A spiral is plane curve that, in general, unwinds around a point while moving ever farther from the point.


While there are many kinds of spirals, two most important are the
Archimedean spiral and the
equiangular spiral.The Archimedean spiral is described
in polar coordinates by
It
was discovered by Archimedes in about 225 BC in a work On
Spirals. It has been used to trisect angles and to square the circle.
As you can see, the radius increases by a constant amount each revolution.
The equiangular spiral given by
The reason it is call equiangular is because the angle it makes with each
radial line is the same. angle
Indeed, we have
Note the diagram
below
The famous equiangular spiral was discovered by Rene Descartes, and its properties of selfreproduction by Jacob Bernoulli (16541705) who requested that the curve be engraved upon his tomb with the phrase "Eadem mutata resurgo" ("I shall arise the same, though changed.")
Spirals of both types and others can be found in nature. The familiar
chambered Nautilus shell is in the form of an equiangular spiral. So are the
spirals of some galaxies, including our own Milky Way galaxy.
Many kinds of spiral are known, the first dating from the days of ancient Greece. The curves are observed in nature, and human beings have used them in machines and in ornament, notably architectural. Other plane spirals are Euler's, or Cornu's, or Clothoid; Cotes', Fermat's, or parabolic; lituus; Poinsot's; reciprocal, or hyperbolic; and sinusoidal.
In this example we will digitize the equiangular spiral found in the
construction of the chambered nautilus shell and attempt to determine the
constant.
As
you can see in the image below we have scaled it to be larger than the
original. This will help with the accuracy of the digitization. Click
here for a larger picture.
In the chart below we have pasted the data into an Excel spreadsheet, split it
into columns and then computed the radii, the natural log of the radii, and
the radial angles in both radians and
degrees.
Next we make a plot of the log(radius) data versus the angle in radius. This
plot with linear trend line is shown below.
We
have assumed a model of the form
.
To determine
and
from this data we can proceed by either using nonlinear regression or linear
regression (i.e. linear least squares) to the logangle data. That is, noting
that by taking natural logarithms to
we
obtain
which
is of course a linear equation in
and
.
Suppose that the data is given in pairs
.
The normal equations for the solution
are then
Note
that we've used the standard symbols
for an ordered pair. Don't confuse these with the
and
in the chart above. Taking just the bottom thirty columns from the chart
above, this system
becomes
The solution is , which gives and ,which gives or
Unlike some of the other examples, the analysis of the spiral required more mathematical machinery.
Exercises.
Repeat the example above for the peatendril
Determine the constant
for the Archimedean
spiral.
In
this case with the model
,
one can estimate
from the formula
Wjhat should you do to improve accuracy.
Here is the Fermat spiral
Develop
a method to determine what the value of
is.
Perform an investigation on the spirals in the sunflower
below.
The
information is not as great as desired, but try to find the angle if is an
equiangular spiral or the coefficient
if it is an Archimedean spiral.
Perform an investigation on the spirals in the daisy
below.
The
information is not as great as desired, but try to find the angle if is an
equiangular spiral or the coefficient
if it is an Archimedean spiral.
For the seashell below, determine if the angles between the peak have any
mathematical relationships, such as equal
angles.
If you are a calculus student you may try to derive the formula for the angle
between the tangent to the spiral at any point and the radial line that passes
through the point. The appropriate picture is shown above. (Hint. Use
and
.
One way to derive this formula and then compute
)