THREE
We
might well want to call this sacred geometry, on account of its high levels
of symmetry and its inner concordance. Three circles are drawn around an
equilateral triangle. Thereby three Vesica Pisces are made. Because
of this, we recognise that root three is going to be involved in the scale
of things: if the triangle has sides of unit length, then the long axis
of the Vesica will be 3.


The
Sphere Within 2^{nd}August 1998, Wheely
down, Hants.
Can
you show that the little circle is onetwelfth of the area of each of the
others? That circle touches the 'latent' triangle, is held within
it (Hint: if the circle has unit radius, then show that the triangle sides
are each 23. Squaring that, gives
the area of the circles). The Circlemakers have a propensity to give us
these integer solutions.

A
year later, the fine symmetry of 'Meonstoke' appeared, having the same
basic geometry as the 'sphere within.' Do you agree that its inner circle
just touches the central 'latent triangle'? If so, show that the extra
circle is four times larger than the central one. Hint: the height of that
triangle is thrice that of the radius of the circle inside it. The next
design may also help here.

A Web of Triangles
20 June 1999, Pennyquick Hill, Bath.
Show
that the area of these four triangles compares to that of one of the circles,
as 3 to .
The outer corners of these triangles just touch the extra circle we looked
at earlier.


The
Harlequin 1 June 1997 Winterbourne Basset
This
classic design has stimulated three different websites to analyse its
structure! On the right is Martin Keitel's depiction showing the unique
pattern of its floorlay (how the wheat was flattened). Each of its three
circles would fit exactly in the triangle  i.e., their diameter was half
that of the big, surrounding circle: that is the key to solving it. But,
how to find where the centres of these are circles located? That is its
enigma.

Martin
Keitel has given us a solution which is rather brilliant,
http://martinkeitel.com/cropcircles/harlequin/harle.page3.html
and has what might be an advantage of not requiring any work by way of
calculation to appreciate  one can merely contemplate it! We start with
the two little concentric circles at the centre, and these are in a 3:4
ratio. Both of these are able to pack the triangle  four of one
and ten of the other, will exactly fill it! That's the first thing we notice.

Taking
the larger of these two, doubling its size gives us the three circles of
Harlequin, trebling it gives the invisible circle on whose circumference
these stand, and quadrupling it gives the big surrounding circle, around
the triangle. So there is a pleasant 1:2:3:4 ratio here. The
diagram shows the position of one of Harlequin's circles (in purple), as
resting on the unseen circle. The
height of the triangle is equal to the diameter of the invisible circle,
Keitel observed, and four of the smallest circles in line will stretch
across this. If we started off with a 3:4 ratio, then the unseen or 'latent'
circle could be viewed as 12 units. Keitel called his study 'the perfect
circular geometry of the Harlequin Triangle.' By contemplating its deep
harmonies he was led to the idea of perfection, which is of interest.

Bert
Janssen's construction is a bit more like hard work,
www.bertjanssen.nl/content/cropc/winterbourne97.html
in a manner perhaps comparable to the above 'Meonstoke' formation, that
appeared two years later. His triune template is woven around the very
small circle at the centre of the design. It has the advantage of finding
the angle bisectors, that feature prominently in this formation. On this
construction, one has to believe that there would be a great deal of 'latent
geometry' that would just have to fade away into nothing, leaving only
the final design.

The
Borromean rings
Three interlinked circles
are called, 'Borromean rings.' For their rich history, see eg www.liv.ac.uk/~spmr02/rings/
none of which are, one is bound to say, quite as good as the Circlemakers'
design. For example, the Vikings 'Walknot' ('knot of the slain') was a
triangular version of these rings. The
three rings taken together are inseparable, but remove any one ring and
the other two fall apart. Because of this property, they have been used
in many fields as a symbol of strength in unity.
Bert Janssen's triune
'template' is needed for this design, plus three circles whose centres
stand on the small, inner circle  i.e., the compass is placed on that
inner circle to draw them  so that these cut the other circles at each
triangle corner.

Bert's
diagram (kindly provided) highlights the circle centres.This construction
uses the ratios 1:7:12 in circle radii  and that is a lot more complex
than they were using a few years' earlier. To
help find these ratios, a diagram is here given that just fits into the
central triangle (Its unit length = radius of centre circle). If the sides
of the latter are 12 units, then can you show that these extra circles
have radii of 7 units? 
Tangents
Touching (4 June 1998, Cheesefoot Head)
Each line makes a tangent
to all three of the circles, i.e. it just touches each of them. Once the
distance between each of the three centres is fixed, then their radii have
to be 3/4 times this.
This design, the last
to appear in the 1997 season, has a quite blissful feeling to it. In that
year a whole series of fractal masterpieces had been laid down, and the
Circlemakers had every reason to be pleased with themselves. The triangles
are here reverberating to the oddnumber scale factors 1, 3, 5 and 7.


Triune
Mandala
As the millennium turned,
an image of perfect harmony appeared, next to Silbury Hill. It took two
nights to develop. A hexagon rotated a halfturn to give a twelvesided
figure, making a twentyfourfold division of the circle, i.e. 15° intervals.
At the centre was a sixfold or ninefold division of the triangle. To quote
from Freddie Silva's website concerning this mandala: '… its elements effectively
encoding the harmonics of light  3, 6, 9, 12, 24.' I felt it was so mysterious,
that every line of an inner triangle pointed towards a corner of an outer
triangle: every sharp corner (30°) of the outer triangles, was being
pointed to by an edge of an inner triangle. The big triangle is delicately
held, with the circle around it about 10% smaller than that at the perimeter.
This
triangle gives us an opportunity to clarify some elementary matters: one
can see that its height is three times the radius of a circle inscribed
inside it; and that the lines bisecting the angles which meet in the middle
are dividing each other in the ratio of 2:1. for one who has difficulty
grasping that the circle around an equilateral triangle has twice the radius
of that inscribed within it, this diagram could be just what isneeded.
This
pattern expresses the Tetraktys, eternal symbol of the Pythagorean
order.
Whirling
Moons 12 June 1996 Littlebury Green, Essex
These
three 'whirling moons' can be reconstructed using the Tetraktys,
as Gerald Hawkins discerned. The arcs here required are of circles having
a 3:4 area ratio.
Hawkins presented this
insight at the 1998 meeting of the American Astronomical Society at Washington
DC, where he gave this construction. I had previously used a hexagon construction,
where adjacent tips of the hexagons formed the centres of the arcs. Each
lunarcrescent tip gets pointed to by lines, either from the hexagon or
in the tetratkys, twice.
A
circle is drawn around the whirling moons, to touch them. A hexagon around
this will reach the centres of three outer circles. These will be on the
slightly larger circle, that was marked into the crop. The larger arcs
of the crescentmoons just reach this circle. It's a complex, symphonic
design.
If
three circles touch, with a fourth at their centre, and they are all the
same size, then an enveloping hexagon around the centre circle can be drawn
through the three centres. That is the harmonious theorem expressed in
this construction. I wish I'd been taught that at school.

Radiant
Pattern 29 August 2005 Marden Wilts
With
a circle divided into 18, draw arcs within it, setting the compass to four
of its divisions, without removing the pen from the paper, as it were.
That gives one a ninepetalled design. One then gives that a halfturn
rotation, and presto! There
is the 18fold design in all its splendour. One rather lacks a name
for this formation. I was foxed by this design, until Allan Brown explained
it to me.
Here you may (or indeed may not) wish to
meditate upon the significance of these numbers: nine concerns the process
of incarnation, whereby the Sun and Moon meet nine times as the foetus
grows in the womb; whereas 18 concerns the synchrony of the Saros cycle,
whereby the Sun and Moon have their rhythms interlocked to generate the
synchrony of eclipse repetitions. 