Sezione Studi su Rennes-le-Château

Analysis of alignments in R-Environment - Alignments on pentacles

Two points define four different pentacles. Given the coordinates of the first two points (A1,A2) and (B1,B2), with a simple mathematical formula it is possible to find the coordinates of the three other vertices in the pentacles.

Let's fix the first two points A(10,10) and B(40,50):

A1 <- 10
A2 <- 10
B1 <- 40
B2 <- 50

The other points can be calculated with these formulas:

C1<-(B1-A1)*((sqrt(5)-1)/4)+(B2-A2)*sqrt(sqrt(5)/8+5/8)+B1
C2<-(-1)*(B1-A1)*sqrt(sqrt(5)/8+(5/8))+(B2-A2)*((sqrt(5)-1)/4)+B2
D1<-(C1-B1)*((sqrt(5)-1)/4)+(C2-B2)*sqrt(sqrt(5)/8+5/8)+C1
D2<-(-1)*(C1-B1)*sqrt(sqrt(5)/8+(5/8))+(C2-B2)*((sqrt(5)-1)/4)+C2
E1<-(D1-C1)*((sqrt(5)-1)/4)+(D2-C2)*sqrt(sqrt(5)/8+5/8)+D1
E2<-(-1)*(D1-C1)*sqrt(sqrt(5)/8+(5/8))+(D2-C2)*((sqrt(5)-1)/4)+D2

The five points should be stored in an array:

x1 <- c(A1,C1,E1,B1,D1)
y1 <- c(A2,C2,E2,B2,D2)

A second pentacle can be found with a second formula:

C1<-(A1-B1)*((sqrt(5)-1)/4)+(A2-B2)*sqrt(sqrt(5)/8+5/8)+A1
C2<-(-1)*(A1-B1)*sqrt(sqrt(5)/8+(5/8))+(A2-B2)*((sqrt(5)-1)/4)+A2
D1<-(C1-A1)*((sqrt(5)-1)/4)+(C2-A2)*sqrt(sqrt(5)/8+5/8)+C1
D2<-(-1)*(C1-A1)*sqrt(sqrt(5)/8+(5/8))+(C2-A2)*((sqrt(5)-1)/4)+C2
E1<-(D1-C1)*((sqrt(5)-1)/4)+(D2-C2)*sqrt(sqrt(5)/8+5/8)+D1
E2<-(-1)*(D1-C1)*sqrt(sqrt(5)/8+(5/8))+(D2-C2)*((sqrt(5)-1)/4)+D2

The five points should be again stored in a second array:

x2 <- c(B1,C1,E1,A1,D1)
y2 <- c(B2,C2,E2,A2,D2)

The third and fourth pentacles can be calcualted with these two formulas:

C1<-B1-(1+sqrt(5))*(B1-A1)/4-sqrt((5-sqrt(5))/2)*(B2-A2)/2
C2<-sqrt((5-sqrt(5))/2)*(B1-A1)/2+B2-(1+sqrt(5))*(B2-A2)/4
D1<-C1-(1+sqrt(5))*(C1-B1)/4-sqrt((5-sqrt(5))/2)*(C2-B2)/2
D2<-sqrt((5-sqrt(5))/2)*(C1-B1)/2+C2-(1+sqrt(5))*(C2-B2)/4
E1<-D1-(1+sqrt(5))*(D1-C1)/4-sqrt((5-sqrt(5))/2)*(D2-C2)/2
E2<-sqrt((5-sqrt(5))/2)*(D1-C1)/2+D2-(1+sqrt(5))*(D2-C2)/4
x3 <- c(A1,B1,C1,D1,E1)
y3 <- c(A2,B2,C2,D2,E2)

C1<-A1-(1+sqrt(5))*(A1-B1)/4-sqrt((5-sqrt(5))/2)*(A2-B2)/2
C2<-sqrt((5-sqrt(5))/2)*(A1-B1)/2+A2-(1+sqrt(5))*(A2-B2)/4
D1<-C1-(1+sqrt(5))*(C1-A1)/4-sqrt((5-sqrt(5))/2)*(C2-A2)/2
D2<-sqrt((5-sqrt(5))/2)*(C1-A1)/2+C2-(1+sqrt(5))*(C2-A2)/4
E1<-D1-(1+sqrt(5))*(D1-C1)/4-sqrt((5-sqrt(5))/2)*(D2-C2)/2
E2<-sqrt((5-sqrt(5))/2)*(D1-C1)/2+D2-(1+sqrt(5))*(D2-C2)/4
x4 <- c(B1,A1,C1,D1,E1)
y4 <- c(B2,A2,C2,D2,E2)

In order to print the result, let's fix the two source points:

plot(c(x1,x2,x3,x4),c(y1,y2,y3,y4),col="light gray")
points(A1,A2,pch=19)
points(B1,B2,pch=19)
text(A1,A2,"A",pos=1)
text(B1,B2,"B",pos=1)

The four pentacles can be easily drawn with the function polygon:

polygon(x1,y1)
polygon(x2,y2)
polygon(x3,y3)
polygon(x4,y4)

Excluding the two source points, we have a total of 12 points on the vertices of the pentacles. The list of them can be stored in an array and printed:

vx<-c(x1[2],x1[3],x1[5],x2[2],x2[3],x2[5],x3[3],x3[4],x3[5],x4[3],x4[4],x4[5])
vy<-c(y1[2],y1[3],y1[5],y2[2],y2[3],y2[5],y3[3],y3[4],y3[5],y4[3],y4[4],y4[5])
points(vx,vy,col="dark red",pch=19)

Any point at a distance from one of these 12 points smaller than a tolerance value E can be considered "on a pentacle" defined by the two points A and B.

Please, download the complete R file from here: pentacle.txt

Using these formulas, a function aligned can be defined to verify if the point (C1,C2) is on a vertex of a pentacle defined by points (A1,A2) and (B1,B2). The function can be downloaded from here: pentacle_aligned.txt

This function recognises as "aligned" to A&B-generated-pentacles all the points inside the yellow circular areas, each of which has a radius equal to E:

By using this function on Rennes-le-Château map and selecting La Tour Magdala and the Château de Blanchefort, with a tolerance of 400 meters (!) you can identify a unique point fitting one pentacle, the Bezu Chapel:

Relevance

Has a result like the one mentioned any relevance? In order to prove it, you should compare the number of alignments found in a particular map with the mean number of alignments found in a large set of RGMs (randomly generated maps): "chi square" analysis will show if the map you are considering has any (sacred?!) "relevance". By now all the studies failed to show that there's any relevance in Rennes-le-Château area, but any analysis which pretends to affirm it has to follow the steps provided in this short guide.

© Mariano Tomatis Antoniono