“Wonderful spiral” indeed! This, of course, is a chambered nautilus shell (“Nautilus pompilius”).
The posted image was a happy accident. Of course I photographed the entire shell; I couldn't help but do otherwise. But the spiral’s the thing, and when I accidentally zoomed in, I found this image, which reveals more of the shape than any image of the entire shell. I should have remembered my rocking chair post! “Heirloom”:
http://365project.org/rhoing/365/2012-01-20
Of course, my interest is the mathematics! The spiral that the nautilus creates is a natural example of a “logarithmic spiral” —
http://mathworld.wolfram.com/LogarithmicSpiral.html — or “equiangular spiral” (Descartes, 1638); the equation is given most conveniently in polar coordinates,
r = Exp(t),
where r is distance from the origin,
Exp is the “natural” exponential function (using the “natural” base, e = 2.718···) and
t is the angle to the horizontal-axis.
The geometric shape of the logarithmic spiral is intimately related to the so-called “golden ratio” —
http://mathworld.wolfram.com/GoldenRatio.html — i.e., the irrational (but algebraic) number known as “phi” (phi = 1.618···), as noted by skygirl in her wonderful post of an embedded nautilus fossil —
http://365project.org/skygirl/365/2012-07-25 !
See also—
•
http://mathworld.wolfram.com/GoldenSpiral.html
•
http://mathworld.wolfram.com/GoldenRectangle.html
•
http://mathworld.wolfram.com/GoldenTriangle.html
But not all spirals are “logarithmic”:
•
http://mathworld.wolfram.com/ArchimedeanSpiral.html
“The chambered or emperor nautilus, “Nautilus pompilius” (Linnaeus, 1758), reaches about 20cm in length. The spiral shell of this chambered nautilus is thin and smooth with a brown and white pattern. Up to 30 chambers are created in the shell as the animal increases in size and moves to occupy the outermost chamber. The chambered nautilus maintains buoyancy through a gas passed through a tube in the shell tube called a siphuncle. This also helps the animal stay upright in the water. The chambered nautilus has about 90 small suckerless tentacles found on the body close to where it is attached to the shell. This mollusk has eyes but they do not have a cornea or lens. Nautilus and Allonautilus are the last living genera of externally shelled cephalopods.
“Nautilus pompilius first appeared around 550 million years ago during the early Paleozoic era. According to fossil records, some had shells between 6-9m long when uncoiled.
“Unfortunately the nautilus shell is widely collected by humans and therefore commercial fishing for this animal is too common. Since 1987, export of chambered nautilus shells has been widely banned, but many countries continue to sell them” —
http://marinebio.org/species.asp?id=168
One year ago (“Yellow = ‘Caution’”):
http://365project.org/rhoing/365/2012-01-12
Two years ago (“Lights, camera, no-action!”):
http://365project.org/rhoing/365/2011-01-12
![Sigh. [Travel-day]](https://media.365project.org/1/9428160_bfilmqru48_sq.jpg "Sigh. [Travel-day]")