Four Round Churches on Bornholm
Abstract:
It is shown that the positions of the four round medieval churches on Bornholm are connected by a simple hexagonal geometry.
The geometry would dictate the exact positions of the churches within a tolerence of 3.5 meter under specific influence from inverse mapping equation.
The plane geometry pattern between the theoretical locations of the four churches in figure 1 can be calculated as seen in figure 2.
Here the theoretical location of Østerlars is called Øl, Nylars is called Nl, Olsker is called Ol and Nyker is called Ny - and the pattern's geometrical relation to a point on Christiansø is called Chr,:
The plane geometry in the quadrilateral figure, hypothetically controlling the location of the four round churches, is shown in Figure 3. This makes it possible to compare the actual location of each of the four round churches in the landscape with its hypothetical location,
calculated from the geometry in figure 2.
2: BASIS FOR CALCULATIONS
The basis of the calculations is the set of coordinates of the centers of the rotundas of the round churches. These coordinates have been measured and published by Kort og Matrikelstyrelsen (Table 1
With these coordinates it is possible to calculate any distance between the churches and the bearing of the lines connecting the churches using the program KmsTrans (lines between locations in the landscape in this study - unless otherwise mentioned - will always be a chord in a geodesic between the locations which, in practice, is the same as a sightline, since it is the shortest distance between two points in the landscape). Table 2
C (identical to Chr in figure 2) on Christiansø. These additional locations are considered in Section XX. The data in Table 2
There are many ways of making this comparison, scaling and orienting the quadrilateral. The conventional approach is to minimize the error in the four distances (the least squares method) as a function of the scale and orientation parameters. A simpler method is used here.
3: THE METHOD:
One way to see how well the church locations fit the geometry is simply to fix Øl at the location of Østerlars, and to scale and orient the quadrilateral such that Nl coincides with Nylars.Then the unit distance in the geometry of Figure 3 corresponds to the distance between Østerlars and Nylars (14335.51 m). This mapping of Østerlars and Nylars on the geometrical pattern in Figure 3 makes it possible to compare the actual locations of the remaining two round churches Nyker and Olsker with the hypothetical locations of Ny and Ol
4: EXAMPLES:
Angle 1:
Using table 2 we find the real angle in Østerlars between the line Østerlars-Olsker and the line Østerlars-Nylars is
305.04114 - 220.83219 = 84.20895 degree.
We call it angle 1 in the quadrilateral figure, formed by the four round churches.
Deviation between theoretical and real angle (84.2105147 - 84.20895) = 0.002 degree.
Angle 1 is surprisingly accurate.
We do not find the same accuracy, when we compare the three other theoretical angles in figure 3 with their corresponding real angles:
In Olsker the real angle between the line Olsker-Østerlars and the line Olsker-Nyker is
190.51665 - 124.90869 = 65.60796 degree. We will cal this angle 2 in the quadrilateral figure.
Deviation between theoretical and real angle 2: 65.7894853 - 65.60796 = 0.182 degree.
In Nyker the real angle between the line Nyker-Olsker and the line Nyker-Nylars is
158.06862 - 10.49093 = 147.57769 degree. We will cal this angle 3 in the quadrilateral figure
Deviation between theoretical and real angle 3: 147.57769 - 147.438369 = 0.139 degree.
In Nylars the real angle between the line Nylars-Nyker and the line Nylars-Østerlars is
360 - 338.10625 + 40.71181 = 62.60556 degree. We will cal this angle 4 in the quadrilateral figure
Deviation between theoretical and real angle 4: 62.60556 - 62.561631 = 0.044 degree.
[Control: Angles of real quadrilateral figure measured between Østerlars. Olsker, Nyker and Nylars - from above figures: (84.20895 + 65.60796 + 147.57769 + 62.60556) degree = 360.00016 degree. This confirms the reliability of both the coordinates and the KmsTrans program - and demonstrates the Earth's spherical geometry has no significant influence on angles within the area, we are examining.]
4.1: DISCUSSION:
If landscape geometry between the medieval round churches is intended, it is expected to find deviations in the accuracy of the lay out, since the churches are constructed sometimes (+/- 50 years) around year 1200.
The question is what kind of accuracy one might expect?
This (I believe) cannot be answered; because we do not have any similar landscape geometry lay out between Christian medieval churches to compare with - as little as we - to my knowledge - have registered similar landscape geometry between any medieval structures within Christian Europe.
Thus: If we decide to believe - or at least decide further to investigate if the presented geometry is intended and not a result of chance, we are totally depending of the archeological evidence of the accuracy in order to answer the question of what kind of accuracy, one might expect.
The average accuracy of the four angles in the landscape geometry can be calculated to +/- 0.09 degree.
4.2: DISAGREEMENTS
But - one angle out of the four angles is remarkable more accurate than the other three - 45 times more accurate than the average inaccuracy.
Also one distance between the four round churches is remarkable accurate:
When compared to the geometry in figure 1 and the distance Nylars-Østerlars = 1 = 14 339.505 m (table 2), the distance Olsker-Nyker is accurate within:
[14 335.505 * ½ *(sqr7/sqr3)]m - 10 944.83) = 4.09 meter. Deviation = 0.4 promille of the distance.
Notably the accurate distance Olsker-Nyker is not depending on the accurate angle Olsker-Østerlars-Nylars, since the inaccuracy in angles occur mainly through a prolongation of the line Østerlars-Nylars, which influence the angle Olsker-Østerlars-Nyker. Thus, the surprisingly accurate angle Olsker-Østerlars-Nylars cannot explain the surprisingly accuracy of the distance Olsker-Nyker, see figure 4:
These, I believe, is extraordinary, and whenever something extraordinary is observed, it is necessarily to consider the reason.
In the following, I shall present a hypothesis, which might explain, why one angle and one distance out of four angles and four distances between the round churches of Bornholm are so remarkable accurate:
5: TRIANGULATION:
Today (or at least recently), measurements and practical layout of geometry in the landscape would be done by the use of triangulation.
Since we have no records to confirm such possibility, one might doubt, triangulation was used on Bornholm by the end of the 12th century. On the other hand we cannot dismiss this possibility, simply because we do not have such records. That would be unscholarly because:
Triangulation is not a complicated invention. For those familiar with geometry it is in fact a very simple thought. I think any bright 12-years old with sufficient knowledge of geometry could get the idea of the advantage of combine triangles in the landscape with the purpose of measuring distances.
If the landscape geometry on Bornholm is the result of intentions, it would not have been possible to obtain the documented accuracy of distances by measuring with chains or tape measure through the undulated landscape. Thus, if the geometry is found to be the result of intentions, it could only have been achieved through carefully measuring angles. Those doing it must have been very familiar with geometry and might, to my opinion, have understood triangulation.
In fact, there are some indications triangulation was used in the layout.
Here we shall look at three such indications: 1: C on Christiansø, 2: Klemensker and 3: Rutsker.
5.1: C on CHRISTIANSØ:
The tiny island of Christiansø is visible from almost any location on the coastline following the northeast side of Bornholm.
But since Bornholm has a ridge down the middle of the island from northwest to southeast, parallel to the northeast coastline, it is possible - in addition to the coastline - to make observations to Christiansø from the ridge and from most part of the slope on the northeast side of the island.
Thus an observation post on Christiansø is extremely well qualified for the use of triangulation.
The geometry in figure 1 involves a location on Christiansø, in the following called C.
Today nothing can be found on Christiansø to confirm the location of C.
But the theoretical location of C (in accordance with figure 1) can be calculated from the position of the round churches of Østerlars and Nylars using the program KmsTrans:
The geometry in figure 1 dictates the distance from Østerlars to C being (sqr7/sqr3) * the distance from Nylars to Østerlars. We can prolong the line Nylars-Østerlars with this factor and use the program KmsTrans to calculate the coordinates to theoretical C.
The result of this calculation can be found in table1.Table 2 makes the foundation for the calculation of distances and angles to and from C in table 2.
The location of theoretical C on Christiansø is shown below in figure 5 (Sattelite foto):
The two red lines from C indicate sightlines to the north and south of Bornholm respectively and thus indicate the frame for observations from C to Bornholm. C is located on flat bedrock approximately 10 meter above sea level, while the island Frederiksø, crossed by the northern sightline, has its peaks limited to approximately 3 meter above sea level. C thus has free sight to Bornholm within the frame. Even today nothing on Christiansø blocks this free sight from C to Bornholm (the present houses on the sightline are all below the horizon). Seen from Bornholm, C is found right in the middle of the profile of the tiny island.
C is aligned with Østerlars and Nylars (described above).
In table 2 we find, how C is aligned also with the towers of two other churches: Rø and Klemensker.
Rø and Klemensker are unfortunately no longer the original medieval churches. These churches were knocked down in the last decade of the 19th century.
Both present churches are about 10 meters longer than the originals; both of them are to some degree an architectural copy of the originals with the same kind of a western church tower (a Romanesque tower in four stocks, connected to the church in the western end. An outstanding architecture attached to 10 medieval churches on Bornholm).
Both new churches are supposed to be situated on the original location, which can be proved for Klemensker, due to records of a crypt.
Using table 2 we can calculate the accuracy, with which the present towers of Rø and Klemensker are aligned with C:
C-Rø: 236.74583 degree, C-Klemensker: 236.74157 degree.
Deviation: 236.74583 - 236.74157 = 0.004 degree.
5.2: KLEMENSKER:
It is not difficult to place two church towers, Rø and Klemensker, on the same sightline from C, since there is a free sight from C to both towers.
But amazing then is the accuracy of Nylars on the line from C through Østerlars. One can see from C to Østerlars, but one cannot see from Østerlars to Nylars or from C to Nylars.
In order to find the exact location of Nylars (if based on C and Østerlars) I suppose it would have been necessarily to triangulate the geometry across the ridge.
(A suggestion for a possible practical procedure in constructing the geometry will follow.)
Considering doing so in practice, the tower of Klemensker could become important, since the tower not only is aligned through Rø to C on Christiansø but is further remarkable located on top of a hill positioned on the Bornholm ridge. Thus the church tower of Klemensker has extremely good observation possibilities to both side of the Bornholm ridge.
Remarkably Klemensker is one out of only two of the 15 medieval churches on Bornholm being located on a hilltop!
The rest, 13 medieval churches including the round churches, are specifically not located on a high point in the surrounding landscape, which is remarkable, since medieval churches in general are found placed on hights, and since the Bornholm topography is undulated ground.
5.3: RUTSKER:
The second church on a hilltop is the church of Rutsker.
Like Klemensker, Rutsker's hilltop is located on the ridge with extremely good observation possibilities to both sides of the island.
In my opinion this could indicate both towers of Rutsker and Klemensker has been used in a triangulation over the ridge to find the exact position of Nylars and Nyker.
This hypothesis is supported by finding the same distance from Østerlars to Rutsker as from Østerlars to Nylars.
If we examine the triangle Østerlars-Rutsker-Nylars, we can use table 2 to calculate its angles.
The angle Østerlars-Rutsker-Nylars = 165.21299 -109.76858 = 55.44441 degree;
The angle Østerlars-Nylars-Rutsker = 360 -345.26631 + 40.71181 = 55.4455 degree.
Deviation: 55.4455 - 55.44441 = 0.001 degree. Illustrated in figure 6
This then makes it necessary for consideration in relation to the observed accuracy
5.4: CONSIDERATION:
1: The angle 1, Olsker-Østerlars-Nylars, is - when compared to the theoretical geometry in figure 3 - accurate within 0.002 degree.
This can be compared to:
2: From the alignments of the four churches Nylars-Østerlars-Rø-Klemensker with C, the inaccuracy of the location of any of these churches - when compared to the location of Østerlars and Nylars - are found to be equal or less than an angle of 0.004 degree.
3: The accuracy of angles between the three churches Østerlars-Nylars-Rutsker is found to be less than 0.001 degree.
2 and 3 are in good agreement with the accuracy of angle 1.
I believe this makes it necessarily to ask the question:
If the design in figure 1 is deliberate, then:
Were the experts responsible for the landscape geometry capable of measuring angles down to 0.01 degree or better - (as reflected in relations between Klemensker, Rø, C, and Østerlars, reflected in relation between Østerlars, Nylars, Rutsker, reflected in the distance Olsker-Nyker, reflected in the distance Østerlars-Ibsker (explained later) and reflected in angle 1 between Olsker-Østerlars-Nylars?)
Or were they only able to measure angles down to 0.1 degree - as we find, when we compare real angles 2, 3 and 4 in figure 4 above with the corresponding hypothetical angles?
A part from it is difficult to accept how accuracy in measurement down to 0.01 degree at this time and place of history could be done; the above indications for this kind of accuracy are too many to dismiss this possibility!
On the other hand if we accept, the constructors were able to measure with this accuracy; one would expect this accuracy first and foremost to be found in the angles between the four round churches (figure 4), because these churches seem to define the geometry.
Thus we have a disagreement.
It would be easy to say - as some historians have said in general about my hypothesis: It is all due to coincidence.
The common argument has been twofold:
1: It must be due to coincidence, because when you have enough combination possibilities, geometrical designs like the Bornholm Geometry are likely to appear from coincidence.
2: Nobody could do this on Bornholm by the end of the 12th century.
But both arguments are unscholarly, if they are not sustained by scientific evidence.
Using scientific methods Lind has examined the possibility of coincidence (Bayesian analysis of a landscape geometry, Niels Lind, Institute for Risk Research, University of Waterloo, exhibit A) Lind concludes: "In this case the evidence strongly supports the hypothesis of deliberate design."
In fact, if you - during further examinations of a real layout in the landscape - discover extremely high degree of accuracy between the hypothetical layout and the real landscape geometry, this diminishes the possibilities for coincidence - at the same time as it in it selves then proves, what they (the constructors) were able to do at this time of history.
Thus is seems worth to consider how to explain the above disagreement, which I will boil down to:
1: We have four round churches in the landscape.
2: When we measure each of the four angles between these churches and compare the measured angle with its corresponding hypothetical angle, we find one angle with an accuracy of 0.002 degree and three angles with average accuracy of 0.122 degree.
3: Why this significant difference?
Could it have been easier to measure the Olsker-Østerlars-Nylars angle than to measure the other three angles?
No - the fact is that two of the four churches (Olsker and Østerlars) are on one side of the ridge, and the other two (Nylars and Nyker) are on the other side of the ridge. Thus it is impossible to se from one of the round churches to two other round churches. Thus it should be considered if triangulation has been used in the layout.
We do seems to have indications for the use of triangulation - but these evidence all have accuracy better than 0.01 degree - which we find in agreement with the accuracy of the angle Olsker-Østerlars-Nyker - but in disagreement with the other three angles between the four churches.
6: A DOOR TO UNDERSTANDING:
We can open a door though, which could lead us to understand how all four round churches are located in the landscape within the same accuracy (all better than 0.01 degree).
At the same time this might sustain a hypotesis of the reason, why these extraordinary round churches were build and located within a specific and accurate landscape geometry.
That door connects to an analysis of the upper floor in Østerlars (will be provided later: Erling Haagensen, exhibit B). This analysis do not find a single argument - neither in the original architecture nor in the original arrangements - to support the purpose of its construction had anything to do with defense. Instead the analysis finds astronomical reasons behind the architecture and arrangements to be the most likely explanation for the purpose of the third floor of Østerlars.
It is not likely to have one purpose for one of the round churches and another purpose for the others, since they are so alike in original design.
Thus if the purpose was astronomical observations, what might be the reason to construct four astronomical observatories on the same little island?
6.1: KEY TO THE DOOR:
This might be explained from a desire to measure the curvature of the earth.
How measures from astronomical observatories within a limited area like Bornholm/Christiansø are able to provide information of the earth's curvature is explained by Lind (The Bornholm Method, Niels Lind, Distinguished Professor Emeritus, University of Waterloo, exhibit C).
An important factor is measuring the meridian convergence.
The meridian convergence might, as we shall see, be the key, that opens the previous mentioned door to an understanding.
6.2: ORIENTATION
First, let us look at one factor of the geometry, we have not yet examined: Its orientation.
It is shown in figure 1, how the plane geometry is orientated towards true north.
But this is not possible to mirror exactly when laid out in the landscape, which is due to the meridian convergence, illustrated in figure 7: Earth with meridians.
If it is desired to orientate a geometrical pattern on earth, like the Bornholm geometry, towards north, as I believe it was, it will be necessarily to make a choice. The orientation will only be accurate along a chosen meridian.
So which of the meridians in the Bornholm geometry would it be logically for the constructers to chose?
In order to answer this question, it is of interest to see how it is possible in a landscape under influence of the meridian convergence to calculate measure and fix desired geometrical proportions.
Let us assume, we in the landscape want to create a specific geometrical relation between the length of a fraction AB of a meridian and the length of a fraction YX of a parallel crossing AB in X.
Issue A: Since AB extents north-south along the meridian and are crossed by an infinite numbers of parallels, all with different circumferences, we will have to fix the point X (a latitude) on the meridian, in which "fraction AB" should have the desired proportion to "fraction YX".
The following is a method to obtain the above mentioned desire:
For reasons to be discussed later we assume in the following example, the desired relation AB/YX should be:
AB = (2/sqr7),
YX = (sqr3/sqr7)
The starting point is to select two measuring locations with free sight to each other.
Examples are Ø' (on Bornholm) and C (on Christiansø).
The first step is to measure the meridian convergence between Ø' and C.
Figure 10 illustrate the real meridian convergence between Ø' and C.
If we - in plane geometry - desire to construct a triangle Ø-Nl*-QNl* with sides 1, 2/sqr7, sqr3/sqr/, the angle QNl*-Ø-Nl* would be (arctan sqr3/2) = 40.89339 degree as illustrated in figure 9:
We want to construct triangle Ø-QNl*-Nl* so that (QNl*-Ø) follows the meridian through Ø - and (Qnl*-Nl*) follows a parallel through Nl* (which makes QNl*-Nl* a fraction of one of the parallels crossing AB). We want further the relation between Ø-Nl* and Ø-QNl* to be 1 / ( 2/sqr7).
Then:
Step 2 is to decide, the distance between Ø and C = (sqr7/sqr3).
Step 3 is to find Ø by moving along the meridian through Ø' until the angle between north and C equals:
40.83249 degree.
The reason for this angle is explained in figure 11 below:
Because only then Ø-ØNl* will be equal to the desired fraction AB, see figure 11:
Thus the angle Ø-Nl*-QNl* should be [(arctan sqr3/2) - m/2], where m is the meridian convergence between the meridian through Ø and the meridian through Nl* as illustrated in figure 11:
6.4: IMAGINARY POINTS IN THE BALTIC SEA
C is a point located on Christiansø and Ø is a point located on Bornholm. Nl* and QNl* will be imaginary points, located in the Baltic Sea, which help us to calculate the above angle between the meridian through Ø and the sightline from Ø to C.
Ø-QNl* = 2/sqr7 will be (an imaginary) fraction of the meridian through Ø = the desired AB. Because we assume the earth is a perfect sphere, the fraction Ø-QNl* = AB will have the same value for any latitude of Ø (=A). Thus X, mentioned in Issue A, is the important issue.
If we further analyze the imaginary geometry in figure 11, we find an additional imaginary geometry illustrated in figure 12:
Ø* have the following two qualities:
1: A parallel through Ø* will be intersected by the meridian through Nl* and the meridian through Ø. The fraction of this parallel between the two meridians will be the desired (sqr3/sqr7) in relation to the fraction QNl* - Ø of the meridian through Ø being (2/sqr7) = AB. The latitude of Ø* will thus be the latitude (equal to X), which we can determinate as the answer to the above mentioned issue A.
2: The meridian through Ø* crosses the sightline Ø-C in the exact angle from the geometry in figure 1, that is (arctan sqr 3/2) = 40.89339 degree.
From this I suggest it would be logical for the constructors to orientate the geometry from the meridian through point Ø*.
I suppose the location of Nylars has been determined from C, Ø and G by use of triangulation (G, which is calculated in table 2 and 3, will be explained in a following exibit, which gives a suggestion to how how the geometry in practice could be measured and constructed).
Nylars is thus a manifestation on Bornholm ground as a mirror of Nl* in the Baltic Sea.
In the following we shall see how orientation of the geometry from a meridian through Ø* locates all four round churches in the landscape with accuracy in accordance with the accuracy of previous mentioned angle 1.
Orientation of the geometry from the meridian through Ø* as described above is (perhaps?) reminiscent to a known map projection shown in figure 13:
inverse mapping equation transforms mathematically the planar Cartesian coordinates (x,y) of a point on the map plane to a set of geographic coordinates (f,l) on the curved reference surface:
(f, l) = f (x, y)
It seems the Bornholm geometry is based on inverse mapping equation. Because, as we shall see, the real geometry, pointed out by the location of the four round churches, become identical to the hypothetical geometry in figure 1, if the prime meridian through Ø* is a straight line, and meridians through C, Østerlars and Olsker are "twisted" to be parallel to the prime meridian through Ø*, as illustrated in figure 14.
The usual way to use a map projection is to create a (flat) map as a reduced image of the Earth's spherical surface. But in this case, it seems the intention has been to use projection know-how to create an enlarged image of a (flat) geometry pattern onto the spherical surface. This hypothesis is supported below.
In order to find the angle of the "twist" in figure 14, it is necessarily to calculate the meridian convergence between Ø* - Ø (Ø = Østerlars), Ø* - C, and Ø* - O. (O is theoretical Olsker under the influence of the meridian convergence).
Prolonging the line Nylars-Østerlars with 0.5 * distance Nylars-Østerlars, KmsTrans help us to find the coordinates to Ø* (Geo euref89):
55 13 12.69774
15 02 07.17040
Again by help from KmsTrans we can calculate the above mentioned meridian convergence
Ø* - Ø = 0.0614; Ø* - C = 0.1248; Ø* - O = 0.1929 ( Ø*-O is in praxis measured and calculated from O - C after O has been located - I will explain later how it might have been possible to do the layout in praxis). Illustrated in figure 14.
We can make the following adjustment of the geometry in respect of figure 14.
First we look at triangle Ø-C-O:
Angle O-Ø-C = (N-Ø-O - 0.0641) + (N-Ø-C + 0.0641).
This means the theoretical angle O-Ø-C is identical to the theoretical angle Ol-Øl-Chr in figure 2
This could explain the accuracy of angle 1.
Angle N-C-Østerlars = 221.01769 (table 2). Theoretical angle Øl-Chr-Ol (figure 2): 28.1712837
Angle O-C-N (when adjusted for the "twist"): 221.01769 + 28.1712837 + 0.1248 = 249.3137737
The real angle N-C-Olsker: 249.31199;
Deviation: 249.3137737 - 249.31199 = 0.002 degree.
The same calculation can be done by adding the "twist" 0.1248 to the theoretical angle Øl-Chr-Ol in figure 2.
Thus theoretical angle Ø-C-O = 28.1712837 + 0.1248 = 28.2960837 degree
Real angle Østerlars-C-Olsker (table 2) 249.31199 - 221.01769= 28.2943 degree
Deviation: 28.2960837 - 28.2943 = 0.002 degree.
Coordinates to O can be calculated with KmsTrans (using the above information and the real coordinates to Østerlars):
O (Geo euref89):
55 14 09.73736 sx
14 48 01.39666 sx
From Olsker (center of rotunda) to O:
0.868 m
333.90387 dg
153.90387 dg
Point O can be demonstrated with help from Google Earth:
Figure 15:
Using the same method we can calculate theoretical location of Nyker, Nk, illustrated in figure 16:
From the above calculation of angle Ø-C-O we realize, that we just have to subtract the "twist", 0.1929, from the theoretical angle Øl-Ol-Ny in figure 2, to find the theoretical angle Ø-O-Nk.
Thus Ø-O-Nk = 65.7894853 - 0.1929 = 65.5965853
Real angle Østerlars-Olsker-Nyker (table 2) = 190.51665 - 124.90869 =
Deviation: 65.60796 - 65.5965853 = 0.011 degree
The theoretical distance O-Ny = ½ * (kv7/kv3) = 0.763762616 (figure 2 and 3).
From this we can calculate the coordinates to the location of Nk, based on the coordinates to O:
Nk (Geo euref89)
55 08 21.60650 sx
14 46 08.64854 sx
from Nyker to Nk:
3.451 m
180.00000 dg
-0.00000 dg
Point Nk can be demonstrated with help from Google Earth:
If we accept a certain "projection" of the hypothetical geometry in figure 1, we can demonstrate how the theoretical geometrical locations of all four round churches are found within the circumference of the respective church's rotunda.
Consideration:
If the above method was used to calculate the location of the four "observatories" on Bornholm, then what was the purpose of doing so?
I cannot answer this.
I can guess it could be connected to the purpose of the four observatories - and may be it strengths the hypothesis of this purpose being connected to the desire of measuring the curvature of the earth?
But I need help from expertise in geodesy and astronomy to examine this question.
Another question, which needs further examination, is the reason why the geometrical relation AB = 2/sqr7 and XY = sqr3/sqr7 seems "to lock" the proportions of the landscape geometry in relation to the meridians and the parallels. Why these specific relations?
Might it have any relation to geometrical proportions between longitude and lattitude?
This proportion is [3/2 * sqr7] on latitude 55.46241621, which is reasonable close to the latitude of Ø*.
Appendix A: RELATION TO "THE BORNHOLM METHOD":
Lind *) has described "The Bornholm Method" in this way:
Suppose we have a pair of points A and B on the Earth's surface. B lies y units North of A and x units East of A. It is important to note that the length of the unit of need not be known; the configuration of the five points can be laid out, and all necessary astronomical observations made, by measuring only angles. By Eratosthenes' method, the difference u in latitude and the value of y gives the local curvature of the earth in the N-S direction as u/y. The local radius of curvature of the meridian equals y/u; this is the distance from Bornholm down to the center of curvature of the meridian locally. Now, if the Earth is roughly spherical, then this center lies on the axis of rotation of the heavens. If, on the other hand Earth is oblong or toroidal the center will lie beyond, or short of, the axis, respectively. This is the question.
The Bornholm Method considers a plane triangle formed by two observatories and a point N on the horizon in the direction of North. In this triangle two angles are measured. The third angle, called the meridian convergence, is calculated by subtracting the two measured angles from 180o. The distance from Bornholm to point N and to the axis is easily calculated using x, y and the meridian convergence. Thus, every pair of observatories gives an answer to the question… It is remarkable that the Bornholm Method uses no measurement of length, and requires only plane, not spherical, geometry.
: In short, Lind explain how it is possible to compare measurements from two locations, A and B, to estimate if the curvature of the Earth is perfect spherical within the area, in which A and B is located.
The conditions are, that B is (A + y) degrees north of A - and (A + x) degrees east (or west) of A. A
and B is located within a known landscape geometry. This makes it possible to measure the difference in latitude and the difference in longitude between A and B, and - by use of the landscape geometry connecting A and B - it is possible to perform and compare two independent calculations of the radius of the Earth.
But as described, A and B cannot be on the same meridian - and A and B cannot be on the same parallel.
Thus: In order to be able exactly to compare the two different measurements, it is necessarily to establish a geometry, based on two locations (Ø and QNl* above, where Ø = A) on the same meridian and a location (Nl* = B) on a parallel through QNl* as described above. From the geometrical relation between Ø-QNl* = y and QØ*-Ø* = x (QØ* and Ø* are described above) it is possible to make an exact comparison between the two measurements of radius of the Earth as described by Lind.
A landscape geometry constructed with the above method seems to secure the fundamental exact relation between x and y?
Appendix B RELATION TO BANDHOLMS PROJECTION
Or could the relation sqr7/sqr3 be important in relation to the obliquity of the ecliptic, as suggested by Niels Bandholm (Niels Bandholm, "The Celestial Key: Heaven Projected on Earth", pp. 95-116 in Nexus VII: Architecture and Mathematics, ed. Kim Williams, Turin: Kim Williams Books, 2008.)
I need help to answer these questions.
I hope publication of this article on the internet will inspire international geodesist, researchers and other scientist to consider these questions.
©Erling Haagensen, May 2012.
*) Niels Lind, University of Waterloo Distinguished Professor Emeritus, 404-1033 Belmont Avenue, Victoria, BC, Canada V8S 3T4; nlind@telus.net
Abstract
1: Landscape geometry
2: Basis for calculation
3: The method
4: Examples
4.1 Discussion
4.2 Disagreements
5: Triangulation
5.1: C on Christiansø,
5.2: Klemensker
5.3: Rutsker
5.4: Consideration
6: A door to understanding
6.1 Key to the door
6.2: Orientation
6.3: A choice
6.4: Imaginary points in the
Baltic Sea
6.5: Discussion
7: Conclusion
Appendix A Relation to The Bornholm Method
Appendix B Relation to Bandholm's Projection
1: Landscape geometry:
The position of the four medieval round churches on Bornholm island -- Østerlars, Olsker, Nylars, and Nyker - fit into the simple hexagonal pattern shown in figure 1:
