By Adrian Lodge Ph.D.

Section 2

An objective study of geometry in les Bergers d’Arcadie

2.1.  Introduction

Although the previous section suggested the Grail Star geometry was highly unlikely (and the Rigby pentagon likely) to be intended by the artist, one can argue the method of analysis was still rather open to interpretation. In other words, it was subjective. It is little different to what others previously have said and is bound to lead to the same response by Mr Ettinger. Effectively it is the word of one person against another.  One says the geometry exists, another says it does not.  The truth exists in that grey area between, somewhere where it has no place being.  How can we rectify this situation?

There now follows a more objective analysis of both proposals to see whether on a balance of probability the artist intended neither, either or both.  It involves employing a) a rigorous method that determines geometric alignments and intersections, and b) a statistical approach to test them. 

2.2.  Guidelines for determining the presence of geometry in paintings

There will always be some subjective element when analysing geometry.  To limit its effect it would be useful to define what constitutes an alignment, an intersection, a key feature and a good fit.  Examples are shown using the Rigby pentagon on les Bergers d’Arcadie (see Figs 6a - c).

1. An alignment is where a line of the geometry corresponds directly with a key feature in the painting.  For example, in Fig 6a, lines of the geometry align with a) the mark representing the red shepherd’s knee-pit (and thereby obscuring it), and b) the line that demarcates the yellow robe and the blue dress worn by the shepherdess.

Fig 6a 

2. An intersection is where a) a line of geometry cuts two lines that intersect on the painting or b) two intersecting lines of the geometry coincide with a feature in the painting.  In Fig 6b, the vertex of the pentagon (point B) intersects with a) the line dividing a cloud and the sky, and b) the chord that aligns with the red shepherd’s staff.  The side AB also intersects with the shepherdess’ eye.

Fig 6b

3. A key feature may include a part of the human body such as the face, toes or hand.  Pointing fingers probably mean something more if they align with oddly arranged items of clothing or a strategically shaped landscape.  For example, the side AB of the Rigby pentagon (see Figs 5 and 6c) passes through the region where the white shepherd’s arm and a small part of his robe meet. This line then intersects at the point where the red shepherd’s staff crosses a line between cloud and sky.  As noted above (Fig 6b), extending this line leads to it passing through (i.e. intersecting with) the shepherdess’ eye.  The upper line of clouds appears to determine the point where the alignment of the pentagonal chord (BD) with the red shepherd’s staff should end.  In addition, one appears to be encouraged by the artist to draw the axis line (GD) that bisects side AB (Fig 6c).  The axis passes through a point near the blue shepherd’s thumb; the white shepherd’s extended little finger, and a groove in the mountains. 

Another method the artist may employ is the inclusion of a number of apparently extraneous features.  One example is the thick dark line painted upon the shepherd’s white robe, and which is parallel with the staff held by the red shepherd.  There is no apparent reason for it to be there.  This conspicuous feature first drew the attention of its discoverer (Greg Rigby) to the location of this pentagon. [12]This told him to place two lines that turned out to be one side and one chord of the pentagon.

A further consideration is the oddly shaped fold in the yellow robe of the shepherdess.  Could it be that the point of the chevron is to tell the observer that a line should run through (i.e. intersect with) its point?  This line turns out to be the pentagonal chord AC.  In addition, there is interplay between the robe, dress and crutch of the shepherdess such that a pentagonal chord (EC) goes through the crutch of the shepherdess and (as already mentioned) the pentagonal side (CD) runs through the line between the yellow robe and the blue dress. 

Fig 6c

As noted above, features that appear to be merely background detail may imply connection with the geometry.  Further examples include two indentations (or grooves) in the rock the red shepherd’s leg is resting upon (see Fig 5).  One indentation has a pentagonal side (CD) pass through it.  The other is where the axis that bisects CD intersects with the circle.

In addition, the positioning of staffs may suggest the artist wishes to inform the observer that a particular geometry is present.  Two staffs positioned such that they form an angle of 18 degrees, 36 degrees, 54 degrees or 72 degrees may suggest the presence of a hidden regular pentagon.  Alternatively, angles of 30 degrees or 60 degrees probably suggest the presence of a regular hexagon or hexagram.

4. For the geometry to be considered a good fit, it normally must run (i.e. align or intersect) precisely through a particular location.  Normally, a ‘near miss’ should in no way suggest intention.  However, certain alignments may require a degree of latitude. For example, a staff is invariably thicker than the construction lines that define the geometry.  It may be intended that the geometry aligns with first one side of the staff and then the other.  In addition, the curvature of the staff may denote the artist’s intention.  A curved staff may imply both sides of the staff are to be used, a straight staff suggests only the one side.

2.3 A comparison of different geometric constructions on les Bergers d’Arcadie

We now have some idea about what to look for when determining the presence of hidden geometry in a painting.  The following part of the article compares the number of alignments and intersections found for different geometric constructions (see Table 1).  The constructions considered are the Grail Star geometry (consisting of the large pentagon GSP 1, and the small pentagon GSP 2), a Theoretical Irregular Pentagon (TIP, see Fig 7) and the Rigby pentagon (RP, Fig 5). 

The Theoretical Irregular Pentagon is included in order to demonstrate how easy it is to draw an irregular pentagonal geometry and superimpose it on chosen key features.  As with GSP 2 the TIP has one side and one axis aligned with two of the staffs.  One of its chords rests upon the top of these staffs.  In addition, the middle shepherd’s staff may have the axis of a second TIP (not shown) placed upon it (as with the axis of GSP 2).  This gives a construction that is very similar to the Grail Star geometry.  Mr Ettinger may argue that the TIP cannot be applied to the islands of Mahone Bay.  This would be a valid point except that there is insufficient evidence of the existence of the geometry on these islands. 

Fig 7.  The Theoretical Irregular Pentagon on les Bergers d’Arcadie

Table 1.
Alignments and intersections for different geometric constructions on les Bergers d’Arcadie

It is important to note:

1. A circle encloses each pentagon.  All geometric constructions have five sides, five chords and five vertices.  RP has five axes but GSP 1, GSP 2 and TIP have only one.
   
   
2. Some pentagonal sides, axes or chords have more than one alignment or intersection.  Some have none at all. 
   
   
3. Extensions of pentagonal sides, axes or chords that align or intersect with other features on the painting are not included in the analysis.

From Table 1, GSP 1 has a total of 2 alignments and intersections for the 12 lines (plus 5 vertices) that make up its construction.  GSP 2 meanwhile has 4 (out of 12 + 5).  This brings the combined total for the Grail Star geometry to 6 out of 24 construction lines (plus 10 vertices).  TIP has a total of 4 out of 12 lines (plus 5 vertices) and RP 14 out of 16 lines (plus 5 vertices). 

These results give a Confidence Factor (CF) for TIP of 23.5 %, GSP of 17.7 % and RP of 66.7 %.  CF is an arbitrary value used as a simple comparison for each geometric construction.  The higher the value the more confident we may feel that the artist intended the geometry.  It is determined by dividing the number of alignments and intersections by the number of lines and vertices from which the geometry is constructed, and multiplying the value by 100 to obtain a percentage (%). 

The results show we have a low confidence that Poussin intended either GSP or TIP (< 25 %).  There appears little difference between them in the number of alignments and intersections.  It is known that TIP was constructed by using the staffs to fix its position.  Could it be that this technique was used to create the Grail Star geometry?  No other alignments or intersections were found for either geometry.

It appears that the Rigby pentagon (RP) provides the greatest confidence with regard to artistic intent.  However, this level of confidence does not imply statistical significance.  Although it appears high, it may be that the design of a regular pentagon is conducive to finding alignments and intersections in this particular painting.  The next part of the article presents a way in which we may address this problem.  It sets out to determine whether the number of alignments and intersections are greater than that expected by chance.

2.4 A statistical comparison of geometric constructions on les Bergers d’Arcadie

The first part of this section determines the number of alignments and intersections a regular pentagon (RRP) has when it is randomly positioned on the painting (see Table 2). [13] The pentagon used is of identical size and shape to the Rigby pentagon (i.e. enclosed by a circle, and has five sides, chords, axes, and vertices).  The pentagon is randomly placed on the painting a total of five times and the alignments and intersections then counted.  There is then a statistical comparison of the data obtained for the randomly positioned pentagon and the Rigby pentagon.  The Grail Star and the Theoretical Irregular Pentagon are discussed in the light of this statistical approach.

Table 2.
Alignments and intersections for randomly arranged regular pentagons on les Bergers d’Arcadie

From Table 2, RRP has a total (out of 16, plus 5 vertices) of 3, 0, 3, 2, and 5 alignments and intersections.  This is an average (i.e. mean) of 2.6 alignments and intersections per position.  The Confidence Factor (CF) is 12.4 %.  This CF is lower than all the geometric constructions previously discussed.  The main reason for this is that the previous examples were deliberately positioned upon the painting.  They therefore already start with at least one alignment or intersection.  Even allowing for this, the Rigby pentagon’s CF should still be considerably greater than that recorded for the RRP. 

However, is there a (statistically) significant difference between the two geometric arrangements?  Two possible methods may determine whether there is any such difference in the total number of alignments and intersections for RP and RRP.  Either the data for RP can be compared with the mean number of alignments and intersections for the randomly positioned pentagon (i.e. RRP 1 - 5), or with each individual RRP in turn.

The method of statistical analysis employed is the Student’s t-test.  It is useful for comparing the difference between the means of two sets of data.  Here we are trying to establish whether the obtained difference between the means of the two groups of data is too great to be a chance event.  The independent variable is the type of geometry (i.e. a regular pentagon) and the dependent variable is the number of alignments and intersections.  We start with the null hypothesis that there will be no significant difference in the number of alignments and intersections obtained when a regular pentagon is placed on the painting. In other words, there is no geometry hidden in the painting. Of course, any number of random arrangements may be employed.  The greater the sample size, the greater confidence there will be that any result is unbiased.
  
In the case of the Rigby pentagon (RP), there are five types of construction line that align or intersect with features in the painting.  These are the circle, side, axis, chord and vertex.  It is important for the statistical comparison that RP and RRP contain the same number of construction lines (i.e. 16 + 5 vertices = 21).  Since all lines are part of the same geometric construct (i.e. pentagon/pentacle), the sum of the number of alignments and intersections (Σx) can be divided by the number of replicate values (n). 

Therefore the mean number of alignments for RP and RRP can be written as:   Σx1  and   Σx2
                                                                                                           n             n

Hence,  Σx1  =   14  =  2.8, and   Σx2  =  [(3/5)+(0/5)+(3/5)+(2/5)+(5/5)]  =  2.6   =  0.52.
            n           5                      n                             5                           5              

Table 3 shows how the sum of the means and the sum of the squares of the means are calculated. Table 4 shows how the t-test is calculated. [14] 

Table 3.
Sum of the Means and Sum of the Squares of Means for RP and RRP

Table 4. 
Method for t-test calculation

From a t-table [15] for 8 degrees of freedom [16] (4 for RP and 4 for RRP) the t-value is 2.31 (p = 0.05). [17] Since the calculated t-value of 3.62 is greater than the tabulated t-value the difference between the means is statistically significant at the 0.05 (5 %) probability level.  In other words, the position of the  Rigby pentagon produces significantly more alignments and intersections than an identical pentagon placed randomly on les Bergers d’Arcadie.

In addition, comparing RP and RRP 5 gives a t-value of 2.41, which is statistically significant at the 5 % probability level.  RP versus RRP 1, 2, 3 or 4 are all highly significant at the 1 % level. 

In order to test the validity of the Grail Star geometry on the painting one needs to compare it to a randomly placed version of itself.   Again, the more random placements there are, the greater the confidence we have that the positioning of the geometry is not biased.  A minimum of five random placements should suffice.  Alternatively, one may compare the Grail Star geometry and any number of irregular pentagons (e.g. TIP).  This is providing they have the same number of construction lines, e.g. GSP 1, GSP 2 and TIP all have 12 (plus 5 vertices).  Since the Grail Star geometry has a relatively low number of alignments and intersections it is almost certain that any comparison with randomly placed geometry would not yield a statistically significant difference.

General Discussion

This article began with a review of the Grail Star geometry proposed by Brian Ettinger.  He suggested that the geometry was a) related to a formation of islands in Mahone Bay, Nova Scotia, b) used to locate the Oak Island treasure, c) used to compose two seventeenth-century paintings and d) locate the lost treasure of Rennes-le-Château. 

To start with, the evidence presented suggests it is highly unlikely that the Grail Star and the Oak Island mystery are in any way connected.  That is not to say it is impossible; it is just that Mr Ettinger needs to undertake a much more accurate study.  For example, it is difficult to imagine how one can accurately verify a particular location using a satellite image and a low-resolution map.

However, one can be very clear regarding the presence of Grail Star geometry in the suggested paintings.  The major problem the geometry has is that, with regard to les Bergers d’Arcadie, the Rigby pentagon is far more likely to exist.  There are considerably more alignments and intersections with key features, and most notably, its regular shape is in keeping with the geometry found by Professor Cornford. [18] A statistical test showed that the Rigby pentagon had a significantly greater number of alignments and intersections than the same pentagon when randomly placed.  Therefore, the test provides excellent evidence that Nicolas Poussin intended to incorporate the Rigby pentagon in les Bergers d’Arcadie.

The article also described a number of approaches that would help determine geometric alignments and intersections.  It suggested that statistical analysis of geometric alignments and intersections require:

1. An equal number of construction lines.
   
   
2. The size and shape of the geometry is identical.
   
   
3. A comparison between non-random and randomly positioned geometry.

To conclude, this article is a work in progress.  It is hoped that it will lead to a more scientific approach when analysing geometry in paintings and therefore provide a better method of determining artistic intent.

Adrian Lodge Ph.D. © 2007

Notes

1   Ettinger, B. Grail Star Discovered, Journal of the Rennes Alchemist, Vol. 2, Issue  6, June 2004, pp. 104 - 107.  The review of Mr Ettinger’s work is based on this article, the photographs included, and material from his websites.
2   ibid.
3   www.renneslechateau.com/forums/viewtopic.php?t=450.  Posted June 15th 2003.
4   Ettinger, B. Grail Star Discovered, Journal of the Rennes Alchemist, Vol. 2, Issue 6, June 2004, pp. 104 - 107.
5   www.renneslechateau.com/forums/viewtopic.php?t=671.  Posted March 12th 2004.
6   www.geocities.com/arkofzion/location.
7   www.geocities.com/avalon1398/Poussin.html.
8  ibid.
9   http://grailstar.4t.com/poussin.htm.  This comment made by Mr Ettinger does appear to contradict that made in Note 7 (above).
10   www.crackingdavinci.co.uk/forum1/viewtopic.php?t=56&start=15. . .  Posted May 28th 2004.
11  A pentagonal rectangle has a length: height (aspect) ratio of 1.3764:1, or 1/tan 36 degrees; see Lodge, A. Geometry in Poussin’s les Bergers d’Arcadie, Journal of the Rennes Alchemist, Vol. 2, Issue 6, June 2004, pp. 15 - 34.
12  Construction of the Rigby pentagon is based on ideas from a conversation the author had with Mr Greg Rigby while at the Sauniére Society symposium at Newbattle Abbey in November 2000.  The drawn geometry is based on memory (as no notes were taken) and may differ slightly from what was actually shown at the time.
13  Random placing of a regular pentagon will show whether this particular shape is conducive to finding alignments and intersections (A/I) in les Bergers d’Arcadie.  If RP has significantly more A/I than RRP, then it is not the shape of the geometry that is important but its position in the painting relative to certain key features.
14  Adapted from http://helios.bto.ed.ac.uk/bto/statistics/tress4a.html, a method for t-test calculation.
15  http://helios.bto.ed.ac.uk/bto/statistics/table1.html.
16  http://helios.bto.ed.ac.uk/bto/statistics/tress.html.  For the number of measurements (n = 5 for both RP and RRP) only n – 1 of them are free to vary when we know the mean.  The number of degrees of freedom (df) accounts for this.  Therefore for this particular comparison df = 8, i.e. (5 – 1) + (5 – 1). 
17  p stands for the level of probability.
18  Lodge, A. Geometry in Poussin’s les Bergers d’Arcadie, Journal of the Rennes Alchemist, Vol. 2, Issue 6, June 2004, pp. 15 - 34.

This article appeared in its original form in the Journal of the Rennes Alchemist, Vol. 3, Issue 7, October 2004, pp. 9 - 23. It was also published in Italian as Geometria ed intento artistico nelle opere pittoriche in Indagini su Rennes-le-Château, Guigno 2006, Numero 1, pp. 25 - 36. 

Some of Mr Ettinger’s photographs in this article are updates of those he provided in 2004.

The author can be contacted at docadlo@hotmail.com

Relevant websites include:

www.rennesalchemist.com
www.renneslechateau.it