[RascalPuff]

Dear mckirkpatrick & MJA:


According to several famous sets of correct equations, it has been consummately proven that a bumblebee can't fly ("It's body mass and wing surface and shape are not aerodynamically compatible".) - it's been repeatedly proven with equations. Fortunately, bumblebees don't do math. If they did, they wouldn't fly, and if bumblebees didn't fly, about 1/3 of every flowering vegetation that's green, wouldn't be pollinated and therefore couldn't exist - this would be catastrophic to the ecosystem of course. All of this is to clarify that the truly correct equation of the exemplary bumblebee is the morphology of the bumblebee itself. Math is not reality. It describes and represents and presents reality...


Returning to the = sign exhibited and held up as the illustrated theory of everything is not a return to natural equation or math such as is manifest in the exemplary bumblee (the best is yet to come), it does not equal nature for example, the bumblebee is prima facie nature, incarnate. Inexplicably airborne, ineffably beatific - like the tiny spiral shells that Mr Mkirkpatrick discovered on the beach; then enchanted him to bring home in a tissue, to examine under a microscope...

(The following information was extracted from a public domain exhibit, it is my honor to present it as publicly exhibited gift to both of you, Mr. mckirpatrick, and MJA, and anyone else who may encounter it here or elsewhere...


Mathematics and all of it's numbers, equations and symbols is a man made artifact, once removed from and paled by what it may accurately describe or represent. Allow Truly Yours the privilege of presenting the essence - the mother and father - of mathematics: nature incarnate - We heard you have an affinity for spirals, Mr. mckirkpatrick. Please take due note sir, yet once again - rediscover the universe, as it has occurred beneath your feet on the beach, and in the palm of your hand, and under your microscope at home...

The four, five and six dimensional constitution of the entire enchilada... Behold Nautilus Pompilieus Linnae, and her geodesically straight lined <'Space time curvilinear'> relatives... The illustrated - mathematically and geometrically accompanied - Theory of Everything <in living color>: Revisited.)


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.
Sorry,this page requires a Java-compatible web browser.
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/SpecialPlaneCurves_dir/specialPlaneCurves.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/~history/Curves/Equiangular.html. Web page on the history and visualization of the Spiral.
Equiangular Spiral. http://online.redwoods.cc.ca.us/instruct/darnold/CalcProj/Sp98/GabeP/Spiral.htm. Html document on the Spiral.
A Book of Curves. Lockwood, E.H. Source: David Arnold