I would like to present my research on Cerrig Duon stone circle, in a similar manner to the treatment given to Maen Llia. So, I would like to keep this thread just for the research and I will start another thread for discussion, should anyone have something to say about this. All contributions are welcome, on the discussion thread, not here, please.
Please use the other thread to discuss any aspects of the main Cerrig Duon research
You are free to use it in any way you see fit, including commercially. The only caveat is that any work that uses it or builds upon it carries the same benefits for other researchers/users, and must be attributed back to the original source.
I have a brand new discovery to add here which ties a lot of things together from the Maen Llia thread. The re is a common theme running through these sites that takes some explaining, but this new site confirms many things.
I will start with Cerrig Duon circle.
This is a plan of the circle drawn by W Grimes in the early 1960's.
diagram 1
Grimes plan was quite good, but I needed a more detailed and much larger plan so I made my own. I used the method, favoured by Aubrey Burl, of selecting two points 3 feet apart on a North/South line inside the circle, and then making a separate plan from both. They are then overlaid onto one another and compared. Any glaring errors are obvious and can be corrected easily. Between the two plans something worthwhile should show up.
I have obtained a digital copy of Alexander Thom's plan of this circle and it matches almost perfectly, so I'm happy to declare this an accurate plan, and I base some radical concepts on that accuracy.
It turns out that Cerrig Duon is not a very round circle, in any sense. Alexander Thom classified it as a class 2 egg shape, but it isn't a particularly good fit as the stones wander a bit offline all over the place.
This is the circle with Maen Mawr standing stone and it's two small companions. Just to the North West is a small outlier. This little stone, about 11 inches long by 9 inches high is the key to the circle geometry.
diagram 2
Adjoining the circle from the North East is the Avenue. A higgledy piggledy collection of small stones forming two very rough converging lines which stop some 40 feet or so from the circle. Quite a few of these stones are buried and unfound by me, but most are still evident.
The stones in black are unfound today.
The three elements here, the circle, Avenue and Standing stone/row seem to be disconnected, but they are all part of a curious geometric set up.
diagram 3
Thom's plan differs from mine in a few places only. It's otherwise almost identical. I'm very pleased to say that our respective true North's matched perfectly, so you can't get any better than following A Thom on that.
True North is very important with this circle and it is evident in several places.
The differences are minor really, and mostly reflect the passage of time.
There is one stone missing from Thom's plan, on the West of the circle. It is quite small and overgrown now, so maybe it was then too, although it is shown on Grimes plan? I can't say that it appears to be important at all, but it was put there for some reason so who knows?
The outlier I mentioned previously is misaligned on Thom's plan, as he has it pointing squarely to the circle centre, whereas it points away to the East a few feet. I would say it's referencing the large gap in the South East.
The other curiosity is the fallen stone in the North East. It would appear that someone has picked it up and completely flipped it over to its present position. It is easily the biggest stone in the circle so it must have taken some doing. Oddly, in Grimes plan, the stone is shown in its present position, but overlaying the self same position on Thom's plan. I can only assume that the original impression had survived enough to leave an imprint worth noting at the time. There is just a non descript puddle there now. I would say that the stone was almost certainly standing when originally sited.
There is a stone missing from the circle now that was on Thom's plan, quite close to the centre. Grimes hasn't noted it, but I can remember it from 20 years ago. There is just a hole where it once was?
As can be seen, things change over time, so with that in mind any sort of a plan that is proposed has, by definition, to be imposed on the remains of what might have been. It's very difficult to be definite about these things. I intend to show that there was an overall scheme, and it was done on a grand scale.
This is my plan with Thom's differences in red.
diagram 4
This is Thom's type 2 egg geometry construction, faithfully copied from his original onto my plan. As you can see, its a bit of a hit and miss affair.
diagram 5
Something that Thom seemed to miss, even though his axis passes straight through it, is that the N/W outlier is in just the right place to be a corner of an equilateral triangle that neatly encloses the circle. I'm assuming he missed it because there is no reference to a triangle on his plan.
This is where the circle and Maen Mawr become entangled. When the circle diameter of 60.24 feet
(Thom made it 59.8 feet with 0.3 feet tolerance) is drawn from the outlier it lies tangent to a N/S line that grazes the Western edge of Maen Mawr. If this line is continued Southwards till it cuts the circle it will be found that this point is equi distant from Maen Mawr's S/W corner and the N/W face of the outlier, at 91.26 feet.
This position lies right next to one of the circle stones.
diagram 6
When some sizes are added things begin to get very interesting. I have spent many hours trying to find the best for this geometry, both on site and off. These truly are the sizes that work best.
The Equilateral side length of 104.24 feet is the same as the outer diameter for the Stonehenge lintel ring.
diagram 7
The Avenue, when added, fits as below. There is a position, S/E of the circle, where the centre line of the triangle and the Avenue lines converge. This point is directly due South of the Easternmost stone of the circle, and the centre line of the Avenue passes through the S/E corner of the triangle to get to it.
The converging point here is 45 feet from the circle.
Other slight variations on this may also apply. It is difficult to be precise with the rather haphazard nature of the Avenue stones.
diagram 8
Another alternative
To finish this part of the Tawe valley set up it's pertinent to look at Maen Mawr and its two companions. They seem to have been very deliberately lined up. I'm inclined to believe that Maen Mawr was the last part of this puzzle to be sited, and I can only conclude that it must have been necessary to put a big old block just there for some unknown reason. It looks to me like some kind of a balancing addition, but maybe that's just me?
The view looking South. The gouge down the length of the stone is quite intriguing to me.
plate 1
The view looking East. All the measures are multiples of 3.6 inches?
plate 2
Cerrig Duon, Maen Mawr and the Avenue don't sit in isolation within the valley. There are connections to other monuments, although, because of its siting nestled in the cradle between the hills there isn't a view to any of the surrounding summits from this circle, which sets it apart from all the other Breconshire circles, who have wide open vistas?
This is looking up the Tawe valley to the North. The hill directly in front is Moel Feity and its summit is over the brow and unsighted.
The important view here is to the N/E and Bwlch y Cerrig Duon barrow. It is sited in the join between the two horizons, as shown. This is a feature that recurs all over the National park with these monuments. I'm convinced it isn't a coincidence because it is so common.
Also shown in this image is the N/S line from the Western face of Maen Mawr and the previously shown equilateral triangle centre line.
plate 3
The Equilateral triangle apex stone and it's alignment. It points to the central area of the large gap on the Eastern side of the circle.
plate 4
The same alignment to the N/W. Pointing straight at the Easternmost Compass stone.
plate 5
The Circle/Equilateral centre line. It clearly points to where the two horizons meet. Bearing here is 336.5 degrees.
plate 6
The measurement from the far side of the circle. It is showing about an inch long here because I couldn't take the picture and pull the slack out of the tape at the same time.
As can be seen the alignment goes corner to corner of the stone. This is a feature I have noticed elsewhere, notably Datum 1 stone at Maen Llia.
plate 7
View East to Maen Mawr. Set up over the stone S/E face is my faithful Ushikata transit theodolite.
plate 8
Maen Leuci
Maen Leuci standing stone, from the West.
This is the view NE towards Maen Leuci. The theodolite is set up on the crest of a ridge and is inline and intervisible between Maen Mawr and Maen Leuci. This is the closest there is to a sightline between the two stones apart from a spot 1/2 mile South West of Maen Mawr.
The distance between the stones, according to Google Earth, varies between 3015 and 3025 feet. This is a close match for the distance from the circle to Waun Leuci cairns to the East.
This distance pops up in other places too!
plate 9
The same position looking South West, towards Maen Mawr.
plate 10
The view from the Equilateral triangle corner marker N/E towards the ridge along the alignment.
plate 11
This is the marker flag I erected to check the alignment. It's surprising how well it showed up, and worth all the effort of siting it because it proved crucial to finding an otherwise unknown aspect of the circle geometry.
plate 12
Curiously, the bearing of 20.5 degrees is a match for the alignment between the Henge and Maen Llia.
diagram 9
The stone centre line as determined by me, from the North, looking South West, along the alignment towards Maen Mawr ( unsighted over brow )
plate 13
The same centre line, looking North East.
plate 14
The alignment between Maen Leuci and Maen Mawr, when continued on through the circle goes through a circle stone, and on to the South West corner of the surrounding equilateral triangle, as here.
diagram 10
This alignment cuts the circle such that the chord produced is half the length of a side of the surrounding equilateral, meaning it is the whole side of a contained equilateral.
When the other two sides are added it is found that the alignment along the centre line of the surrounding equilateral corner stone point directly down one of them, as shown here.
diagram 11
The inner triangle is skewed from the outer triangle at an angle of about 44.5 degrees.
diagram 12
This is the view to Bwlch y Cerrig Duon barrow through Maen Mawr and, thanks to Harry Potter, the sorting hat stone.
plate 15
Bwlch y Cerrig Duon barrow
There are several, apparently stand alone monuments, within the upper Tawe valley.
The Cerrig Duon complex, comprising stone circle, standing stone and row, Avenue, and the Compass stones.
Waun Leuci cairns and standing stone
Maen Leuci standing stone
Quincunx
Bwlch y Cerrig Duon barrow
Bwlch y Cerrig Duon cist
There are also several cairns quite close to the Quincunx, as well as on the summits of Bannau sir Gaer to the West.
Bwlch y Cerrig Duon barrow, due to its placement directly in the dip of the pass, has some unique sightlines.
The most obvious are the major summits of Pen y Fan and Corn Ddu to the East, and Twr y Fan Foel to the West, the Equinox markers.
The central Beacons, Pen y Fan and Corn Ddu
plate 16
Bannau sir Gaer, and Twr y Fan Foel.
plate 17
These two mountain ranges are the highest in this area ( the Beacons are the highest in Southern Britain) and the barrow is sighted such that, on the day of the Equinox, the sun rises over Pen y Fan and sets over Twr y Fan Foel.
There is about 10 miles or so between the barrow and Pen y Fan, and this means that the sun or moon is a very good fit in the bowl between the two peaks, which is about 1/2 degree wide.
The angle of the slopes to the right of Twr y Fan Foel means that some years the sun will set twice on the Equinox.
The Equinox sunrise over Pen y Fan, and the rise through the bowl on the day before and after the Equinox's, neatly framing them.
diagram 13
Sunset on the Equinox's over Twr Y Fan Foel.
diagram 14
The view down the valley towards Cerrig Duon and Maen Mawr, with Maen Leuci on the left and the Quincunx down in the valley bottom close to the river.
plate 18
Zoomed in image of the circle
plate 19
Reverse view, North to the barrow, showing it's position in the notch between two horizons. A feature repeated elsewhere in the Beacons.
plate 20
The barrow itself is in a sorry state now. In several places it has been quarried, and driven over. The mound that is visible now is probably the Western half only, if that.
It does have its own driveway though, and commanding views all around.
plate 21
Some of the quarrying and traffic erosion
plate 22
There is line of sight with at least 4 stone circles. Cerrig Duon to the South, Ynys Hir 10 miles to the NNE, Banc y Celyn 19 miles to the NE and Pen y Beacon 25 miles to the E.
Rhyd Wen Fach to the West is just unsighted over the brow.
Quincunx is the name given to a 5 stone setting, comprising a central stone with 4 outliers. Normally only found in Exmoor, this is the only one known to exist outside of that.
I found this almost 20 years ago while trying to cross the river Tawe after some heavy rain. This normally entails walking back upriver until a place can be found where the genuine threat to life has faded until its only going to be a thorough drenching. On this day the river really wanted to kill me so I trudged back to the car on an unfamiliar route and stumbled onto the Quincunx.
I didn't know what it was so I reported it to the RCAHMW, who confirmed it was a genuine prehistoric setting and it's links to Exmoor to boot. Quite a find, which generated hardly any interest whatsoever. I still haven't been interviewed about it!
This is a rough plan of it.
diagram 16
The Quincunx, looking East.
plate 23
I must admit that I find it totally baffling. I have noticed some peculiarities it has, which I will share, but what was going on here is completely beyond me.
This is the largest stone of the 5. It's the only pillar type stone, and it has been set up such that it's flattish Western face, shown here, is aligned to true North.
plate 24
The stone from above, showing it's Western face at the bottom here.
plate 25
The Northern alignment, taken against the Western face.
plate 26
The central stone, from above. A big lump, like a prop forward.
plate 27
The Southernmost stone, resembling a footprint. Probably a broken off stump now, but no sign of any other parts.
plate 28
The Westernmost stone. A semi circular leaning slab.
plate 29
The Northermost stone. Another slablike stone, with a double lean. Reminiscent of the "Hafen stone pair" There is evidence of frost damage and the stone is quite fractious.
plate 30
As can be seen, they are 5 very different stones, which I believe is significant, but I have no real idea why?
A particularly puzzling aspect of the set up is the alignment s of the stones high points. While the damage to the stones makes the North/ South alignment a little dubious there is no real doubt about the East/West alignment.
Again, I have no idea what this is about, but it repeats what was found with the Compass stones to the South, and various stones in the Llia valley.
This is the North/South alignment, looking North.
plate 31
North/ South alignment, looking East.
plate 32
The East/West alignment, looking East.
plate 33
East/West alignment, looking South
plate 34
The crossed alignments, looking East.
plate 35
The crossing position on the middle stone high point. Not quite a right angle.
plate 36
There is a little more to come for the Quincunx.
The Quincunx is part of an alignment of monuments that span the valley floor from Maen Leuci in the East to a collection of cairns just over the river to the West.
Here they are on Google Earth
Diagram 17
The view from the cairns looks like this
Plate 37
And a zoomed in view of the Quincunx from the cairns
Plate 38
Tawe valley Triangle N 51.88036 W 3.66971
This is a three stone triangular setting.
This is a new discovery, as of June 2021. Although I was aware of these stones from when I first came across the nearby Quincunx (https://coflein.gov.uk/en/site/404184 ) almost 20 years ago, I never had much confidence in their authenticity until recently.
A few months ago I had an inkling to leave my work with Maen Llia and go back to where I began my interest in megaliths, in the upper Tawe valley, specifically, the Quincunx. A monument I hadn't visited in a long time.
While I was there I had another look at these three stones, and it was only then I realised just what they really were. When I showed the Quincunx to the RCAHMW inspector in 2006 I pointed out these stones, but with very little conviction, and that seemed to be that.
These are the three stones, looking North.
Plate 39
Again, just like the Quincunx, these are three quite different stones.
This is the Corner stone. It has possibly suffered some erosion or damage to its N/W top edge, but otherwise it seems ok. It is a rectangular flattish slab, set upright and pointing approx. 150 degrees S/E.
Plate 40
Note the angle on the left edge here. It matches the corner angle of the triangular set up.
Plate 41
This is the Pillar stone. It's quite a lump, standing just under two feet tall and about the same long, with an arched profile.
Looking S/W.
Plate 42
Looking West
Plate 43
This is the Apex stone. When I saw this again, after my work with Maen Llia, it was obvious what it is, and not difficult to see why it didn't mean anything much all those years ago. Without a good grounding in Garsalian geometry (https://www.megalithic.co.uk/modules.php?op=modload&name=Forum&file=viewtopic&topic=8258&forum=4 ) it was just a curious triangular shape with no real recognised precedent to compare it with.
Now it had a meaning, and finding it as the apex stone of a triangular arrangement, whose proportions match the self same triangle, stunned me a little.
I don't see how this could be more obvious, now.
Plan view
Plate 44
Looking South.
Plate 45
Looking West
Plate 46
I thought this was astonishing enough, an example of stone age geometry that matches other sites and really confirms the findings at Maen Llia regarding the interest in this particular triangle, but that was just the start. What is a very modest monument holds all sorts of implications regarding megalithic science. When I investigated further the links to other sites started to pile up.
This is the view to the East, towards Maen Leuci standing stone ( the small black dot in the sunbeam just above the theodolite ) just under a thousand feet away, taken while undertaking some early measurements of the triangle. Maen Leuci proved to be a good reference point for the theodolite.
The actual triangular alignment passes just to the North of Maen Leuci.
Plate 47
Looking West, towards Bannau Brycheiniog.
Plate 48
Looking N/E, over the Apex stone, Pillar stone, and onto Bwlch y Cerrig Duon barrow.
Plate 49
Lookin South from the Corner stone, over the Apex stone. The theodolite is set up precisely over the N/W corner of the Corner stone here. This is the Datum point of the surveys.
Plate 50
An image taken through the pillars of the theodolite, showing the view South, grazing the Western edge of the Apex stone and on to Maen Mawr standing stone and Cerrig Duon stone circle.
Plate 51
Every leg of the triangle is pointing towards another monument within the valley. Those relationships are telling the same story as found in the Llia valley. The builders here were investigating the same geometry and the self same Garsalian triangle.
The triangle identified within these stones is determined by the N/W corners of each stone. This gives a triangle of 50.34 ft x 54 ft x 19.8 ft, with angles of 21.47 degs, 68.53 degs and 90 degs.
This is the Garsalian triangle, sized to fit the Stonehenge sarsen ring mid diameter and its Bluestone Horseshoe mid diameter, as identified at Rhyd Uchaf cairn in the Llia valley. It's the same triangle!
Here is the plan of the Tawe Valley Triangle.
Diagram 18
This is the geometry of the Apex stone
Diagram 19
This is how that triangle fits Stonehenge
Diagram 20
The Tawe valley Triangle would fit like this, size for size.
Diagram 21
The distances from the Triangle to the other monuments is also linking this to other important sites.
The alignment from the Corner stone past the Apex stone Western edge would graze the Eastern face of Maen Mawr and cut the Cerrig Duon circle diameter at a point right next to a stone on its Southern arc, as here.
Diagram 22
The alignment looks like this
Plate 52
A bit closer its like this
Plate 53
The distance from the Corner stone to the point where it cuts the southern arc of the circle diameter is 3020 feet, according to Google Earth. This is 60 x the 50.34 feet length of the Triangle leg that is part of that alignment.
Diagram 23
The distance from the Apex stone to Bwlch y Cerrig Duon barrow is about the same. The barrow is rather a damaged and an indistinct feature so there is no actual position that can be pointed at, just that the alignment grazes the Western edge, as seen here.
Diagram 24
This is where the Triangle sits within the upper Tawe valley and its monuments.
Diagram 25
Something to note here is that the leg that is aligned with Bwlch y Cerrig Duon barrow is 54 feet long, and 56 of those is 3024 feet, which is also a good fit for the barrow distance.
These are particularly curious measurements. I'm sure a lot of the researchers on the Portal will recognise that 3024 feet is the distance around the base of the Great pyramid of Giza, 4 x 756 feet( per side)
The leg lengths of the Triangle, 50.34 ft + 54 ft = 104.34 ft, the side lengths of the equilateral triangle that surround Cerrig Duon circle.
The distance to Maen Leuci is the same as the centre line distance from the midpoint recumbent at Maen Llia to Rhyd Uchaf cairn.
Another curiosity is that, if the true North bearing from the point where the southern arc of Cerrig Duon is cut by the aforementioned alignment is followed North back to the region of the triangle, a 60/30 triangle constructed using the divergent bearings would work out with sides of 90.36 ft x 52.17 ft x 104.34 ft, which are the principal dimensions of the Cerrig Duon geometric scheme.
Just to the side of the Corner stone is another small set stone. This stones Northern edge is 52.17 feet from the Apex stone N/W corner position.
Plate 54
All in all, quite a lot from 3 rather anonymous stones. The implications here are challenging a lot of my thoughts on the capabilities of the builders and their technology, as well as confirming my belief that most of this was beyond them. It is certainly beyond me because I still have no idea what they were doing?
cerrig
cerrig
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New Message Posted!2021-06-17 20:31
This is a video I made a few years ago to describe the sequence of discoveries in the Llia valley, up till about 2017 or so. There are some significant additions missing from this, but it illustrates the basic layout reasonably well, and I get to show off my advanced ceramics skills.
Another video detailing the "compass and square edge" geometry that defines the monument placings within the complex.
The basic geometry that defines the monument layout in the Llia valley
The first method I came across to construct the Garsalian triangle. There are many others.
Summer Solstice sunset shadow, not quite making it to the stream now.
The Winter solstice sunrise shadow.
Early April "false Equinox" sunset shadow, making it into the stream for the first time of the year.
Sunrise shadow, a couple of days after the "false Equinox". Note how quickly Maen Llia's full shadow is revealed from inside the shadow cast by the hill behind it.
Early April sunrise shadow. A couple of days before the perfect date.
A timelapse of the partial eclipse made on the Spring Equinox 2015.
The midday Equinox shadow, looking South.
Maen Llia in the snow.
[ This message was edited by: cerrig on 2023-04-06 17:41 ]
cerrig
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New Message Posted!2021-05-31 15:07
[ This message was edited by: cerrig on 2023-04-06 17:47 ]
PAB
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New Message Posted!2021-03-19 23:36
"twinkling, along, flagged"
I think you must have used special influence to have your Main Llia designated thus....!
cerrig
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New Message Posted!2021-03-19 17:17
"What 3 words" co-ordinates for Maen Llia are "twinkling, along, flagged"
Seems appropriate to me.
cerrig
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New Message Posted!2021-01-19 02:13
Quick Reminder
[ This message was edited by: Runemage on 2021-01-19 02:16 ]
cerrig
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New Message Posted!2020-06-04 19:33
[ This message was edited by: cerrig on 2023-03-21 11:30 ]
cerrig
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New Message Posted!2020-06-01 15:13
This is why the pivot point geometry is important. The position where the line drawn from the apex to the intersection of the definer with the baseline circle cuts the baseline at a point marked by RyleeJacks rock.
This rock is one of several in the valley that has a distinctive pointed shape, and this one is pointed in the direction of the cairn.
RyleeJacks rock surrounded by my minders, with Maen Llia in the background. This is looking along the triangle baseline.
RyleeJacks rock looking along its point towards Rhyd Uchaf cairn and the triangle apex's. This is NRG doing some photogrammetry.
Another way of finding the pivot point that works with all Isosceles triangles is to combine the baseline corner angle bisector and the triangle leg perpendicular bisector. Where they meet is the pivot point.
[ This message was edited by: cerrig on 2023-04-11 08:10 ]
cerrig
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New Message Posted!2020-06-01 11:55
I will try and make this as painless as I can for all for those who don't like maths but who want to learn a bit more about this. Unfortunately for those same people geometry was still geometry thousands of years ago, so it will all look depressingly similar, but while there is nothing new here in the subject what is new is the way it has been used and marked out on the ground.
Somewhere here there is a point where human endeavour and human investigation meet. A major part of my interest is finding where that point lies. I'm still looking?
The geometry that can be found in the Garsalian triangle is the main reason that I can't link the original triangular patterns to a human invention, but rather it seems to me that it can only be down to human investigation of a detectable phenomena. I don't know what this could consist of or how it was detected, but after years of trying to make the human made theory match I finally had to give up when the depth of the geometry and the way it was marked out on the ground made that theory unsustainable.
Below are examples of the Garsalian triangle geometry. This is a very small sample of it and it mainly shows some basic properties of the geometry and how it works and builds. The fractal nature of it is the killer for a human involvement from 5000 years ago.
This is the first method of drawing the triangle I ever came across. As can be seen here it is intimately tied to the Vesica Piscis.
The sequence of construction follows the letters, from drawing a circle AB on the line, then arcs BC and CB, forming the Vesica. The vertical line is drawn through A and D, and then the most important circle within the whole "style"of Garsalian geometry is drawn. This circle, which I call the "definer", opened up the whole can of worms when I moved it to the corner of the triangle, as shown, and then drew a line from the corner B through the intersection F to cut the centre line at G. This formed the first Garsalian triangle seen for thousands of years, if my research into geometry is correct that is, and the field is so vast that I can't really be certain of that, but I believe it is correct. Time will tell.
This is another way of drawing it. Using the same method but this time instead of transferring the circle a line is drawn through intersection D until it cuts the definer circle at F, the distance BF is the radius for the arc that cuts the centre line at G.
Here's another method. The same line/circle/vesica arcs construction is followed as before, this is the basis for all garsalian geometry. The centre line is extended down to F. Another vesica arc is drawn from F through E to cut the baseline at G. Another arc, centre C radius CG is drawn up to cut the centre line at H.
These are just 3 of the many ways to construct this, and the end result is exactly the same triangle.
Another feature of the geometry is its ability to grow, and the need to include all the construction lines in their entirety. For clarity I have left out the construction lines from the earlier examples overlaid onto the map because it quickly becomes a mess otherwise. When drawing otherwise I try to include all the lines as this reveals a lot that would be hidden otherwise. This is where a computer is invaluable.
Following on from the last method a bit of a tangent can be followed, because this is how I learnt by following the geometry rather than generating it myself.
If a line is drawn from A through B till it cuts the outer arc at C a new triangle exactly the same size as the original is formed, rotated through 111.4707 degrees.
Following this further, a line is drawn from A through intersection E to cut the new triangle baseline at its centre point F, and its baseline circle is drawn in. This circle cuts the original triangle leg at B and G, matching the vesica arc. This relationship shows up time and again.
This geometry is all about intersections and the way a developing construction generates new ones.
It occurred to me after this rotated triangle showed up that there would be a point inside the two triangles that served as a kind of pivot point that the rotated triangle spun around in order to reach its new position. This point lands on the baseline circle at H.
There are other Isosceles triangles with pivot points but this is the only one whose pivot point lands on the baseline like this.
This pivot point also has other properties missing from other triangles. There are several ways its position can be found by using the existing intersections.
In this one a line drawn from G to I passes through it.
When a definer circle is added this gives another method of finding the pivot point. A line drawn from the apex K down to the intersection J also goes through the pivot point, and if this is repeated for the rotated triangle to point L the same thing applies, confirming that the two triangles share the same pivot point.
cerrig
[ This message was edited by: cerrig on 2023-03-21 12:06 ]
cerrig
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New Message Posted!2020-05-14 14:26
Something that keeps on recurring in multiple ways is the concept of the first tier and its associated triangle within a triangle. The Garsalian triangle, above all others, uses this constantly. The first tier is an important element is the recumbent stone row, Rhonwens row.
This is a comparison of the first tier triangle and the triangle it came from.
I'm using my usual convention of ratio's for the lengths, with the baseline being equal to 1. As can be seen here this gives some interesting ratios, and the link between this triangle and the vesica piscis is shown in the sqrt 3( 1.73205 ) showing up all over the place. The first tier baseline length being sqrt 3 - 1 = 0.732.
When the differences in lengths between the full size and first tier triangles are compared the ratio is 1.366 : 1 . ( 1/ 0.732 )
When the areas are compared its 1.866 : 1 .( 0.6356 / 0.3406 )
Curiously, 1.866 / 1.366 = 1.366?
Another important element in Rhonwens row (extended) is the incircle. This defines the position of the "Dor stones" by it's tangent point with the leg of the greater Garsalian. This position is always precisely half the length of the baseline, (or the radius of the basline circle) so using ratios will give you the lengths of all the other components of a Garsalian triangle from knowing this one distance. The main problem is deciding which is the relevant length because of the uncertainties of the end points, a feature of investigating rough stones. I have come to the conclusion that there is a range of possibilities available, and they probably all apply. I tend to link these to known prehistoric features in order to give some context, but this is just a guide.
The image below shows various ways of finding the incentre, namely; bisecting the triangle corner angles, where they meet, on the centreline, is the incentre; drawing an arc with its centre on one of the baseline corners, equal to the baseline circle radius, that cuts the leg and then constructing a right angle from that point to cut the centre line, this is the incentre; In the case of a Garsalian, and only a Garsalian, where the first tier line cuts the centre line is the incentre, and it can also be found, only in the Garsalian, by joining the "pivot point" to one of the baseline corners. There are other ways too.
The incircle tangent point is the baseline centre of a small Garsalian when joined to the incentre, as shown here. When this centreline is extended it cuts the left hand base tier leg precisely half way between the first tier point and the corner. Again, this is only true with the Garsalian.
As can be seen, there is another correlation between lengths and areas with the incircle radius and the the first tier triangle area?
When the smaller Garsalian is replicated, as shown, it is found to be a first tier triangle of its own, defined by the full size Garsalian. Expanding and developing the geometry in this way is the essence of Garsalian geometry, and as can be seen there are numerous other Garsalian triangles evident in this diagram, just by extending existing lines. This process has no end.
The problem with this is deciding which direction to follow. This is just one of an infinite number of options that could have been taken here.
Just to show an example of an expansion, these Garsalians share the same first tier line in a way defined by the previous diagram, and the ratio between them is, again, 1.366 : 1.
[ This message was edited by: cerrig on 2023-04-10 12:46 ]
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New Message Posted!2023-06-15 13:21 
