**Comparing patterns in Equiangular
Spirals for**

**#1
The Nautilus shell,
**

**#2 The
Fibonacci Numbers
(the Golden Mean),
**

**&
#3. Equal Tempered Chromatic
Music Scale**

**This work
started with the study of the growth of the Nautilus shell by Don and his
students on the rubbing of the shell below.**

**They took 4 measurements
of the radius vectors to get 4 ratios OA/OB and averaged these growth ratios.
Then they measured angle A in 4 places and averaged these. The results they
obtained were confirmed in the book
"The Language of Mathematics" by
Frank Land.**

**On 9 September 2004, an old friend,
Bud, emailed Don
to ask if there was a relationship between the
Fibonacci numbers
and the Nautilus shell.
Don didn't know this, but
found
the polar equation for the Fibonacci spiral
in the book "Introduction to Geometry"
by H.S.M.
Coxeter. **

**Don also worked with a student
finding geometric means. This led to the geometric sequence for the
Equal Tempered Chromatic Music Scale **
where the constant ratio is 2^(1/12)=~1.059..**
and led Don to the polar equation
for its spiral.**

**
The spirals below were
rendered in
Mathematica with Needs["Graphics`Graphics`"]
and the PolarPlot commands.**

**#1
Nautilus shell**

**Polar
Equation: r =
E ^{x'Cot[79.5
'p}^{/180]};**

**Ratio of
radius vectors every 360'
is ~3.2; **

**Smaller
angle between tangent to curve and radius vector is ~79.5'**

**PolarPlot[E^(x*Cot[79.5*Pi/180]),{x,0,4*Pi}, AspectRatio
-> Automatic, PlotStyle**

**->{Thickness[.011],RGBColor[1,0,1]}];**

** **

**#2
Fibonacci Spiral **

**Polar
Equation: r=((1+Sqrt[5])/2)^(2*n/Pi);
Note: The limit of the sequence of ratios of
successive Fibonacci numbers is The Golden Mean =
(1+Sqrt[5])/2.**

**Ratio of
radius vectors every 360' is ~6.7; **

**Smaller
angle between tangent to curve and radius vector is ~74'**

**PolarPlot[((1+Sqrt[5])/2)^(2*n/Pi),{n,0,4*Pi},
AspectRatio -> Automatic, PlotStyle ->{Thickness[.01],RGBColor[0,0,1]}];**

** **

#3. Equal Tempered Chromatic Music Scale:

Polar Equation: r =261.63*2^(n/12); Note: 261.63 Hertz is the frequency of middle C, and the constant ratio in the sequence is 2^(1/12)=~1.059...

**
Ratio of radius vectors every 360'
is ~1.4; **

**Smaller
angle between tangent to curve and radius vector is ~ 87.3'**

**PolarPlot[261.63*2^(n/12),{n,0,13}, AspectRatio ->
Automatic, PlotStyle->**

**{Thickness[.013],RGBColor[1,0,0]}];**

**Notice, within each spiral, the
growth rate for 360'
is a constant and the angle is constant. Each is called an equiangular spiral or
logarithmic spiral.**

**
Try the measurements**
**on these spirals yourself. Please let Don know
what you find.**

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