Filling a Plane with Leaves

A mathematician designs a plant

Material

A 360 degree protractor
A manila folder
scissors
paper
A copy of the protractor circle marked with degrees.

Assembly

Draw two 10 cm diameter circles on a piece of cardboard such as a manila folder, mark the center of each circle.

Cut out both circles.
Cut an arc of 137.5 degrees out of one circle. Like a pie slice. Be precise.

To Do and Notice

Think about how a mathematician might design a plant for an equatorial country where the sun is directly overhead.

For simplicity assume that the plant produces long narrow leaves as it grows.

Assume the plant sends out leaves from a source that spirals around one central stem, at equal time intervals. This will produce leaves all spaced apart by the same angle.

What is the best angle between leaves to choose to prevent one leaf from shadowing older leaves?

Experiment with different angles and see what patterns you can find.

Notice. If you choose 180 degrees then leaf one and leaf two are on opposite sides of the plant, but leaf three shadows leaf one completely. (180 = 1/2 x 360)

If you choose 90 degrees then leaf 5 shadows leaf 1 completely. (90 = 1/4 x 360)

In fact, if you choose any rational fraction of 360 degrees, a later leaf will always shadow an earlier one.

So you want an irrational fraction of 360 degrees. But which irrational fraction?

One over the square root of 2 (0.707...) ? 1/p, 1/ e ?

It turns out the "most irrational" fraction is the golden ratio f= 0.618... .

The golden ratio fraction divides the 360 degrees of a circle into two arcs one of which is 137.5 degrees .

360 (1- f ) = 137.5... degrees.

A quick method to make leaves

Draw a line from the center of the circle to the rim. This is the youngest leaf.

Label it 1.

Use the arc to mark the older leaf, number 2, at an angle of 137.5 degrees from leaf 1.

Then continue to leaf 3 and so on to leaf 16.

Compare the patterns produced around the class.

This method allows error to build up at each step so that after a few leaves the patterns from different groups start to diverge.

Using Mathematics

Instead of using an segment of a circle to increment each leaf you can use mathematics and create a table of angles each of which is 137.5 degrees greater than the previous angle (see the table below.)

Copy the master circle with angles.

Then draw the leaves at the angles in the table, Notice that each leaf does a good job of avoiding previous leaves.

A Table of angles
each incremented by 137.5+ degrees

Leaf Square root Angle
number

 leaf # sqr root Angle 1 1.0 0 2 1.4 137.5 3 1.7 275 4 2.0 52.5 5 2.2 190 6 2.4 327.5 7 2.6 105 8 2.8 242.5 9 3.0 20 10 3.2 157.5 11 3.3 295 12 3.5 72.5 13 3.6 210 14 3.7 347.5 15 3.9 125 16 4.0 262.5