Spiral

SPIRALS

Thegeometric spiral appears in sea-shells, e.g. the Nautilus here shown, andis called a logarithmic spiral. It never reaches its centre, and its shape always stays the same however much it is enlarged. Itsdistance from the centre increases by a fixed proportion per turn, or foreach quarter-turn. The spiral shown increases by phi, the golden ratio, every quarter-turn: the squares as they revolve shrink by phi every time. The two diagonals drawn in show the radius distance of this spiral: theyare at right-angles to each other, and we can see how each successive radius-distance shrinks by phi as the spiral revolves inwards.

TripleSpiral

A huge triple-spiralappeared in 1996. Each of its three arms expanded by the square of thegolden ratio per 90° turn. Thus,it moved outwards, or shrunk inwards, faster than the above 'golden' spiral- it expanded by one extra power of the golden ratio, compared to the abovefigure. A geometric spiral always holds the same angle to its radius-line (or strictly, a tangent to the spiral at the point where the radius linecrosses, keeps that same angle). I found that this triple-spiral more orless did that, for most of its layout (its angle was 59°), using adiagram of it prepared by Paul Vigay. By contrast, a spiral formation whichappeared three weeks earlier opposite Stonehenge ( www.lucypringle.co.uk/photos/1996/uk1996ay.shtml),while no less magnificent - it became the most visited crop circle ever,with over 15,000 visitors - wasn't a geometrical spiral, i.e. it did notmaintain a fixed angle to its radius-line. It was more biological in itsshape.

Sacred Spiral 17^thJuly 2002 White horse, Pewsey, Wilts

Six years later, the circlemakers came up with a slow, meditative spiral, whichmade three rotations, expanding by phi at each rotation (as near as onecould tell). We saw how the above 'golden spiral' expanded by per quarter-turn, or ⁴percomplete rotation. So, this 'sacred spiral' was expanding by six powersof phi more slowly than the 1996 triple spiral.

GalaxySpiral
The same geometricspiral has here been fitted, both to a galaxy spiral, and to a formation (13 June 1999, Newton St Loe). It roughly fits, except for the outermostcircle. This spiral is doubling insize in just under one rotation.

Lily spiral 21 May2002 West Overton, Wilts

A nest of concentric circles keep halving in size. They go as a half, quarter, eighth and sixteenth of the outer rim, and we can hardly see the last one. They point to an infinity within. Through these there move continuous geometrical spirals, and they cut through each circle per sixty degree turn. For a spiral that halves its radius for each sixty degrees of turn, its equation is
                     r = R/2^/60

where R is the outer rim radius and the turn angle is in degrees. The figure shows the spiral circling inward, on its infinite journey, ever moving towards the centre which it never reaches. Each dot marks sixty degrees of turn, halving its radius. I guess that five of these circles were visible. There are twelve of these spirals and they combine in pairs into six ‘leaves’.

After constructing the logarithmic spirals, the Author would have needed to rub out quite a few sections. You may need several attempts to get this right! Thereby we discern four concentric rings of a lily-pattern, containing an infinity within. Alternate layers rotate in opposite directions, and so the shape has rotational but not reflective symmetry: it lacks any axes of symmetry. As a work of art, one feels that its process of becoming is as interesting as the final product. See also Freddy Silva ’s discussion of this deeply original little masterpiece.

Spiral Mandala           13 Aug 2000 Woodborough Hill, Wiltshire
There is a different kind of spiral, that moves not by any ratio or proportion, but by a fixed amount at each step. This is called an Archimedes spiral. I always thought the Archimedes spiral was boring, until I saw how the Circlemakers used it. Twenty-two equilateral triangles surrounded the open circle. Then another ring of 22, at the same distance, and so on, whereby the triangles gradually grow flatter as they move outwards. The basic structure was 11-fold. Without taking one’s pen off the paper, tracing out these spirals will yield the 11-fold design as shown. One then has to perform a half-turn rotation of this entire design to obtain the full, twenty-two fold mandala.
An Archimedes spiral does reach the centre, unlike the geometrical spiral we looked at earlier: if one here extrapolates the lines back to the centre, a perfect heart-shape is revealed. This formation is woven of twenty-two ‘hearts’. Its construction implies a division from the centre into forty-four equal angles. Some saw the ratio in the way a circle divided into 44 expanded through 14 concentric rings, which reduces to a 22:7 ratio.

Others compared it to the sunflower’s spiral, which is likewise an Archimedes-type spiral, but is not symmetrical in the same way.

The first four rings were contained or implied within the central circle, i.e. the formation began on its fifth ring. There were 22 triangles in this ring, and one can show that each must therefore have had a base angle of 60.3° (1) - i.e., they were as close to equilateral as made no difference. Strictly speaking, as Michael Glickman observed, the sides of each of the triangles were gently curved.
A Star-heptagon fits well into this formation – as a couple of croppie-geometers have observed (2). That will define its inner and outer limits to within an astonishing 0.1% (3).

These concordances built into this formation, may help us to apprehend the notion of perfection. ‘Sacred’ geometry may mean that various symmetries are somehow deeply present within it, and that they affect us in some way. The number seven features in its basic ? ratio (44 divisions and 14 rings), then again in the star-heptagon. See if you can show that the design has equilateral triangles around the centre.

References
1. The base angle for these 22 triangles, standing on the inmost circle of 4 units radius, is given by
                               tan-1 (11/2 ) = 60.3°
2. Allan Brown and John Mitchell, Crooked Soley, A Crop circle Revelation 2005 p.13.
3. The inner and outer radii of the Spiral Mandala are 4 and 18 units, and therefore the equation of star-heptagon fit can here be written as
                      sin (/14) = 2/9,                  to 99.9%

DNA Helix

Sineand cosine waveforms (i.e., out of phase by 90°) went through two cycles.But, the formation did have an impressive 3-D appearance and it did haveten steps laid out, and these inevitably reminded croppies of the human DNA chain, which makes one rotation through ten steps (each step twisted36° from the previous).

Allhuman DNA turns the same way, clockwise like a corkscrew (called, a 'right-handed' turn). Now look at the 'DNA' agriglyph design and which way is it turning? You will probably see it as turning anti-clockwise (or, a 'left-handed'turn). Clearly, this is something to reflect upon.