MEASURING THE ANGLES

The students need to be shown how to recognize the generative spiral when it can be seen.  In tracing the generative spiral it is a good idea to proceed upward toward the growing tip.  The generative spiral will proceed either clockwise or counterclockwise upward.  Each element (i.e., branch, twig, or bud) lies approximately 137.5 degrees around the spiral from the preceding element. 

Thus the student needs to be able to recognize the golden angle f = 137.5... degrees.  (It is slightly more than half-way between 90 degrees and 180 degrees.) 

Counterclockwise Angle Diagram

A diagram of this kind is very useful for recognizing the angular positions of the successive bontanical elements. Arm number n + 1 is f = 137.5... degrees counterclockwise from arm number n. Thus arm number 4 makes an angle of 3f with arm number 1.

Clockwise Angle Diagram

Obviously, if the generative spiral is clockwise, we need a clockwise angle diagram, as on the left.

In looking for a generative spiral, it is a great help to look for the following features:

 4th element is approx. 52.5 degrees in advance of the 1st
 6th element is approx. 32.5 degrees behind of the 1st
 9th element is approx. 20 degrees in advance of the 1st
14th element is approx. 12.4 degrees behind of the 1st

Assuming that the generative spiral is counterclockwise, we see that these values agree with the Counterclockwise Angle Diagram.

Problem XXX: Calculate the four numbers in the table to an accuracy of 3 decimal places.
 
 

Artichokes have the advantage that the generative spiral can be exhibited by numbering the petals in the order of growth.  Just write the numbers on the artichoke itself with a ball point pen.

 
 

Here is a bottom view.  The generative spiral is clockwise.  Petals 5 to 11 are just about where they are supposed to be in relation to petal 4. This can be seen by drawing from the clockwise angle diagram (or, at least, radii numbers 4 to 11) on a sheet of cellulose, which can be laid on the photograph.

The deviations from the exact positions predicted by the clockwise angle diagram are mainly due to the fact that the stem is not quite perpendicular to the plane of the photograph.

 

 Here is a head of broccoli.

It is easy to number the branches in order of growth.  Branches 1-5 are visible in the bottom view (left).  The generative spiral is counterclockwise.

On the right is the same broccoli after surgery.  In relation to branch 4, branches 2, 7, 9 ,12 are approximately in the positions predicted by the diagram.

Here is the vine with sharp thorns.  It is a rose vine, I think. The generative spiral is clockwise upward.  Starting from branch 1, at the bottom of the photograph, it is easy to count up to branch 14, which is in the position predicted by the diagram.
 
 

SUMMARY OF STEP 1

Once the students know what to look for, they should embark on some field work; in other words, they should collect some examples of plants with Fibonacci spirals, and examples of plants in which the generative spiral is visible.

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