## A Program for Finding the Minimal Variation Center of a Two Dimensional Phyllotactic Pattern.

#### Instructions for using the program.

The code uses only low level input/output routines so it can run in several environments. But a unix like operating system with built-in input/output redirects will make it easier to use the program.

Save the source code in a file. A good file name would be "find_center.c". The program needs to be linked with the standard math library during compilation. Under a unix like operating system the command is:

% cc   find_center.c   -lm   -o   find_center

While the program can read and write to a terminal window it is convenient to place the data to be read into a file and use input redirect. For example suppose the data is in a file called "perturbC". The command to run the program with this data would be:

% find_center < perturbC

The program will then proceed to look for the minimal variation center. The input itself is simply a list of x and y coordinates for the data points separated only by white space characters. A sample data file looks like this:

42.23944217     -0.4917291616
-20.87269253     18.59472133
2.253700853     -20.23047981
8.656859039     11.37945103
-10.44990849     -1.977633283
6.291805073     -5.339319904

Each line has the x and y coordinates for a data point. The program reads input until it reaches the end of the file, it receives a cntrl-D character, or it has read in 500 x and y coordinates. Once it is done reading in data it proceeds to compute the minimal variation center.

The ouput is displayed on the terminal window. The program outputs the starting estimate for the minimal varaition center (which is the center of mass of the data set). Once every million iterations it outputs the most updated estimate for the center. When it is finished it displays the final estimate for the center. A typical run would look something like this:

% find_center < perturbC
Starting estimate for center: (4.686534, 0.322502)
Estimate for center: (3.116342, -0.066155) after 1000000 iterations
Estimate for center: (2.174011, -0.328605) after 2000000 iterations
Estimate for center: (1.558815, -0.466284) after 3000000 iterations
Estimate for center: (1.131538, -0.526998) after 4000000 iterations
Estimate for center: (0.822636, -0.547612) after 5000000 iterations
Estimate for center: (0.593575, -0.549673) after 6000000 iterations
Estimate for center: (0.420937, -0.544380) after 7000000 iterations
...................
...................
...................
Estimate for center: (-0.161164, -0.504152) after 39000000 iterations
Estimate for center: (-0.161211, -0.504149) after 40000000 iterations
Estimate for center: (-0.161248, -0.504147) after 41000000 iterations
Estimate for center: (-0.161276, -0.504146) after 42000000 iterations
Estimate for center: (-0.161299, -0.504144) after 43000000 iterations
Estimate for center: (-0.161316, -0.504144) after 44000000 iterations
Estimate for center: (-0.161330, -0.504143) after 45000000 iterations
Final estimate for center: (-0.161339, -0.504142)
%

The variable "epsilon" affects the distance by which the estimate for the center is incremented. The larger epsilon is the further the estimate will be incremented each iteration. A larger value of epsilon can allow the program to find the center more quickly. However an excessively large value for epsilon can cause the algorithm to fall into a specious periodic orbit in which case the program will not exit on its own. The variable "tolerance" is the minimal change in both the x and y coordinates for the estimated position of the center. So long as the change in at least one of the coordinates is above the tolerance level the program will continue to try improve the estimate. The smaller tolerance is the more precise the estimate for the center will be when the program is done. Of course a small value for tolerance will cause the program to run for a longer amount of time. But very precise values can generally be obtained in a matter of minutes.

An explanation of how the algorithm implemented by this program works is in:

Finding the Center of a Phyllotactic Pattern (Scott Hotton, J. Theor. Biology , Vol. 225, Issue 1 (2003) pp. 15-32) (pdf 341 K)

#### Data Files

Files for the data used in the paper are provided below. There are ready for use by the program.

The values of epsilon=0.00001 and tolerance=0.00000000001 in the source code are good for the patterns in section 2. For the patterns in section 5 setting epsilon=0.0001 will safely speed things up and setting tolerance=0.0000000000001 will provide enough precision for the program to obtain the origin to six digits as the minimal variation center of the unperturbed patterns.

## A Program for Finding the Minimal Variation Axis of a Three Dimensional Phyllotactic Pattern.

Very similar to the previous code (see instructions above). The input is now a list of u , v , and w coordinates for the data points separated only by white space characters. The output is the set of four numbers (φ0, ψ0,x0,y0) that specify which line is the minimal variation axis of the the data points. (φ0 and ψ0 are expressed in degrees.)

A sample data set is provided. The data points lie on the surface of a cone whose axis is the w-axis of the coordinate system. The coordintes of the points are given by the formulas

uj = (200-t/4)cos(δ t)
vj = (200-t/4)sin(δ t)
wj = t

where where t takes on integer values 0 to 99 and δ=137.5 degrees. These coordinates are then rotated by the matrices

( 1           0             0   )     ( sin(φ0)     0   cos(φ0) )
( 0     sin(ψ0)   cos(ψ0) )     (     0         1       0       )
( 0   -cos(ψ0)   sin(ψ0) )     ( -cos(φ0)   0   sin(φ0) )

Where φ0=π/2-0.2=78.540844 degrees and ψ0=&pi/2+0.3 = 107.188734 degrees. This makes the axis of the cone point in the direction (cos(φ0), sin(φ0) cos(ψ0), sin(φ0), sin(ψ0)). Next all the points are translated by the vector (-0.9,0.7,0). The reader can verify that the axis of the cone now passes through the point 0.8409623589 ξ0-0.6687355420η0.