"The Spiral Dance - Nature at Work and Play"

Overall Winner of The RDS/Technology Ireland - Young Science Writers Competition, April 1994.

A common thread linking cosmology, natural sciences and certain aspects of technology - one which has inspired our ancestors to create designs to ornament and commemorate their most sacred beliefs - obeys the beautiful geometry of the spiral. This shape first aroused my curiosity a few years ago when I participated in a survey of neolithic standing stones near Athgreany, Co. Wicklow (1) on behalf of my grandmother, Helen O'Clery. Her purpose was to establish the 'solar calendar' possibility of this stone circle by photographing shadows cast at dawn and dusk, with reference to pilot stones, on the eight pre-Celtic feast days. During the course of these investigations we visited Newgrange to see the standing stones there, and I became fascinated by the spiral decorations on the kerb stone set at the entrance to the cairn. I was intrigued to read some time later in 'Technology Ireland' (2) that the remarkable triple spiral figure could have been the locus of shadows cast by some of the standing stones at equinoxes and solstices approximately five millennia ago.  
  

 Fig. 1: Whirlpool Galaxy (M.51) as (A) Rosse Sketch (1845),  and (B) recent photograph.

A more recent Irish discovery in the context of the spiral form happened in 1845 when Co. Offaly landowner William Parsons, the third Earl of Rosse, built a huge reflecting telescope known as 'The Leviathan of Parsonstown'. It contained in its 58ft. length a 72 inch speculum metal mirror weighing 4 tons which remained the largest in the world until the opening of the Mount Wilson 100 inch in 1917, (3). Using the Leviathan, Lord Rosse became the first to see that some of the cloudy objects known as nebulae - actually other galaxies - were spirals. His 1845 drawing (Fig 1-A) of the Crab Nebula, also known as the Whirlpool Galaxy (M.51), is a remarkably accurate version of this astronomic object (Fig 1-B); this has since been shown to contain the first supernova remnant, the first x-ray source and the first known site of a pulsar (4). The Hubble classification suggests that 30% of all known galaxies are in spiral form (60% elliptical), and it is thought (3) that only in the spirals is star formation still ongoing.  
 The ubiquitous spiral form is to be found in most aspects of accelerating organic growth, and a seminal study earlier this century by D'Arcy Wentworth Thompson gives a remarkable account (5) of this phenomenon - he quotes from Roman poet Pliny on molluscan shells as 'magna ludentis Naturae varietas', 'the vast variety of nature at play'.  
    Fig. 2: (A) The Spiral of Archimedes (6); (B) The Equiangular (Logarithmic) Spiral.   The study gives a clear and concise mathematical description of the two main types of organic spirals i.e. the equable spiral, or spiral of Archimedes (Fig 2-A) and the equiangular or logarithmic spiral (Fig 2-B). The former may be roughly illustrated by the way a sailor coils a rope upon the deck, each whorl of the same breadth as its neighbour. This curve may be compared to a cylinder coiled up, with its radius vector increasing in arithmetical progression and having the formula R = a q,i.e. a constant times the whole angle through which it has revolved. In contrast, the whorls of the equiangular spiral continually increase in breadth in a steady and unchanging ratio; the length of the radius vector increases in geometrical progression as it sweeps through successive equal angles and the equation is R = aq , or q = k log R. The figure may be considered as a cone coiled upon itself, such as the coiled trunk of an elephant or chameleon's tail. The French philosopher Descartes established many properties of the curve, including the key concept of self-similarity i.e. that sectors cut off by successive radii, at equal vectorial angles, are similar to one another in every respect and that the figure may be conceived as growing continuously without ever changing its shape. A nice instance of the equiangular spiral (5) is the route which certain insects follow towards a candle (Fig 3-A) "owing to the structure of their compound eyes, the insects do not look straight ahead but make for a light which they see abeam, at a certain angle. As they continually adjust their path to this constant angle, a spiral pathway brings them to their destination at last". This influences organic growth in such structures as snail shells (Fig 3-B), the lovely shell of the cephalopod 'Nautilus pompilius' (Fig 3-C) and the swirling spiral of the cochlea in the human inner ear (Fig 3-D) which retain their form in spite of asymmetrical growth i.e. at one end only. Nail and claw, beak and tooth all grow in this way; the graceful curves of foraminiferal shells offer the least resistance to the wave motions that maintain them on the ocean floor, and the florets of sunflowers and tree bark also obey the spiral incremental growth pattern. The deadly 'sticky trap' spiral woven by a spider is a remarkable feat of construction and is described by Nobel Laureate Karl von Frisch (6) in fascinating detail. It is not surprising that the Swiss scientist Jacob Bernoulli called the equiangular spiral the 'spira mirabilis' and asked for it to be engraved on his tombstone.  
    Fig. 3: The Equiangular Spiral in Nature - (A) An insect seeks the light; (B) The edible Snail ('Helix pomatia'); (C) The Cochlea in human inner ear; (A) Nautilus Pompilus.   Technological man has adapted the principles of spirality and vorticity in various ways such as:-  

• Spiral and spiral bevel gears (7) in which the cross-rubbing action of gear teeth for linking non-parallel shafts, such as in automobile oil pumps, distributor drives and rear axle gearing allows smoother and quieter power transmission at high speeds;
• classical Greek designers adapted the logarithmic spiral (8) to Ionic capitals (Fig 4-A), while Celtic illuminations (1) mirrored the Newgrange triple spirals (Fig 4-B);

 Fig.4: The Ornamental Spiral - (A) 6th c. B.C. Ionic, Cyprus; (B) Book of Kells 9th c. A.D. - triples within triples.

• Design of modern clover-leaf junctions (8) can be realised by transition spirals, which may be increased or reduced from a master figure, thus maintaining a constant change of curvature. (Figs 5-A and 5-B) This allows a smooth change of pace for vehicles and a constant and minimum centrifugal effect;

 Fig.5: Clover Leaf curves realised with spiral transitions.

• Wings and fuselages of aircraft utilise spiral curves of smooth acceleration to find the most organic profile; studies of vortices in fluid dynamics which "serve as a paradigm to illustrate how patterns in nature become organised" (9), are of significant value to the meteorologist in understanding the genesis of tornados and Arctic hurricanes, and "may well be the key to understanding turbulence, one of the last frontiers in classical mechanics." An update of Jonathan Swift's verse about 'little fleas' seems appropriate (10):

 "Big whorls have little whorls, Which feed on their velocity, And little whorls have lesser whorls, And so on to viscosity."

Not all technology is gastropod-friendly however (Fig 6) as the Hargreaves Snail can attest !  
    Fig.6: 'Spira mirabilis' meets 'Tracto-spoor vulgaris'.

 

The concept of similitude which emerges as the central theme of D'Arcy Thompson's studies of biological structure and function incorporates not only the equiangular spiral, but also such scaling devices as 'the golden mean' and 'Fibonacci proportionality'. This approach cannot however adequately describe the full range of structural variability apparent in the lung and other organs (11). The Thompson assumption of biological processes as being continuous, homogeneous and regular does not accord with modern observations of most biological and physical systems which are discontinuous and irregular. Between 1950 and 1970 Benoit Mandelbrot (10) evolved a new type of mathematics "capable of describing and analysing the structural irregularity of the natural world, and coined the name fractals for the new geometric forms... Recently, fractals have found their most important use in describing the dynamic shapes associated with chaos theory". Fractals describe the peculiar geometry of irregular surfaces which look the same on all scales of length; the shapes derived include not only complex spirals but also wonderful forms such as snowflakes, seahorses, rabbits, stardust and the Mandelbrot sets known as 'gingerbread men'. The oscillating self-organising reactions in inorganic chemistry known as 'chemical clocks' (Fig 7-A) which are an 'excitable' aspect of dynamic chaos theory yield spiral waves (12) "which bear more than a passing resemblance to those formed in heart attacks, primitive slime moulds (Fig 7-B), waves of star formation in spiral galaxies and hurricanes". There is also an uncanny echo of the Newgrange spiral incisions (Fig 7-C) which had triggered my interest initially.  
  

 Fig.7: (A) Oscillating 'chemical clock' reaction; (B) Slime Mould Aggregation; (C) Incised Spirals at Newgrange.

There is a sad irony in the realisation that the logarithmic spiral, called by Bernoulli 'the curve of life', might also, literally, be in at the death. A recent report suggests that the onset of fibrillation as a prelude to cardiac arrest "is marked by a break in the stable spiral pattern of the heart muscle into a series of excitatory spirals that meander across the heart" (Fig 8). Fractal geometry is providing computer-generated models of these patterns (13) and preventive medicine could benefit; 'the curve of life' lives on !  
 

References
 1) “Athgreany Stone Circle - The Stones Of Time” - Helen  O’Clery, publ. Morrison, 
    N.Y. 1990.  
  
2) “New Data On Newgrange” -  F. Prendergast, ‘Technology Ireland’, March 1991.  
  
3) “Guinness Book Of Astronomy Facts and Feats” - ed. Patrick Moore, 1980.  
  
4) “Illustrated Encyclopedia of Astronomy” - ed. John Man, Carl Sagan, publ.  
    Hamlyn 1989.  
  
5) “On Growth and Form” - D’Arcy Wentworth Thompson, abr. J.T.Bonner, publ. 
    Cambridge Univ. Press 1961.  
  
6) “Animal Architecture” - Karl von Frisch, publ. Hutchinson, London 1975.  
  
7) “Kempe’s Engineering Yearbook” - London 1975.  
  
8) “Form, Function and Design” - Paul Jacques Grillo, Dover Books, USA, 1963.  
  
9) “Vortices and Vorticity in Fluid Dynamics” - Hans J. Lugt, ‘American Scientist’ 
    Vol 73, March/April 1985.  
  
10)  “Does God Play Dice - The New Mathematics of Chaos” - Ian Stewart,  
      Penguin 1989.  
  
11)  “The Arrow of Time” - Peter Coveney and Roger Highfield, Flamingo Books, 
      Harper Collins, London 1990.  
  
12)  “Physiology in Fractal Dimensions” - West/Goldberger, ‘American scientist’  
      Vol 75, July/Aug. 1987.  
  
13)  “Spiral Heartbreak” - Mike May, ‘American Scientist’ Vol 81, May/June 1993.