Trisection of an angle, Avebury aug. 11, 1994
Trisection is the devision of a given angle in three equal
parts. This is not possible to construct with a straight edge and compass (most
angles that is, angles that are devidable by both even and uneven numbers are
possible. Such as 12 which is 3x4 and 2x6)
It is one of three unsolvable problems in geometry as it was
practised in ancient times. The other two being Squaring the Circle and
Doubling the volume of a Cube. None of these can be solved because they deal
with irregular numbers. They can only be solved aproximately.
One could ask why such obscure riddles survive all the
cultural changes throughout history. I think the answer to that is that these
problems delt with real situasions that needed a sollution. Not in mundaine,
every day life perhaps, but in archtitecture and construction. A mason had to
use the same methods of construction both on the drawing board and on the
building site. Which was just a bigger board, with ropes and pinns in the
ground instead of straight edge and compass. They often encountered these
problems and must have taken pride in showing how they solved these
aproximately in a way that was only visible to other initiated. This
knowledge was handed over from master to aprentice and ceeping it secret was
paramount from a compettitve point of vieuw. Hence the secrecy and symbolism in
free mason tradition. (Which, i think, is a much more plausible explanation
than all the metafysics and mysticism surrounding it these days.)
Initial design of the
st. Peter by Bramante
Anyway,
Squaring the circle was discovered as an integral part of
crop circle design in the late 90's. A lot is written about this ( by Michael
Glickman and Bert Janssen mostly). if you're new to the subject, you should really
check this out. It is an aspect of crop circles that has been shown countless
times and it has the same function as it once had regarding architecture; To
really apreciate the cleverness and intelligence of the design, you have to
know what you're looking at.
Squaring the circle
with equal circumference and equal area.
Doubling the cube has not been in the search light where it
involves crop circles. The reason to this is that when a cube has a volume of
1, it will have a side length of 1. When the volume is doubled to 2, the side
length will be 1,259921.... which is very close to other propoprtions that are
otherwise often used in crop circles; such as a squared circle with equal area
1:1,253315.., A perfect 5th or1:5/4, a consonont proportion in music, and
1:1,2360696... which is the proportion that determines the inner- and outer
radius of a pentagon.
Pentagonal proportion,
doubling of the cube, 5/4 proportion and squared circle with equal area
Hidden pentagonal proportions too are a regular part of the
crop circle portfolio. It was discovered in the early 90's by John Martineau
and Wolfgang Schindler. It doesn't have the attention it deserves nowadays, but
this too has been concistent throughout the last two and a half decade. They
found out that crop circles where placed so that the tramlines coincided with
pentagrams and pentagons. The reason for this, probably, is that pentagonal
shapes are defined by ɸ, or the Golden mean, which is visable trhoughout
nature. We are hardwired to unconsciously regard these proportions as beautifull.
Crop cricles become more pleasing to look at when they are harmonious with
there surroundings, such as tramlines.
But it makes it harder to discerne the proportion nescasery
for doubling the volume of a cube.
Reconstruction made by
Wolgang Schindler.
Trisection has not been recognised previous. The reason
for this is probably because it is not so exciting compared to a squared
circle. I will not go on in length about this. There are many good websites and
books and videos about this subject. It speaks to our imagination on a totally
different level because squaring the circle involves not only geometry, but
astronomy and metaphysics as well and weaves them together. It gives a sence of
order to an otherwise chaotic and incomprehensible world. The same goes for
pentagonal geometry, Phi proportions and the Fibonacci sequence. We look at
pictures of DNA, spiral galaxy's and the face of Scarlet Johanson and feel a
sence of awe that between all that there is place for uss as well. It has a
spiritual dimension that trisection lacks.
Trisection is boring.
From a cultural historic point of vieuw however it is just
as fascinating. It acknowledges a part of our history. And it takes a lot of
creativity and intellect to take it in to the patterns laid down in a field together
with squared circles and hidden pentagonal proportions.
The presence of trisection in crop circles however does not
give an instruction as how to do it. In most of the cases trisection is shown
by one tramline crossing the perimeter at a trisection point. In some cases it
is part of the design but does not allways give away clues as how it is done. I
want to make that clear, it has no practical use. It just is there as a
recognition of our cultural inheritence. And i believe it has been consistently
so, as i am going to show in the following entry's
In most cased
trisection is shown by one tramline crossing the perimeter at a trisection
point. Liddington castle june
24, 2001
Avebury crop circle photo by Lucy Pringle, taken from
http://www.lucypringle.co.uk/photos/1994/uk1994cg.shtml#pic3
st. Peter design taken from
https://quadralectics.wordpress.com/3-contemplation/3-3-churches-and-tetradic-architecture/3-3-1-the-form-of-the-ground-plan/3-3-1-2-the-cross-shaped-plan/3-3-1-2-1-the-greek-cross-type/
Schindler's reconstruction taken from
http://www.kornkreise.info/schindler/texte/text03.html
Liddington castle crop circle photo by Steve Alexander,
taken from
http://temporarytemples.co.uk/crop-circles/2001-crop-circles
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