Friday, 24 August 2007

A special pentagonal figure, after David Woods in GENESET - consider this Part One

This is the penta-shape discovered by David Woods in the Rennes Le Chateau area, with a controlling circle of 15 divisions, of 24 degrees each, or 4 clock-face-minutes(cfms). I only have GENESET of his works, and this figure is presented there but with subtle alterations which do not concern us here. This is a symmetrical firgure with all angles equal to 36 degrees.
This little session is vital for the next stage, describing it in the landscape of Lothian and its implications for the development of the geometry in total. It was described first by W.S.Buehler, and I did some follow up work on it with interesting developments. A special point of intersection was found which allowed a tightening up of the original pent system as it was found that it and this interlinked.

The hexagon in the circle.

The penta and the hexagon, with common horizontal chord.

The perfect pentagon is found by extending the short chord, and drawing a horizontal at 42degrees, or 7 minutes on a clock-face. I believe this to be correct but shouldn't get upset if it was slightly off. My geometry ain't fool-proof, and I am the living proof!

The centre of the pentagon is found by drawing in of axes.

An interesting construct on the pentagon gives a golden ratio rhombus, using a half penta side as unit. The golden ratio rhombus is a figure I first heard about from the postings of W.S.Buehler, who I have mentioned previously, and shall be covering common geometric topics.
It has axes of phi(FC&CG), and phi-squared(HC&CI), with the half penta-side as unit, equal to BC&CO.
This post explains the geometry that is based on Bill's centering this figure on Borthwick church. To follow.

Next, I will describe this system as it is in the landscape of Lothian and the Borders. It gets very complex, and this and the original penta system need to be understood seperately to keep track!

strange lights - msytery solved, I guess!!

On the probability that the lights where a 'ghost' lense reflection I went back and took another set of five, from the same positions, with the same camera.

The same lights appeared in some of them to convince me that was the explanation. Lower in the photograph, due to less light more camera shake.(The flash function is 'lost' on my digi-camera). Although I took this around the same time of night, the light has gone from the north at midnight. The nights are fair drawing in, are they no!? Sin be christmas!

Thanks to 'anonymous' for pointing it out!

why Phi-latitude!?

Geoff Simmons asked the question. A fuller answer is given here.

I've been a bit distracted for the past couple of weeks, re-reading Robert Temples' The Sirius Mystery for the first time in a few years, especially the Oracle Centres octaves, with their omphalos stones, and carrier pigeons/swallows. This led me to ploughing my way through Peter Tompkins' The Great Pyramid, and even more so through Livio Stecchini's material at the end on ancient measure. I came up with some interesting stuff, related to the general topic of geodetics and ancient Man's knowledge and abilities. Fascinating but time consuming.
One tantalising snippet is that Thebes/Karnak is at latitude 27.5degrees north. Hadrian's Wall is at 55degrees north, twice the number. The Antonine Wall straddles the 56 degree latitude.

The area between is the main focus of my investigations.

Phi, the Golden Section, is well known to those who dabble in pentagonal geometry. It is inherent in the perfect pentagon, being the ratio of side to chord. It is considered the perfect proportion, in words, the ratio whereby a line is divided such that the short is to the long as the long is to the whole. Numerically, it is 1.618(034). The reciprocal is 0.618(034). The square is 2.618(034).

The square, 2.618(034) times 6/5 is within 99.9985% of pi, 3.1416408, against calculator pi, 3.141592654. Using the short version, 2.618, 6/5ths gives 3.1416, a version of pi used by engineers as it is a neat number, even closer at 99.999766%.

It is astonishingly easy to construct phi, phi-squared, and pi, as a straight line, without having to know the numbers at all.

Draw a two by one rectangle, draw in a diagonal, extend the diagonal by one unit, (the short rectangle side is one unit), by use of compasses, half this length is phi in relation to initial unit. (sq.rt 5 plus one, divide by 2). Add one unit to this constructed phi, gives phi squared. (phi + 1 = phi-squared). Divide the phi-squared length into five parts, extend the line by one of these parts, gives pi.

So, a very practical proportion. Very Pythagorean too, in that only straight edge, compasses, and pencil, (and a sheet of paper, or even sand) are necessary.

So when I went to stay in Selkirk, in 1999, I soon became aware that I was living very near to the phi-latitude. In that, 90/Phi is 55.62306degrees. This latitude is straddled by the River Tweed. It runs east to west, from Ross on the Northumberland coast, just to the south of Lindisfarne, very close by Smailholm, the Eildon, Innerleithen/Walkerburn, Cademuir, Stobo, Broughton, Biggar, Thankerton, Lesmahagow, Ardeer on the west coast of the mainland, Goat Fell on Arran, between Carradale and Grogport on the Kintyre peninsula, and between Port Ellen and Risabus on Islay. I did this in Google Earth.

Now this assumes a perfect spheroid for the northern hemisphere. I have been aware that this latitude would have to be corrected, and Livio Stecchini provides a table of the lengths of all latitudes in his section in Tompkins. Pages 329 and 330.

So today I calculated the total of these 90 degrees, and the phi of that total came out at 55.75799 degrees. Again in Google Earth, I plotted this across the breadth of Scotland. From east to west, The very mouth of the Tweed, actually just on land at Tweedmouth, Fishwick, Sinclair's Hill, Westruther, Corsehope Rings Fort, near Borthwick Hall near Heriot, Northshields Ring Fort, Lamancha and West Linton, Auchengray, East Kilbride, and Fairlie on the coast. Millport, Kilchattan Bay, Claonaig on Kintyre, and Bowmore on Islay.

So in a 10 mile band the two versions of phi are contained. To check Google Earth I measured from the equator to the north pole, with the intention of taking the phi value and comparing with what I had just done. A bit disconcerting to find that the distance was only 9900740 metres. Compared to 10,001,987 metres for the Heimert figures given in Stecchini, and other sources where the length is slightly more than the ideal distance of 10,000,000 metres assumed for the French metre. The calculated phi distance is equivalent in Google Earth to 55.7547degrees, agreeing with the figure derived from Stecchini. The discrepancy is some 365.7 metres or 1200 feet.

On a planetary scale I see this as meaningless, and am happy enough to consider the phi-latitude as that lying between 55 and 56 degrees, that is between the two Roman Walls. As stated above, the core of the geometry I have been investigating is included there-in.

And that was the domain name I secured, so used it here as a title on advice from someone who knows more than I do about the internet.

So, that hopefully covers why the name is used!