|
|
|
|
|
|
|
|
|
|
|
PENTAGRAM
 The pentagrams blossomed in 1998 - although one had first appeared in '93 www.lucypringle.co.uk/photos/1993/uk1993de.shtml - and croppies had to start studying golden-ratio geometry. All proportions of the pentagram express this, the 'divine proportion or 'golden ratio,' phi,  . If a pentagon
surrounds a pentagram, as shown,
is in the ratio of the sides of each of them, i.e. by AE/AB, AB/BD and
BD/CD. has the value of 1.618… |
 Double
Pentagram August 8, 1998, Beckhampton Wilts A
half-turn rotation of a pentagram appeared in 1998, together with a scale-factor
enlargement given by half of the golden ratio,
/2.
This had the effect of setting up a surrounding implied or latent pentagon,
just touching the ten corners. That has been a quite distinctive feature
of Circlemaker art. |
 The
enlargement scale- factor is 24%. To ascertain this scale-factor, one has
to draw in a right-angled triangle which forms the one-tenth angle at the
centre of the pentagram, and reaches out to touch two adjacent corners;
then the cosine gives the ratio in size of the two pentagrams, roughly
4:5. It happens that cos  /5 = /2
(NB Equations having both pi and phi are rather special, as they are both
'transcendent' numbers which go on forever and never repeat.) where /5
is one-tenth of a circle or 36°. A
pentagon coincided with the main pentagram, which meant that the angles
at the corners had been trisected, i.e. each was 108°, comprising three
lots of 36° angles.
|
  The
central circle seemed to indicate a latent pentagram, just fitting into
the larger one. This pentagram has to be shrunk by the square of the
golden ratio to just touch the inside of the big one. |
Nest
of Pentagrams July 24 2000, Silbury
Hill
  A
similar scale-factor here applies, where an invisible pentagram is the
framework, and six little ones are reduced as the square of the golden
ratio. From the outlying pentagon, an inner one is reduced by 2,
and finally the inmost pentagon is reduced by 4
as compared to the outer one. NB, there is a humorous touch in this formation,
of an error in flattening too much wheat at one triangle, with a different
kind of error in an adjacent one, as if a triangle had been wrongly dropped.
These triangles are sometimes called 'divine triangles' with 36° at
the apex and side-lengths in the golden ratio. |
|
Windmill
Hill June 2003 6 July
 I
will always remember walking into this lovely formation, with the summer
barley blowing and rustling in the wind: one could see the five crescents
a-swirling at the centre, but not the overall design. That only appeared
from up in the sky! There were five interlocking crescents. Drawing a line
bisecting one of these crescents, one found that it touched the tips of
all the others. That seemed clever. Then, drawing in all five of these
made a big pentagram, the framework for this formation. The inner pentagram
marked the centres of the five circles, shown in the diagram by Allan Brown.
See if you can construct a second pentagram inside the first, to define
the central circle marked by the five small crescents. Can you show that
its magnitude is smaller than around the outlying pentagon, by the fourth
power of the golden ratio? |
 |
 |
 |
 Seven
Pentagrams 6 July 2003 Green Street
Avebury, Wilts
'It was like a kaleidoscope
of shimmering fragments and I realised that the 36° angle, the generator
of the Golden Section proportion, was repeated no less than 45 times,'
to quote Michael Glickmann.This
formation has a pentagram inscribed within a pentagram, which means, as
we have seen, that their proportions are as the square of the Golden Ratio.
Five smaller ones are cleverly tucked away: recalling our initial definition
of the golden ratio, show that these diminish simply by the golden ratio,
compared to the central pentagram. They are therefore smaller than the
big pentagram by
3.
See if you can show that smallest pentagrams have pentagons at their centres
smaller than the outlying one (bounding the pentagram) by 5.Let
us now look at the two outer pentagons. The corners of one are on
the mid-points of the other, so their sizes will be as cosine of
/5
(36°) which is half the golden ratio. So, the very outermost pentagon
is larger than the very smallest (there are five of these), by half of
the sixth power of the golden ratio. Starting from the smallest ones, the
scale of these pentagons was thus increasing as , 3, 5, 24So, in wheat and barley
fields around Avebury in June-July 2003 an astonishing manifestation of
powers of the Golden Ratio took place.
 |
| A Comparison of Pentagram Designs 6 July 2003 Green Street, Avebury |
This formation developed an
expanding series of pentagrams/pentagons, reaching up to the fourth power
of phi.   |
The construction (here in blue), shows how an infinite sequence of pentagrams can be drawn, each one decreasing by a scale-factor of phi. The Circlemakers have shown us the first step in this sequence, so that five pentagrams surround a central one. |
 Theorem:
if two circles are inscribed inside and outside a regular pentagram, as
shown, then their radii are in a phi-squared ratio to each other, i.e one
is 2.618… larger than the other. This theorem should enable you to handle
all these phi-ratios! The pentagram is a figure which always gives phi-ratios.
26
July 2007, Chute Hill, Wiltshire
Extending the sides of these five pentagrams makes another one at the centre, reduced in a phi-ratio. This is a quite magical geometry that has been inscribed here. No-one ever imagined this shape before. In each of the five directions there are three prominent parallel lines, and can you show that the distances between them form the phi-ratio? They are, but showing this isn’t easy! We may also discern large rhombi, diamond shapes, and these subdivide to make golden parallelograms, whose sides are in the golden ratio. |
  |
Thanks
to Mark Fussell for use of Windmill Hill picture, Zef Damen for Triple
pentagram image.
|
|
|
|
|
|
|
|
|
|
|