Fig. 1 Part of Avebury Stone Circle

Neolithic people left no conventional written records, no words to be read and interpreted in the future. But they did leave behind a monumental body of work in the form of geometric shapes dotted all over the landscape. The construction of more than one thousand stone circles across Britain testifies to something meaningful, an organised building programme on a grand scale in which geometrically arranged stones define a point in the landscape for posterity. These stone circles are found from Orkney to Land’s End and their locations initially appear to be random and uncoordinated; however the locations for these monuments were clearly chosen for a reason. Stones were often moved several kilometres to create a circle and we can safely deduce that if the location itself was not significant, and the purpose was simply to make a circle, then all the circles would be found where the stones originated. But this is not the case. So the choice for the position of a circle was evidently an intellectual decision, one apparently based on finding the correct location in the landscape for construction. If a common factor could be found that links these apparently arbitrary points in the countryside then the reason for choosing the site might become clear, and the motivation for doing all this work may be more fully understood.

When the topography surrounding Neolithic stone circles in southern Britain is studied, a common factor emerges regarding the majority of these locations. The circles are evidently placed at specific points in the landscape in order to create geometric patterns with the surrounding natural high points. Measuring straight lines between these high points and the location of a monument results in a series of isosceles and right triangles; and this result is found from one monument to the next. These patterns very rarely occur among a cluster of random points. At Avebury the probability of creating this unusual geometry with surrounding hilltops by chance is less than 0.02%. This evidence supports the hypothesis presented by the late Professor Thom (1894 – 1985) that Neolithic people surveyed the land with skills equivalent to our own surveyors today. But what was the purpose of these surveys? The result of studying the Neolithic monuments in this way is to recognise the landscape itself in the context of a geometrically ordered object. It appears that the motivation to create a geometric point in the landscape was therefore part of a larger picture, one that defines the landscape in terms of pattern. Avebury provides just one example of how a monument’s precise location was chosen in order to reveal a hidden topographical pattern.

**The Obelisk at Avebury.**

The stone at Avebury, called ‘The Obelisk’, was destroyed by local farmers in the 18^{th} Century but fortunately it was still intact in 1723 when William Stukeley measured and sketched it lying on its side. It was 21 ft long with a diameter of 8 ft 9 inches, a similar size to a trilithons upright at Stonehenge. The Obelisk once stood upright ‘rearing its spiring head’ over all the Avebury stones as witnessed by the poet Charles Lucas. The stone was broken up and completely removed, but its location, at the centre of a stone circle within Avebury Henge, was marked with a concrete plinth by the archaeologist Alexander Keiller.

Fig 2. The Obelisk at Avebury. William Stukeley. 1723

Why did Neolithic people decide to drag Avebury’s tallest stone to this particular point and no further? Was it a careless act, or did the specific location on the Marlborough Downs have any significance that we can recognise today?

If this Neolithic stone ‘high point’ at Avebury is treated as a geometric point in the landscape and five of the highest natural summits on the surrounding Marlborough Downs are also treated as geometric points, straight lines can then be drawn from the summit points to the Obelisk. This results in a geometric pattern comprising four isosceles triangles, with each triangle having one corner (or vertex) located on the Obelisk. The probability that this pattern might arise at random has been estimated by computer programs developed by Oliver Bentham. Using the appropriate line tolerances the probability is less than 0.02 % and consequently it is highly unlikely that the Obelisk at Avebury was located randomly. It is far more likely that the location was carefully chosen in order to mark this unique geometric point in the landscape. The pattern at Avebury is described in the following section, it is a fine example of Neolithic landscape geometry, and yet it is only one of numerous examples that can be found on this website and in the book Sacred Geometry of the Earth.

**The first Obelisk triangle (Tan Hill and Milk Hill).**

The Obelisk at Avebury was located equidistant from the two highest summits in Wiltshire; an isosceles triangle can therefore be drawn between the Obelisk and these two natural predominant high points.

Fig 3. On Google Earth the Obelisk is equidistant from geometric points located on the summits of Tan Hill and Milk Hill.

The twin summits of Tan Hill and Milk Hill are 294 and 295 meters above sea level and both summit areas are relatively flat. There is a modern surveyor’s concrete ‘trig point’ fixed on Tan Hill summit which is visible on Google Earth. The distance between this trig point on Tan Hill and the Obelisk can be measured at 5,620 meters using the Google Earth ‘ruler’. In turn the summit of Milk Hill is marked on Google Earth by a single point, a green triangle, (a ‘labels/geographic features’ option). The distance from the Obelisk to this summit symbol on Google Earth is just two meters short of the first measure from the Obelisk to the Tan Hill trig point. Hence the Obelisk and these two summit points in Wiltshire create an isosceles triangle accurate to two meters on the ground when measuring these distances on Google Earth.

Alternatively, on the Ordnance Survey Map (1;25000 only), the summit of Milk Hill is not identified by a single point, instead it is defined by a closed contour 295 metres above sea level containing an area of about 40 x 60 metres. At the southern limits of this contour is a point which is also 5,620 metres from the Obelisk. This point on the map is all but identical to the summit symbol marked on Google Earth. (The green triangle summit symbol is actually 5,618 metres from The Obelisk.)

Although it may be impossible to define a hill summit precisely it is true to say that the locations of Wiltshire’s two highest summits, defined and measured on Google Earth and the Ordnance Survey Map, are equidistant from the Neolithic Obelisk, accurate to a few meters on the ground. The line tolerance required to generate the exact isosceles triangle between these three points on Google Earth is less than 0.1%.

If the tolerance on the line lengths is increased to 1% (approximately 50 meters) it has been estimated by computer simulation that three points randomly distributed in the landscape will create an isosceles triangle on about 3.0 % of occasions.

This equidistant relationship could simply be a chance event, but the calculated nature of The Obelisk’s location is discovered through the repetition of this isosceles triangle motif from one summit point to the next.

**The second Obelisk triangle (Cherhill).**

Cherhill summit is the highest point on the Marlborough Downs to the west of Avebury. This summit is marked on the Ordnance Survey Map with a point height of 262 meters. On Google Earth the summit area around this point height is identified as a patch of level ground all of which is 261 meters above sea level. Using Google Earth it is possible to place a geometric point on this patch of land that is exactly equidistant from Tan Hill trig point and The Obelisk. This geometric point on Cherhill’s summit is about 20 meters from the summit point height marked on the Ordnance Survey Map.

Fig 4. shows the second isosceles triangle sharing a side with the first.

Fig 4. The points placed on the three hill summits on Google Earth create two isosceles triangles with the Obelisk at the common corner of both triangles.

It is clear from the Ordnance Survey Map that the three summits identified in this geometry are the three highest independent summits on the Marlborough Downs to the south and west of Avebury. The unlikely probability that this geometry will occur by chance can be judged by randomly dispersing three fine points on an A4 sheet of paper. It will require an outstanding stroke of luck to introduce a fourth point at random and find that it has the equidistant qualities of The Obelisk. The line tolerance required to create both these triangles on Google Earth is less than 0.5%; using this tolerance the computer estimates the probability of a fourth random point creating two isosceles triangles like those described to be around 0.1%. And the same pattern continues with the next high point.

**The third Obelisk triangle (Martinsell Hill)**

The highest area of land in Wiltshire is a pronounced ridge that rises at Tan Hill and runs eastwards to Milk Hill; the ridge then continues in an easterly direction for about ten kilometres. Point heights on this ridge are all below 270 meters until the land rises to the highest summit on the Marlborough Downs to the east of Avebury, Martinsell Hill at 289 meters above sea level. Like the other hill summits discussed so far a summit point on Martinsell Hill Is not easy for the rambler to define, but there is a concrete trig point on the summit area (now covered in moss in a copse full of bramble and bluebells). A geometric point located on this overgrown trig point is 9,687 metres from The Obelisk. The same geometric point is 9,706 meters from the trig point on Tan Hill. The two modern day trig points and the Obelisk form an isosceles triangle accurate to 20 metres on the ground measured on Google Earth; a line tolerance less than 0.25% fig c.

Fig 5. The base line for both triangles extends from the Obelisk to the trig point on Tan Hill. The two apexes are located on the highest summits on the Marlborough Downs to the east and west of The Obelisk.

Through its location The Obelisk illustrates two interesting natural phenomena. In figure d. the Obelisk is located at the point where the two arcs meet, but it may seem odd in a chaotic landscape to find that this geometric point is also equidistant from Wiltshire’s two highest summits, Tan Hill and Milk Hill.

Fig 6. The two arcs meet at The Obelisk and this geometric point is equidistant from Tan Hill and Milk Hill summits. (diagram not to scale).

Secondly The Obelisk defines carefully measured geometry in the landscape because using a Cosine calculator the angle created at Cherhill is twice that at Martinsell Hill, accurate to 0.2 degrees.

If all four hill summits are considered as random points, the computer simulations estimate the probability of introducing a fifth point (The Obelisk) that will, by chance, create three isosceles triangles with the other points. The probability that the Obelisk will create these isosceles triangles with a line tolerance of 0.5% (as it does) is calculated to be less than one chance in one thousand. This is such an unlikely probability that it becomes reasonable to assume that the people who located the Obelisk were exemplary surveyors who adopted hill summits as geometric points. They are people who both create and describe natural topographical geometry and by inference suggest that these hill summits are ‘ordered’ in some way.

**The fourth Obelisk triangle (Morgan’s Hill).**

At 262 meters above sea level the summit of Cherhill overlooks all of the western down-land and there is only one other hill of comparable height overlooking Avebury from the west, this is Morgan’s Hill which also has a trig point on the summit, 260 meters above sea level.

On Google Earth the distance from the trig point on the summit of Morgan’s Hill to the Obelisk is 7,925 meters. This compares with a distance of 7,923 meters measured from the same trig point to the summit symbol on Milk Hill. In light of the three isosceles triangles already discovered between high summits and the Obelisk the extreme accuracy of this fourth isosceles triangle created on Google Earth is remarkable.

Fig 7. The trig point on Morgan’s Hill summit is equidistant from the Obelisk and the geometric point established on Milk Hill summit. The isosceles triangle joining Milk Hill, the Obelisk and Tan Hill is also shown.

All four isosceles triangles are shown in figure 8.

Fig 8. Four isosceles triangles created between the five high points all sharing a corner at the Avebury Obelisk.

The five high points cannot be joined in any combination with each other to create a single isosceles triangle; The Obelisk provides the crucial geometric point for all four isosceles triangles.

A computer simulation with five random points estimates the probability of a sixth random point creating four isosceles triangles with line tolerances of 1%; the probability estimate is approximately 0.02%. The maximum line tolerance required to create the triangles in figure f. is less than 1%.

This evidence suggests that The Obelisk was carefully situated using accurate measurement. Its location reveals wisdom in art and science, a record of topographical geometry wrapped up in a single geometric point.

Professor Thom (1894-1985) was the Emeritus Professor of Engineering at Oxford University and he believed that Neolithic people around 3,000 BCE were surveying the land with skills equivalent to our own surveyors today. He made more than 300 surveys of Neolithic sites to reach this conclusion. Thom’s work reminds us that Neolithic people were not aware of the attitude we would have towards them in the 21^{st} Century.

**A fifth triangle (Hackpen Hill and Windmill Hill).**

To the north west of Avebury the landscape is relatively flat with the high point on Windmill Hill at 196 metres marked with survey point on the OS Map. The summit earthworks on Windmill Hill date back to 3,800 BCE. Directly to the north of Avebury the land rises considerably to the highest point at 272 meters marked on the map by the trig point on Hackpen Hill (another hill with a flat summit). Using dividers on the map the distance between Hackpen Hill trig point and The Obelisk appears to be the same as the distance between Hackpen Hill trig point and the summit point marked on Windmill Hill. Using Google Earth there is a discrepancy of about 20 meters between the triangle’s two leg lengths.

Fig 9. A fifth isosceles triangle with The Obelisk on one corner and high points on the other two corners.

With this fifth triangle the isosceles geometry now includes the highest summits to the north, south, east and west of Avebury in a radius of ten kilometres, along with Windmill Hill.

There are now seven hilltops involved in this geometry creating five isosceles triangles, each with a corner on the Obelisk. It is tempting to think that these types of triangles must be commonplace, however if the Obelisk point is removed it is possible to create only one isosceles triangle between the seven high points independently, and this triangle requires a line tolerances near 1.0%. In other words the location of the Obelisk is a remarkable and unique point in the landscape because of its extemely unusual geometric relationship to the surrounding high points. By placing the Obelisk so precisely at this location the Neolithic surveyors define the landscape in terms of ordered geometry, moreover they reveal a point where this ordered geometry can be recognised.

It cannot be overlooked that the Obelisk is located only 1500 metres from the largest Neolithic earth mound in Europe, a clear topographical symbol, Silbury Hill. But before moving to this summit the sceptic will ask whether our choice of natural high points is biased and may suggest that more high points should be included around the Obelisk. Looking closely at the Ordnance Survey map it could be argued (at a stretch)that two or perhaps three high summits have been missed out, Golden Ball Hill for example.

The computer calculation was therefore made with an additional three random points included. The probability (to a tolerance of 0.5% ) of at least four isosceles triangles sharing a corner remains below 0.2 %.

The sceptic would also question the method of measurement. A perfect isosceles triangle is hypothetical and has zero probability so a tolerance on the line length is essential with regard to lines meeting geometric points located on the hilltops. Firstly the geometric points need to be established and Google Earth is ideal for this purpose.

(Google Earth has a ‘time slider’ allowing access to several satellite images taken over a period of decades. The older images compared with the current image can reveal a ‘slippage’, where a precise point on an old image appears to minutely shift its latitude and longitude on the newer image. The maximum slippage found at Avebury amounts to 7 metres between the 2005 image of Windmill Hill and the 2015 image. To achieve such accuracy with this unique global digital map is a great credit to the people who produce Google Earth; it gives confidence that measures over relatively short distances in one location will be consistent and accurate within small parameters.)

Zooming in on Google Earth it is possible to define a point on the ground using the ‘ruler’ option; this point has an immeasurably small diameter. A line can be extended from this point to any other and these lines are measured automatically in ‘Ground Length’ and in ‘Map Length’ with negligible difference, in the context of the Avebury triangles.

To create the five triangles discussed above a placemark was located on the summits of the hills. Four of these placemarks were located on the Ordnance Survey summit trig points, only three of which are visible on Google Earth; Morgan’s Hill, Tan Hill, and Hackpen Hill. The concrete trig point on Martinsell Hill was located on site, and by referring to cross hairs created on the OS Map 1:25000 and transferring these to Google Earth. The remaining three summits were identified with placemarks using cross hairs to define the summit ‘point height ‘on the paper map, or highest contour in the case of Milk Hill. Having created these placemarks on Google Earth they were then treated as geometric points with one further point located on The Obelisk plinth at Avebury which is also visible on Google Earth.

Here is the list of placemark coordinates followed by a list of the relevant distances between the placemarks:

**Coordinates of Geometric Points located on Google Earth:**

The Obelisk at Avebury: 51.428039. -1.853392.

Morgan’s Hill summit: 51.400891. -1.958728.

Cherhill summit: 51.418925. -1.924776.

Tan Hill summit: 51.381233. -1.883784.

Milk Hill summit: 51.377547. -1.851159.

Martinsell Hill summit: 51.373189. -1.744891.

Windmill Hill summit: 51.441551. -1.876405.

Hackpen Hill summit: 51.468303. -1.815705.

**Distances between these placemarks are in given in meters**:

Hackpen Hill summit (trig point) to Obelisk: 5189

Hackpen Hill summit (trig point) to Windmill Hill summit:5162 (5189 to summit tumulus)

Obelisk to Tan Hill summit (trig point) 5620

Obelisk to Milk Hill summit (summit symbol) 5618

Martinsell Hill (trig point) to Obelisk 9687

Martinsell Hill (trig point) to Tan Hill (trig point) 9706

Cherhill summit (OS point height) to Obelisk 5062

Cherhill summit (OS point height) to Tan Hill (trig point) 5091

Morgan’s Hill (trig point) to Milk Hill (summit symbol) 7922

Morgan’s Hill (trig point) to Obelisk 7926

The maximum difference between the five paired distances is 29 metres, the minimum is 2 metres. The maximum line tolerance required to create triangle legs of exactly equal length is about 0.5% in one case, and less than 0.5% in the other four cases.

**How the probabilities at Avebury were calculated. **

By Oliver Bentham

The fundamentals of this situation were modelled by computer program where millions of triangles were created at random. In the simulation an individual triangle was flagged as being isosceles when the two “equal” sides where measured to be equal in length to within a tolerance of 1%.

By this single criterion 3% (to the nearest degree) of the randomly generated triangles where measured as isosceles. When we add a second criterion, requiring the difference in the size of the “equal” (base) angles of the triangle to be less than or equal to 0.5 degrees, then the percentage of triangles measured as isosceles was reduced to 1.5% (to the nearest 0.1 percent).

We will use the higher figure of 3% as a safe over- estimate of the probability that three random points will form an isosceles triangle by chance alone. In the calculations that follow this probability is denoted by the symbol p.

Using the assumed figure of 3%, the probability that the five (random) hilltops and the Obelisk will form a pattern comprising of at least 4 isosceles triangles can be shown to be 0.000147 (0.015%). This result is explained below.

Generally if we have a point of interest X in the landscape (in this case the Obelisk) and 5 other points, then using the point X and two points chosen from the other 5 points we can construct a total of:

triangles in the landscape.

If T is the number of triangles out of the 10 possible triangles which are measured to be isosceles, and assuming the triangles are arranging themselves at random, then T is a random variable with a binomial probability distribution:

The probability that T takes a particular value t, (0 ≤ t ≤ 10) is given by:

For our assumed value of p = 0.03 the probability for each value of t are shown in the table.

**The Null Hypothesis**

Mathematically we wish to test the notion or proposal that any configurations of isosceles triangles we observe in the landscape are appearing strictly randomly, and we initially assume an underlying explanatory hypothesis denoted H_{0}: p = 0.03, (called the null hypothesis) that the probability any three random points in the landscape form an isosceles triangle by chance alone is 0.03.

In the situation we are considering at Avebury, if T is the number of triangles out of the 10 possible triangles which are measured to be isosceles, and assuming the triangles are arranging themselves at random then T is a random variable with a binomial probability distribution:

The validity of the explanatory null hypothesis is tested by simply noting the number of isosceles triangles that we actually observe in the landscape. Very large numbers of isosceles triangles will cast doubt on the validity of the null hypothesis, and the null hypothesis may be rejected in favour of an alternate explanatory hypothesis H_{1}: p > 0.03.

We can define what qualifies as very large to our own convenience. What we require statistically, is that the probability of making a mistake by rejecting the null hypothesis when it is in fact true, is kept to a level which is acceptably small.

The probability of rejecting the null hypothesis when it is true is called the significance level of a statistical test. A commonly used value for the significance level of a hypothesis test is 5%.

The last column in the table gives the probability that T will take a value which is as large as or larger than t by chance alone assuming p = 0.03. Looking at the table, if we had rejected H_{0} when the number of observed isosceles triangles T was 2, then the probability that we had made a mistake in doing so would have been 0.0345.

The probability that we are mistaken in rejecting H_{0}: p = 0.03 when T = 5 is 0.0000.

This probability is so small that we can with very great confidence reject H_{0} in favour of the alternative explanatory hypothesis H_{1}: P > 0.03. We need p to be very much larger than 0.03 to explain why we are observing so many isosceles triangles, and consequently we need to seriously consider the possibility that the configuration of the hilltops and the Obelisk is not at all random.

An obvious criticism to this analysis could be that the number of points, other than X we should have considered was in fact larger than 5 and that we may have arbitrarily excluded other geographical high points of note or significance. However it can be easily shown that, observing 5 or more isosceles triangles configured by chance alone in any localised area of a landscape comprised of M high points would be a rare event for less than say 14 high points. This is demonstrated below.

Generally if we have a particular point of interest and we want to investigate its spatial relationship with a further M points, then the point of interest will form a total of

triangles with the M ancillary points.

If p is as defined as before; T is now a binomial random variable with probability distribution:

The probability that T = t is given by the formula

The results for various values of M are shown in the table below.

M = the number of high points in the localised area of the landscape

∆ = the total number of possible triangles that can be formed using these high points.

P = the probability the number of isosceles triangles we observe is greater than or equal to 5

**Summary of the Obelisk Geometry**

Whether looked at on a sheet of paper, or calculated with a computer program, it is clear that the decision to place the Obelisk at this precise point in the Wiltshire landscape is unlikely to be one made by chance. On the contrary, it can be said with considerable confidence that to find this location required either intelligent survey and calculation, or a method of reckoning unknown to us.

Neolithic people have found a point in the landscape illustrating how the high hilltops on the Marlborough Downs can appear to be geometrically ordered. This hidden topographical geometry reveals a purpose dictating the location of the Obelisk. To find a point in the landscape where this extremely unlikely geometry does occur and to mark it so emphatically for posterity amounts to a clear signal that landscape geometry of this kind was considered significant to the people responsible. It speaks of a motive for creating geometric points in the form of monumental circles and mounds in the landscape. The Obelisk at Avebury was surrounded by a stone circle 113 meters in diameter.

Further evidence of this well measured homage to the high hills is found elsewhere at Avebury in the form of the largest Neolithic mound in Europe, Silbury Hill. This huge man-made hill is located less than 1,500 metres from the Obelisk and, with the use of hilltop geometry, it can be seen that there was a very good reason for locating the high point of Silbury Hill at this specific location.