SLSL Questions: Spiral

[David Morris]

--- The Great Quail wrote:
> 10. Bolingbroke's shack is in the middle of a labyrinth within a spiral, some
powerful symbolism. Any comments?

---Labyrinths and spirals are somewhat related.  Both proceed indirectly from a
perimeter to a center (assuming movement in that direction - which is the
natural path of the eye).  Both have ancient pagan conotations of a journey
toward a sacred center.  Implicit in the form of the spiral is the concept if
infinity - one never reaches the center, but forever approaches it.

---Spiral Jetty  http://www.diacenter.org/ltproj/spiraljetty/

---http://www.seds.org/messier/m/m051.html
Spiral Galaxy M51 (NGC 5194), type Sc, in Canes Venatici
http://antwrp.gsfc.nasa.gov/apod/spiral_galaxies.html

---A whirlpool is an apt form of spiral for a man who dreams to be moved by the
tides of the sea.

---http://www-gap.dcs.st-and.ac.uk/~history/Curves/Equiangular.html
The equiangular spiral was invented by Descartes in 1638. Torricelli worked on
it independently and found the length of the curve. 

If P is any point on the spiral then the length of the spiral from P to the
origin is finite. In fact, from the point P which is at distance d from the
origin measured along a radius vector, the distance from P to the pole is d sec
b. Jacob Bernoulli in 1692 called it spira mirabilis and it is carved on his
tomb in Basel. 

It occurs naturally in many places like sea-shells where the growth of an
organism is proportional to the size of the organism. In his book Growth and
Form, D'Arcy Thompson devoted a whole chapter to this curve and describes its
occurence in Nature as the result of coiling a cone upon itself, contrasting it
with the Spiral of Archimedes which is formed by coiling a cylinder as a sailor
coils a rope upon the deck. 

You can see this shell of the many-chambered Nautilus or the shell with the
curve super-imposed. 




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