use a standard unit of measurement and at the same time apply the number PHI in our measurements. The number PHI cannot be measured out in whole numbers of (any) standard unit of measurement. The Pythagoreans discovered this fact about the ‘square root of two’, to the detriment of their beliefs. For the length of the diagonal of a square and the side of that square one cannot find a common measure ! Thus, attributing to the side of the square a unit length, using that same standard unit, for determining the length of the diagonal we find that it cannot be expressed as (ratio) of whole numbers of that unit of length. Its value is 1.41421……. units. When conversely assigning to the diagonal the unit length, then we will find the side in its turn cannot be expressed in a ratio of whole numbers of unit length, and  its length will be 1/1.4142….. equal to 0.7071….. a fraction with an infinitely continuing sequence of digits! This finding devastated the Pythagorean belief that the Universe was built on ‘whole numbers’! This fact, that the diagonal in a square cannot have a common measure with the sides of the square and can only be expressed by an ir-rational number, gives the diagonal and its length a special status. If the square symbolizes the Earth, for which and in which we can establish a standard measuring unit, the Diagonal’s length escapes us, it cannot be grasped in terms of Earth’s measures, like the Heavens being immeasurable. So the Diagonal length has been applied in design of ancient’s Temples as a means to bring (symbolical) connection with the Realm of the Gods (as I believe).

But back to the use of PHI in practice. If this number too cannot be expressed as a ratio of whole numbers we could try at least to find number ratio’s that approximate this irrational number as close as practically needed in terms of accuracy. This same approach had been used already in practice by the ancients for the square root of two and the square root of 3. For example, the square root of 2 can be approached by the ratio 7 / 5 with an error of 1% or by the fraction 17 / 12  (error 0.17% ) or even 41 / 29 (0.03%). Similarly such fractions can also be found for the square root of 3. The Golden Cut can also be approximated by ratio’s of whole numbers: for example by  8 / 5 giving a 1.1% error, by 13  / 8 (0.43%) and  21/ 13 (0.16%) and so on. Using such ratio’s one can construct  two lengths whose ratio approaches the Golden Ratio through measurements, without resorting to geometrical methods.

 

In search for the Golden Mean in the Chartres Cathedral’s floor plan.

After these musings about the Golden Mean, or call it the Golden Cut or Golden Ratio, with my mind now having been energized by all these mathematical and philosophical interpretations and meanings, I was ready to look again at the Chartres floor plan and see if I could find this Golden Mean, this Golden Ratio back in the designed floor plan, who-ever that designer, architect or master mason was. And in my search I would eventually find indications that the Golden Ratio can  be found back in the plan. But I also eventually discovered that -after the “design” fact – a Golden Ratio can be found back, even as it has not intentionally having been used, but it is like a “spin –off “ from another design method. The Ad Quadratum method of proportioning in design can ‘produce’ (seeming) Golden Mean proportions. For example, The sizing, the dimensioning of the piers in the plan of the Chartres cathedral can be done by an  Ad Quadratum construction with the help of a square , a circle plus the square’s diagonal!  Now as John James has found, from his measurements in the Cathedral, the Golden Ratio can be found between the Nave axis and the two inner sides of the piers of the Aisle’s.  I found during my search for the Golden Mean in the floor plan that these piers and their dimensions can be constructed Ad Quadratum.  This Ad Quadratum Construction is shown in the following Diagram : (click to enlarge):

Ad Quadratum Construction of Dimensioning of Nave and Aisle piers while assessing its relation to PHI

From the Geometric construction,  shown above, the dimension of the pier can be found from its relation to the Apse Radius. The radius of the pier can be calculated  from:

Rpier  = (V2 -1) /(2V2) * Rapse  equal to (1.4142 -1)/(2*1.4142) * Rapse

or   Rpiers = 0.14644 * Rapse 

Assuming the Apse has a radius of 28 Roman Feet  this would give for the diameter of the pier 28 * 0.14644 * 2  which is equal to 8.20 Roman Feet. Taking the Roman Foot at 0.295 meter/RF , the pier measures 2.42 meters in diameter! Very close or equal to the actual pier diameters.

Now taking a step back and looking at this Ad Quadratum design construction and question whether the Golden Mean is involved in it (while deliberately ignoring the fact that we definitely did not use the Golden Ratio PHI in this construction) by analyzing the drawing above?  To answer let us look at the two red line L1 and L2. drawn after the construction was finished. If we assume that there is a Golden Mean relationship between these lengths, then we can derive a formula for the Radius of the pier too using these two lengths L1 and L2  (check for yourselves) it reads as :

Rpier = (2-PHI) / ( (1+PHI) * Rapse  equal to ( 2-1.618034) /( 1+1.618034) * Rapse 

or Rpier = 0.1459 * Rapse   

Both yield very close numbers for the pier radius. Under same assumption for the Apse radius, being 28 RF, the pier diameter becomes 8.17 RF.equal to 2.41 meters.

So after the “design fact”, it is difficult to decide on whether the “Square root formula” derived from the Ad Quadratum method of construction, or the PHI formula was  used in the method of design. I guess very accurate measurements are needed , of the type Andrew Tallon has made with his laser measurement techniques.

But with the above description of how Ad Quadratum construction and PHI interrelate,  I am running way ahead of my story of the searching for the Golden Mean in the design of the floor plan of Chartres Cathedral.That story I will tell in the next posts.