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The most mysterious shape ot all

 

A spiral primer

 

SPIRAL IS A CURVE ON THE plane traced by a point that winds around a certain fixed  point (the spiral's pole), ap­proaching or receding from it depend­ing on the direction of motion. The word "spiral" means a coil and sounds almost the same in Greek ( ) and in Latin (spira). Spirality can be regarded as a type of symmetry essentially different from the symmetry of a snowflake, atomic nucleus, or chessboard, which could be called spherical symmetry. While spherical symmetry is characterized by the constancy of the radius drawn from the center as the figure is turned to fit onto itself, spiral symmetry al­lows the radius to change. Spirality is thus a more fundamental property of matter, whereas spherical symmetry is a particular, exclusive case of spirality.

Mathematically, spirals are best described by means of polar coordi­nates. Let r be the distance from the pole O to a point M on a spiral, and let   be the angle

 

 

 

 

between OM and the fixed axis OA (the polar axis) (fig. 1). The interesting case of a spiral that reaches its pole only in the limit as  – that is, after an infi­nite number of windings – can be given by the equation , where k > 0, n > 0 are constant, r > 0, . Spirals with equations of this form are called algebraic. This form can be regarded as the first term in the expansion of a more general function  in powers of r; the re­maining terms are small near the spiral's pole, but may play the main role far away from it. Depending on the exponent n, we recognize three types of algebraic spirals.

A spiral is called hyperbolic if n = I. The equation then takes the form  As , the distance from a point on such a spiral to the polar axis stabilizes, since it's equal to .  It follows that a hy­perbolic spiral has an asymptote – the straight line it approaches as . The hyperbolic spiral was de­scribed by the French mathemati­cian Pierre Varignon (1654 –1722).

If n > 1, then   for small  – that is, in this case the asymptote coincides with the polar axis OA. For n = 2, this kind of spi­ral is called the lituus, which means "crook" (in the sense of a shepherd's staff). The term was used by Colin Maclaurin in 1722, but the curve was first described by Rodger Cotes in 1714.

For n < 1, an algebraic spiral has no asymptote; the distance from the spiral to the polar axis increases ap­proximately as  (as ).

Another kind of spiral is the pseudospiral. In equations for pseudospirals  isn't expressible as a power of r. An example of a pseudospiral is the logarithmic spiral defined by the equation . The angle  here varies from  as  to as  (fig. 2). A remarkable property of the logarithmic spiral is that it meets any ray from its pole at the same angle  (try to prove this yourself). That's why it's often called the equiangular spiral. This curve was described in 1638 by the great French philosopher and mathematician Rene Descartes (1596-1650). It has a lot of applications in engineering: rotating knives, milling cutters, and gears are often made in this shape.

If a logarithmic spiral is dilated A times and at the same time is rotated by an angle of -k In A, the spiral fits onto itself (because the equation re­mains valid). This is only one of the "reproductive" properties of this spi­ral. Many others were found by Jacob Bernoulli (1654-1705), who was so deeply impressed by his discovery that he asked that the curve be chis­eled on his tombstone with the Latin inscription Eadem mutate resurgo ("Though changed, I shall arise the same" in E. T. Bell's translation).

 

 

A spiral can have an infinite num­ber of coils not only in the neighbor­hood of its pole, but also in the "neighborhood of infinity" – that is, as a point tracing it recedes from the pole to infinity. A few examples of this are the Archimedean spiral , the Galilean spiral , and the parabolic spiral . It's also possible that a spiral winds infi­nitely many times about a certain curve approaching it in the limit as . For instance, the spiral   coils around the circle .

On a molecular level, we find spi­ral (or helical – see next page) struc­tures in DNA molecules, and on a galactic scale there are giant spiral galaxies. As to our own world, nestled between these two, we come across spiral structures at every turn. An acceleration eddy forms at the end of an oar thrust into the wa­ter, – it gains strength during the stroke and moves behind the stern as the stroke is finished. Spiral ed­dies are intricately shaped. They can be both algebraic (fig. 3) or logarith­mic near the pole. A logarithmic spiral with a small value of k is shown in figure 4: its coils are prac­tically invisible, because the radius i decreases sharply (in mathematical terms, exponentially) with the growth of . The photographs in fig­ures 3 and 4 were taken in a hydro-dynamic tunnel as an

 

 

accelerated stream of water flowed symmetri­cally around a plate, with pigment fed onto the plate's edges.

Spiral whirling occurs not only when a liquid flows around an ob­stacle, but also when it flows out of a slot. In figure 5 you see a photo of ... an atomic explosion? No, it's only the leading edge of a jet issuing upward from a narrow slot. Initially, the liquid above the plane of the slot was colored with ink. The flow started from the steady state, where spiral structure is hardly seen. But it immediately reveals itself as soon as any pigment (for instance, our ink) is fed onto an edge of the slot. (See figure 6, where the slot has only one edge; the other one is replaced by a solid wall that can be consid­ered the slot's symmetry, so it's a portrait of the flow out of a "half-slot.")

Nature, both organic and in­organic, is full of spiral shapes. We find them in the shells of most ordinary snails and in an­cient fossils – the ammonite in figure 7 (on the next page) is about 180 mil­lion years old. Sunflower seeds in their  pod

 

 

 

form two families of op­positely twisted spirals. In botany, the tendency to spirality is called phylotaxis. This phenomenon often manifests itself in helical arrange­ments, too. A helix is a kind of three-dimensional spiral – it's the curve traced by a point rotating about a certain axis and moving along this axis at the same time. More exactly, this is a cylindrical helix (fig. 8). Branches on a stem of­ten grow along such a curve. If the point that traces a helix, in addition to moving around and along the axis, recedes from (or approaches) it, we get a curve on a circular cone – a conical helix (fig. 9). This curve is found in the arrangement of scales on a fir cone.

A helix can be twisted like the letter S (fig. 10) or Z (fig. 11), where the middle elements of the letters are thought of as lying on the visible side of the imaginary cylinder around which they wind. S- and Z-helices are mirror images of each other, as is seen quite clearly in the horns of ante­lopes and other horned creatures (fig. 12). The helical shape is taken by various lianas – flowering or fern plants unable to keep their stem erect on their own, without prop­ping it up against a rock, building, or other plants. Hops, ivy, wild grapes, and blackberries are all lianas. They developed their knack for winding over the course of evolution as a part of their struggle for light. First a sprout, having emerged from the ground, stretches upward, then its tip begins to perform circular move­ments (in what direction? – make your own observations!) to find sup­port. If a support isn't found, the plant leans back on the ground, grows up a little more, and resumes its "roundabout" exploration.

So far we've been considering curves. But there are surfaces wind­ing about a certain, generally curved, axis (the dotted curve in figure 13) such that their transverse sections (with respect to the axis) are spirals. These are called spiral surfaces. Spa­tial spiral structure is found in certain atmospheric phenomena, such as cy­clones and tornadoes. Figure 14 shows a cyclone over the Indian Ocean photographed by the Kosmos-144 satellite. A waterspout over a lake is seen in figure 15. This destruc­tive column sweeps away everything it meets. In 1905 the oceanographer W. Eckmann discovered spatial spi­ral underwater currents character­ized by a balance between Coriolis forces and frictional forces. The Eckmann current is often observed in the notorious Bermuda Triangle. Similar spiral motions in the upper layers of the atmosphere were inves­tigated by the English scientist J. Taylor in 1915. These ocean and atmospheric spirals result from the Earth's rotation about its own axis.

 

 

Spiral forms are encountered so frequently that it's impossible even to name the variety of their mani­festations. A charged particle in a magnetic field (of the Earth, an ac­celerator, or a thermonuclear reac­tor) moves in a helix whose axis co­incides with the direction of the field (the blue curves in figure 16). Spiral waves are generated by some­thing called spin detonation. They are also observed in the fundamen­tal Belousov-Zhabotinsky chemical reaction. A theory of this reaction has yet to be formulated, even though thousands of specialists are working on the problem. Many bi­ologists think that spiral waves are responsible for heart arrhythmia and other biological phenomena.

 

 

 

Perhaps this is where certain fun­damental philosophical theories of spiral development of the spirit and of nature should be mentioned. What, indeed, is the world around us? Is it expanding? Infinite? Eleven-dimensional? Random?

Protein? These questions have not yet been answered by scientists. As for spirality, the examples given here provide weighty proof that our world is spiral. However, we have a long way to go in explaining the mystery of how spirals emerge and persist.

 

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