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  www.lucypringle.co.uk/photos/1998/uk1998dh.shtml 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 www.lucypringle.co.uk/photos/2000/uk2000cw.shtml 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  www.lucypringle.co.uk/photos/2003/uk2003au.shtml 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 www.lucypringle.co.uk/photos/2003/uk2003bh.shtml '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  1, , 3, 5, 24 So, in wheat and barley fields around Avebury in June-July 2003 an astonishing manifestation of powers of the Golden Ratio took place.