spirplot.html

Experiment #1

In this experiment, we used an ammonite fossil to plot a logarithmic spiral.

To do this we had to:

1. Trace the spiral form as seen on the fossil with tracing paper;

3. Let your starting point be the origin and draw an x and y-axis;

4. Measure every 30 degrees from the x-axis from 0 to 360;

5. Measure the distance from the origin to the point on the spiral at every 30 degree interval;

6. With these measurements, we are now able to plot the results using Maple.

> restart;with(plots):with(stats):plotsetup(inline):readlib(readdata):readlib(readline):

> readline("a:/spirdata");AM:=readdata(`a:spirdata`,2):

"AM" stands for ammonite the name of the fossilized mollusk shell used to collect the data.

The graph is in polar form because we used the distance from the origin and angles, rather than x and y.

> plot(AM,coords=polar);

Using seq allows you to select values separately.

Avalues=angles->we multiplied our values by Pi/180 to convert from angles to radians.

> Avalues:=evalf([seq(AM[i,2],i=1..27)]*Pi/180);

> Rvalues:=[seq(AM[i,1],i=1..27)];

> Lvalues:=[seq(ln(AM[i,1]),i=1..27)];

The R stands for radius, the A for angle theta and the L for linear.

You can take your separate values and make a list of ordered pairs for plotting.

Rvalues=radius->we took ln of the radii to arrive at a linear equation. This will be clarified further on.

(SP, for spiral.)

> SP:=[seq([Lvalues[n],Avalues[n]],n=1..27)];

This is the linear representation of our spiral.

> plot(SP,style=point,symbol=circle);

Now we curve fit the line.

> with(stats):

Using the fit command, we we're able to find the linear equation that best suited the graph.

> eq_fit:= fit[leastsquare[[x,y],y=b*x+a,{a,b}]]([Avalues,Lvalues]);

Knowing that a shell's spiral is logarithmic, we were able to convert the equation to linear form by doing the following: is the equation of a logarithmic spiral where is the distance from the origin, is an angle in radians, and and are unknown constants. By taking of both sides we can arrive at a linear equation as follows:

We can now compare this equation to a linear equation and replace with , with and with . After solving for a and b in the linear equation, we can solve for a knowing and plug those values into the original spiral equation in polar form.

> A:=-1.20603005;

> b:=0.1348843804;

> a:=solve(ln(a)=A,a);

> r:=x->a*exp(b*x);

We're now ready to plot!

> plot(r(x),x,coords=polar);

Sd represents spiral data. We will now superimpose the spiral that we fitted over our data points.

> Sd:=[seq([Rvalues[n],Avalues[n]],n=1..27)]:

> display(plot(Sd,style=point,symbol=circle,coords=polar),plot(r(x),x=-4..20,color=magenta,coords=polar));

The curve fitting was a success and we now have an equation perfectly describing the ammonite fossil that was used.

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