Microworld: Calculus
in Action: Chapter 3, Section 2: Logarithmic Spirals and Planetary Orbits
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above to visit the Microworld.
Author:James
E. White
Nature
presents us with many excellent examples of mathematical concepts. We should
say that mathematics is the art that strives (in its finest moments) to reflect
upon and illuminate our characteristic and human perception of Nature. The
planets are (essentially) spheres, and they move in curves that have an exceedingly
simple description. Flowers, plants and animals grow according to principles
that can sometimes disclose simple mathematical relationships also. As an
example, consider the contour of Oliver Wendell Holmes' "Chambered Nautilus"
shell. This is an example of a construction of Jacob Bernoulli that he called
the "equiangular spiral."
We
can imagine, following an idea of D'Arcy Wentworth Thompson in his wonderful
book, On Growth and Form, that it acquired its shape in the following
way. As the shell grew from the center, the angle that the growing point made
with the radius to the center, O, is always the same angle. This just means
that the principle of growth is the same, and does not differ from time to
time. The figure is then "self-similar." It has the same shape at
any point. Such figures were sometimes called "gnomons" and are
nowadays called "fractals."
We
will see that we can approximate such growth with similar triangles.
The
picture at the top of the page is at the heart of a method for representing
curves by means of differential approximation. That method is called a "differential
equation" model and it does more than represent the curves. It often
offers explanatory principles that can help us understand how complex shapes
can evolve from simple local laws.
In
fact, the picture above, which is built from a growing sequence of similar
triangles, contains the germ of the Euler sequence that we studied in the
previous part and will, by means of that sequence provide a very simple description
of the shape of a Nautilus shell in terms of exponential growth. In that model
we will also find a standard method for solving differential equations in
general. That method is called, unsurprisingly, Euler's method.
Now
the real reason for studying the equiangular spiral is as heuristic preparation
for Newton's deduction of Kepler's laws from universal gravitation. As we
mentioned, Newton set up and solved the basic differential equation that provides
the key to those laws. But his equation took into account the fact that gravity
acts on velocities, and not directly on positions, as Galileo observed. Momentum,
therefore, plays a key role in his equations, and makes them for that reason
somewhat subtler than the spiral equations. In the picture below, we see that
the red acceleration vector (which is determined by the position of the satellite)
directly modifies the green velocity vector in each instant of time. The satellite
moves along the modified velocity vector. Such a differential equation is
called a "second-order ordinary differential equation." We will
examine that process also in the exploration in order to understand Newton's
deduction when we come to it in Harmony of the Spheres.
The
following is the Table of Contents for this 6 page Microworld
- Exponential Functions and Euler's Number
- Natural Exponentials and Infinite Series
- Natural Logarithms and Area Integrals
- Of Nautilus Shells and Planets
- Symbolic Calculator
Each page
of the Microworld, including the Calculator page has the story for that page
under the 
icon.
Just click on this icon to read the story for the page. The Calculator is
quite versatile, and so we recommend you read through the instructions there
to become familiar with it.
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| - James E. White, Ph.D. , Library Director, | ||
| author of this website, Mathwright 2000, MindScapes, | ||
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