Tri-Cycloids 24 May 2000 Kassel, Germany  A cycloid is the curve traced out by a point on the edge of a wheel as it rolls along. This diagram shows one being traced out, by a wheel moving inside another one twice its size. This shape is called a 'trefoil' and its equation is r = sin3. One can express this as a fourth-power equation. In both of these formations the radius of curvature is continually changing, ropes wouldn't make them. The outer curve is rather different and, for the two curves to touch, its equation hasto be 4r = 5 + sin{1.5(-90°)}. The two touch when  = 90°. What are its other two values where they touch?

Cushion  www.lucypringle.co.uk/photos/2000/uk2000at.shtml Try the catenary' Guy Atkinson said to me, over a lunch in Guildford. I had just spent two weeks trying to fathom this projection, in vain. No ratio or arithmetic sequence would give it, nor would any projection using a semicircle, an ellipse, parabola or hyperbola. The catenary is the curve made by a chain hanging under its own weight. It has a flatter bottom than other curves: I found that its projection, and no other, gave the large central square: others eg a parabola gave a too-small central square. The catenary is made with 'hyperbolic' functions: cosh x makes the curve and sinh x gives the equally-spaced points upon it (1). This formation was described by cerealogist Michael Glickman as 'the most assured expression of graphic three-dimensionality' which the Circlemakers had yet given us. From its centre there was sequence of seven squares to the edge, then there was a half-width 'edge' around the perimeter, which helped to give its marvellous 3-D 'rollover' effect.

Bi-Heptagon www.lucypringle.co.uk/photos/1998/uk1998di.shtml The pattern within mirrored that without, although it was subtly different. A heptagon and an alternate heptagon are drawn around seven points on a circle, then one is rotated one-fourteenth with respect to the other and the pentagon is then shrunk in the ratio of cos(/7), or about 10%. The pattern within is the outer silhouette of that pattern. It gives us an effect to which the Circlemakers are partial, where an unseen, enveloping heptagon is touched by 14 points at equal intervals.

The sides of that enveloping heptagon are then extended (here shown in blue) and where they intersect a new heptagon s drawn. That gives us some 'marker-lines' that were lines traced out on the ground, a kind of clue to the design, here shown in Bert Janssen's fine diagram http://%20www.bertjanssen.nl/cropcircles.html. Compare this with Martin Kietel's somewhat different construction: www.martinkeitel.com/cropcircles/tawsmead98/taw.page5.htmlThis formation has all the mystic, unfathomability of the number seven and however long one studies it, it remains a mystery.

 

Vertigo Wheel  July 2000 Bishop's Sutton, Hants www.lucypringle.co.uk/photos/2000/uk2000bz.shtml Twelve touching circles stand on a ring. They increase in size gradually over six steps, with the last six times bigger than the first. You or I might make them increase by a fixed proportion (A) - but that isn't how the Circlemakers did it: their method (B) gives a more organic feel to the process, with the ratio between adjacent circles itself increasing at each step. We might call this, accelerated ratio.Thus the first step, from the first to their second circle, showed a mere 9% increase, while their last step amounted to a nearly 70% increase. With their centres all on a circle, this allowed them just to touch (approximately) an unseen, enveloping circle. Then this whole pattern was shrunk and reflected, and neatly fitted inside.

 

Interference Pattern  1 August 2001 Allington, Hants www.lucypringle.co.uk/photos/2000/uk2000dg.shtml#pic2 Seeking for unusual harmonies, the Circlemakers divided each of three sides of a Star-of-David triangle by 14. Using this, they sat up a three-way interference pattern. Mathematically, the aim of this exercise was presumably to ascertain how well the three 8th arcs (4/7th of the whole) passed through the centre, which they do quite well. There is a strong focus upon this centre-point with three mini-triangles around it. Thus, 4/7 = 1/3 to within 1%. Why anyone would want to show this, in an English wheatfield, is another matter (Also, the 4th and 12th arcs just touched the Star-of-David sides). Around the outside were three sets of 'spokes' 5, 5 and 4, adding up to 14 as an indication of this as the number on which the design was based.

 

Ninefold Star  Cherhill, Wilts17 July, 1999  www.lucypringle.co.uk/photos/1999/uk1999cy.shtml People like wearing a pendant of this well-integrated design www.lucypringle.co.uk/merchandise/necklace.shtml - and no wonder, for it brings together 3, 6, 9 and 12-fold symmetries. An equilateral triangle, rotated through two steps about its centre, gives a three-triangle pattern.  This subtly metamorphoses as it were into nine triangles shown. Each have angles of 60° and 40°, i.e. they are not right-angled). But also, the design within subtly echoes the number nine. How does it manage that?  There are six whirling moons at its centre. No-one could account for these, until Bert Janssen's book of 2004. The Circlemakers, it turned out, had used the same Torus design of two years earlier (section 2) - twelve circles standing on a circle half their radius www.lucypringle.co.uk/photos/1997/uk1997bf.shtml#pic2- and merely shaded in every alternate crescent. What one may find unnerving here, is the sheer extent to which all of the construction lines have faded away; they were quite absent from the ground. Only six 'whirling moons' remain, of the earlier pattern. Its central circle is one-third the radius of the outer circumference, so that 9 is the ratio of their areas. Thus inner and outer reflect each other, and six- and nine- fold designs are here integrated.

Crooked Soley
27 August 2002
www.roundhillpress.com/?page=bookintroduction&id=1

Four interlinked Vesica Pisces formations are constructed around a circle as shown, their radii being 2 times that of the centre circle. An outer circle is then put through the intersection points. The radius of this outer circle is a bit less than double that of the inner circle: it’s actually (1+3)/2 = 193% approx.  These arcs are then rotated at five degree intervals, to give 72 arcs going each way, or 18 replications of our original diagram. All arcs outside the two concentric circles then just fade away (I like that bit) and the basic template remains.

As there are 18 circle-centres per quadrant, so likewise are there 18 ‘squares’ per strip, going from inner to outer surface, and there are 72 of these ‘strips’, giving a total number of ‘squares’ thereby generated equal to the fourth power of 6 (try counting them!). This theme was developed remarkably by John Michell and Allan Brown in their book ‘Crooked Soley, a Crop Circle Revelation’. The radii as specified above have to be quite exact, for this large number to have a meaning.  A sixfold design was inscribed onto this template by the Circlemakers. Here is a different one, which may, admittedly, not be quite so sublime. The field allowed no access to the centre to draw this formation. Our acquaintance with this presentation has depended upon just two people: Steve Alexander, who flew over it and photographed it – for no-one visited it, during its very brief existence - and Allan Brown, who discerned and drew its underlying structure.

The ratio of laid to unlaid ‘squares’ in this torus design, was 11 to 7. We can express this as the area-ratio between a circle and square that just fits within it; and likewise, a quarter of the circumference of a circle, and its radius, make that ratio. The same pi-ratio is expressed in the Great Pyramid slope angle, as its height to base ratio, giving its slope angle of 51 51’.

Nest of  Parabolae
24 July 2005  Aldbourne, Wilts
www.lucypringle.co.uk/photos/2005/uk2005ca.shtml

In 2005, a hitherto unheard-of manner of constructing a parabola was demonstrated, in an English cornfield. A whole nest of parabolae were laid down. Remarkably, these parabolae all had the same focus, i.e. they were confocal. Here’s a picture: a parabolic mirror focusses sunlight onto a single spot, and so confocal parabolae would all focus it onto the same spot - they would all have the same focus. That focus is at the very centre of this diagram.     The pattern was generated by nothing more than a set of concentric circles, with equal increment of radius, and parallel straight lines intersecting them. The equations of the parabolae shown are x2=±2ay+a2 where 'a' has values 1, 3, 5 …, if the circles have radii 1,2,3 etc. NB, we can write the equation of a parabola as x2=2ay, and its focus will then have the co-ordinates (0,a/2); then, shifting that focus to the centre, will give the above set of equations.

Sound Wave interference
9.8 2005  Gosport, Hampshire and 10th August 2004 Shalbourne, wilts
www.lucypringle.co.uk/photos/2005/uk2005cj.shtml
www.lucypringle.co.uk/photos/2004/uk2004cp.shtml

Try striking a tuning fork, and then slowly rotate it; you will then hear the sound coming from it grow alternately louder and dimmer. This shows interference taking place, between the two different sources of sound. The crop-circle diagram shows this. If the two sources are separated by one and a half times the wavelength of the sound, then the lines of constructive and destructive interference will be as shown in this 2005 formation: showing where the sound grows louder and softer. The dark lines in the diagram are zones of silence. It is made from two sets of twenty concentric circles. In a pattern which appeared a year earlier, the two  sources were positioned further apart, at two and a half wavelengths. This created five interference lines (i.e., where the sound fades away) instead of three.   Thanks to Allan Brown for use of his diagrams in  ‘Crooked Soley’ ‘nest of parabolae’ and ‘sound wave interference’ ; and to Pete Howard for his comments on the parabolae of 2005.

Note

1. The catenary is given by y = cosh(x/a) or y = a/2(e^x/a + e^-x/a) where 'a' was equal to 0.83. The distance along this curve is given by sinh(x/a).