A spiral is not just of one kind.




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

spiral__2.png
Archimedean spiral $r=a\theta $
spiral__4.png
Equiangular spiral $r=ae^{b\theta }$




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
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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
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The reason it is call equiangular is because the angle it makes with each radial line is the same. angle $\phi .$ Indeed, we have $b=\cot \phi .$ Note the diagram below
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The famous equiangular spiral was discovered by Rene Descartes, and its properties of self-reproduction by Jacob Bernoulli (1654-1705) 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.
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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. $b=\cot \phi .\ $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.
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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.
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Next we make a plot of the log(radius) data versus the angle in radius. This plot with linear trend line is shown below.
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We have assumed a model of the form $r=ae^{b\theta }$. To determine $a$ and $b$ from this data we can proceed by either using nonlinear regression or linear regression (i.e. linear least squares) to the log-angle data. That is, noting that by taking natural logarithms to $r=ae^{b\theta }$ we obtain
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which is of course a linear equation in $\ln a$ and $b$. Suppose that the data is given in pairs MATH. The normal equations for the solution MATH are then
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Note that we've used the standard symbols MATH for an ordered pair. Don't confuse these with the $x^{\prime }s$ and $y^{\prime }s$ in the chart above. Taking just the bottom thirty columns from the chart above, this system becomes
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The solution is $\ln a=0.073$, which gives MATH and $b=\cot \phi =0.254$,which gives MATH or MATH




Unlike some of the other examples, the analysis of the spiral required more mathematical machinery.




Exercises.

  1. Repeat the example above for the pea-tendril
    MATH

  2. Determine the constant $a$ for the Archimedean spiral.
    MATH
    In this case with the model $r=a\theta $, one can estimate $a$ from the formula MATH Wjhat should you do to improve accuracy.

  3. Here is the Fermat spiral $r^{2}=a^{2}\theta $
    spiral__49.png
    Develop a method to determine what the value of $a$ is.

  4. Perform an investigation on the spirals in the sunflower below.
    MATH
    The information is not as great as desired, but try to find the angle if is an equiangular spiral or the coefficient $a$ if it is an Archimedean spiral.

  5. Perform an investigation on the spirals in the daisy below.
    MATH
    The information is not as great as desired, but try to find the angle if is an equiangular spiral or the coefficient $a$ if it is an Archimedean spiral.

  6. For the seashell below, determine if the angles between the peak have any mathematical relationships, such as equal angles.
    MATH

  7. 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 MATH and MATH. One way to derive this formula and then compute $\dfrac{dy}{dx}.$)
    MATH

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