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SIX
| Pentagram
in Perspective
This formation is woven
of dark and light triangles, that emerge from the three bisecting diagonals.
A unique axis of asymmetry swaps over the dark and light, and is
cleverly divided into four equal parts.The
triangles are all similar, having areas in the geometric series ratio 1
: 3 : 9.
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 Koch
Hexagon Silbury Hill July '97The
Circlemakers developed the 'Star of David' motif in 1997. This design has
an enveloping hexagon around the outside, touching its 18 corners at equal
intervals.
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Koch discovered his
'fractal' process back in 1905, and its first three steps are as shown.
  A
more complicated iteration is of this fractal was published by Gerow and
Holbrook in 1996 - their diagram shows a Koch-star going though three steps
of iteration. A formation appeared in August of 1997, in which the circlemakers
developed the theme by using the same Koch fractal line to generate a pattern
within. |
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The
Koch Snowflake 8 August 1997 Milk hill, Wiltshire
This
always appears to me as one of their supreme masterpieces. It is twelvefold
in its division of the circle and has six axes of symmetry.
 There
are three Koch-fractal bars on the outside and six on the inside. The later
are pointing inwards, and stand on a hexagon which has been rotated at
a 30° angle to the outer structure. Those within are shrunk, and to
help find the scale-factor let's compare the two stars in the diagram.
Can you show that one has twelve times the area of the other?  |
| Dolphins
The
Dolphins (or 'Scimitars') appeared within two concentric circles having
a 3:4 ratio. I found a construction based on dividing the two circles into
6 in the usual way, here shown. I did have to admit, however, that they
came out looking a bit thin by this method.
 
A different version was found by Bert Janssen. This used two concentric triangles one four times larger than the other. Quoting from his instructions:  'Construct
a circle with it's centre in the left corner of the large triangle and
with its perimeter just touching the side of the small triangle. Now do
the reverse! Construct a circle with its centre in the left corner of the
small triangle and with its perimeter just touching the large triangle.
The two circles overlap and form a crescent'.
www.bertjanssen.nl/content/cropcircles/geometry.htmlThis
is neat, and moreover gives us the two circle arcs in the ratio of 2:3
and the Circlemakers do like these integer ratios. There is a problem with
his construction in that it doesn't well connect up to the surrounding
circles.
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 Hemp
Leaf 21 July 2000 Alton Barnes, Wilts  Twelve
circles make up this formation: nine meeting at the centre, and another
three concentric upon it. There are three strong axes of symmetry. The
'hemp leaf' effect comes when two circles of unit radii have one-sixth
overlap of their circumferences; or, a distance of root three between their
centres. This repeating grid, (from Martineau,. Altair Design, 2000) gives
us some insight into the design. |
| Celtic
Knot
Six
arcs here mark out a twelvefold division of the circle - see figure. The
arcs are octagonal, they use a one-eighth division of the circle. The basis
of this design was a twenty-four fold division, as shown in the
second diagram: this includes the centres from which the arcs were drawn.
Using this diagram, draw in the one-eighth arcs. Also, the distance across
the centre, between opposite arcs, should be the same as the one-twelfth
interval. Six and eight are factors of 24. We see how the knot moves round,
clockwise.
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 There
are altogether 13 x 6 = 78 rays which converge towards the 'heart' of the
Cambridge 'angel', so she is in a sense lunar as the number 13 pertains
to the Moon. The centre of these rays is one-third of the diameter from
the top. There seems to be only one straight line going right through this
centre, and that line was horizontal, here shown in blue. I counted 47
rays below this line, from which one may derive a total of 78 as the original
division of the circle. The radius of a circle is further divided into
sixths, to construct her head and shoulders. Let's quote from Charles
Mallett, writing on the Crop circle Connector:
“Gog and Magog have once again been chosen to receive a rather simple,
but nevertheless incredibly stunning crop circle event, this time of such
majestic beauty and harmonious proportion that, for this writer, it's set
far apart from the bulk of the English crop formations in 2001. This exquisite,
overtly feminine work leaves one with a feeling of almost child-like awe.
For me this “angel” is possibly one of the most attractive and inspirational
crop glyphs to have ever graced the magical lands of England.” He
counted 75 'rays' which is correct, insofar as the three at the top are
not visible. He described the 'ray' tracks as being razor-sharp, about
six inches wide, and as having alternate directions of flattening.
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| Hexagon
Lattice 24th July, 2005, Avebury Wiltshire.
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3
if we including their outer layer. The design has six axes of symmetry,
and the three different hexagons which appear have their areas as 4, 12
and 100.
 The
sides of the central ‘Star of David’ reach out to touch corners of the
outer hexagons. This may remind us of the ‘Triune
Mandala’ (June 1st, 2000: West Kennet Longbarrow) where lines used
in the centre design reached out to touch key points on the outer frame.
Thus outer and inner are linked together. A more intricate reconstruction
by Allan Brown is also shown. Note that a single circle touches both outer
and inner designs. A sixfold pattern will normally express harmony, whereas
a twelvefold design is more to do with wholeness and integration.
The Author of this remarkable design must surely also have made the very comparable triangle-hexagon design one month earlier (Windmill Hill, Wiltshire, June 2005), as some kind of practice run. Its hexagon-motifs have a half-turn rotation with respect to each other, making a twelvefold design altogether. That construction had only one axis of symmetry. |
Thanks
to Gerow and Holbrook for use of their Koch-fractal, Lucy Pringle for Hemp
Leaf and Steve Alexander for Celtic Knot.
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