Nautilus Pompilieus Linnae, and her geodesically straight lined <'Space time curvilinear'> relatives? Might this be a naturally manifest mathematically and geometrically accompanied - Theory of Everything? Revisited.)
*"The universe is finite, but unbounded". - Einstein
*Could this mean that living, tubular vectored spirals are finite at any given moment in space, but unbounded in the fact of their growing expansion in time?
*Acceleration out of the infinite microcosms toward the infinite macrocosms; squared?
What does the following information evoke from you with regard to the shape and dynamics of the micro and macrocosmic universe?
What other space, time or dynamics do you think may be reflected here?
Please note the click-on URLs at the close of this post.
Best regards
- RP
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Equiangular Spiral, Logarithmic Spiral, Bernoulli Spiral
by
Darren Tully
The College of the Redwoods
Abstract: The equiangular spiral, a mathmatical curve with polar equation r = r*k^theta, was examined from the definition and the polar equation, parametric equations were derived and shown.
Nautilus Shells
History
The equiangular spiral has a lot longer history than the science of mathematics. The spiral has been produced for thousands of years in the shape of the nautulis shell, the arrangement of sunflower seeds in the sunflower, among various other natural phenomena.
In mathematics, Descartes was the first to discover the equiangular spiral formula around the middle of 17th century. The spiral was further studied by Torricelli and Jacques Bernoulli, later in the 17th century. Bernoulli was so interested in the equiangular spiral's self- reproducing properties, that he had had the curve engraved on his tomb with the phrase ''Eadem mutata resurgo'' (Though changed, I rise again the same.)
As noted by D'Arcy Thompson (1961, 179):
- In the growth of a shell we can conceive no simpler law than this, namely that it shall widen and lengthen in the same unvarying proportions: and this simplest of laws is that which Nature tends to follow. The shell, like the creature within it, grows in size but does not change its shape; and the existence of this constant relativity of growth, or constant similarity of form, is of the essence. and may be made the basis of a definition, of the equiangular spiral.
Today, the spiral is still studied, and is still reproduced in nature. Although, now days the spiral is studied with computers. People don’t have to spend much time on the curve to find what they are looking for. This has given actual applications to the spiral, mainly persuit curves.
Computer generated shell
Pursuit curves
Four lizards are on the corners of a square. Each one starts to chase its neighbor to the right. They all start at the same time and pursue at the same speed. The pattern that that is traced out by the lizards’ paths is called an equiangular spiral (shown below).
This can also be done with three animals and the pattern that is traced is also an equiangular spiral. All of the animals end up in the same spot at the same time, exactly the middle of the area they are chasing around. Therefore an equiangular spiral is defined as a spiral that forms a constant angle between a line from the origin to any point on the curve and the tangent line’s angle at that point and it’s tangent is equal to the original angle.
Mathematics
In the figure below (formulated in Geometer's Sketchpad) ray AB, distance of r is the start of making the equiangular spiral. Another ray, AC, is made theta degrees from ray AB. A perpendicular line is made from point B to ray AC. At the intersection of the perpendicular line and ray AC, point C is placed.
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Then from that ray, the process is repeated until the desired spiral is formed (as shown).
Therefore the polar equation derived from geometry is
r = (r initial) * k ^ theta
Where r initial is the initial radius, k is a constant greater than or less than 1, and theta is the angle.
The parametric equation of the spiral is a little more difficult and having to use a different form of the equation, given;
r = e ^ ( theta * cot (alpha))
The parametric equation then becomes;
x = e ^ (t * cot(alpha)) * cos (t)
y = e ^ (t * cot(alpha)) * sin (t)
The cartesian equation then becomes;
x ^ 2 + y ^ 2 = e ^ (theta * cot (alpha))
Interesting Stuff
The equiangular spiral is also tied into the Fibbonacci numbers and can be the geometrical pattern for that sequence. For a more in depth description of the association of the spiral and Fibbonacci numbers try
http://galaxy.cau.edu/tsmith/KW/goldengeom.html, they have a great "easy to understand" web page.
Fibbinocci rectangular spiral
The equiangular spiral (also known as logarithmic spiral, Bernoulli spiral, and logistique) describes a family of spirals. It is defined as a monotonic curve that cuts all radii vectors at a constant angle. The inverse of these spirals is also the same spiral.
The spiral appears unending whether, proceeding outward or inward; therefore there aren’t any real endpoints, and the curve makes an infinite number of coils around it’s pole. To conflict with the above description of an unending curve, the spiral has finite length , of course, there is more definition to this, like the precise meaning of end.
Eq. Spirals – triangular and rectangular
References
CIGS.
http://forum.swarthmore.edu/sketchpad/sketchpad.html. Web page on Geometer’s sketchpad and a mathematic search engine.
Visual directory of Special plane curves. Xah Lee,
http://www.best.com/~xah/SpecialPlan...aneCurves.html. Web page on Special plane curves.
Some Golden Geometry.
http://galaxy.cau.edu/tsmith/KW/goldengeom.html. web page on creating the spiral from a golden rectangle.
Equiangular Spiral.
http://www-groups.dcs.st-and.ac.uk/~...uiangular.html. Web page on the history and visualization of the Spiral.
Equiangular Spiral.
http://online.redwoods.cc.ca.us/inst...beP/Spiral.htm. Html document on the Spiral.
A Book of Curves. Lockwood, E.H. Source: David Arnold