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.  What questions can you think of about the spirals? There is always more to learn!