Instructions for Spirals and Fractals

 
Software: Spirals

When you start the program, you see a window like this:

which is composed by several panels. The first panel (Kind of spiral), allows to choose between several kind of spirals. The most important is the Fermat spiral (in the mathematical model we are following, it simulates the growth of a flower calatide). You can choose the number of spiral coils and the kind of picture (i.e. you can visualize the simple spiral, the flowers distributed on the spiral or the spiral and the flowers on it). For instance here is what you get when you choose a Fermat spiral with approximately a dozen of coils, with the button coils+flowers selected and with the dimension of the flowers of 5 pixels:

The flowers drawn on the spiral of the above picture are such that the angle formed by two of them consecutive and the center of the spiral is 137 degrees and 30' (the golden angle); the panel Angle between two consecutive flowers gives some buttons to modify this angle. It is instructive to see what happens to the flowers when we assign an angle, for instance, of 90 degrees or of 120 degrees or 100 degrees: we get pictures like these:

It is clear that in the first two examples the distribution of the flowers is far from being efficient. It seems that in the third case the flowers are distributed with homogeneity, but if in this last case we increment the number of coils of the spiral (for instance to 50) (and we select the option kind of picture: flower), we get the following distribution of the flowers:

which again shows that the flowers are not distributed in a homogeneous way. It is interesting to experiment several values for the angle in order to find the best value (in order to have a homogeneous distribution of the flowers on the calatide). What happens is that the golden angle indeed seems to be the best possible angle (the mathematical explanation require the knowledge of the continued fractions and a property of the golden ratio, which says that it is the most difficult number to approximate with rationals).
The panel show secondary spirals (active only when in the panel kind of picture is is selected flowers or coils+flowers) allows to visualize in red color a flower every n on the spiral, where the number n is selected in the text area. This panel allows to see the spiral that are usually seen when we observe the disk flowers of a sunflower. For instance, here is what we see in case the number of coils is approximately 250 (and the angle between two consecutive flowers is the golden angle) and n is 34 (a Fibonaci number). The button next visualizes other spirals.

 
Software: Fractals

This software allows to obtain simple fractals. When you start the program, you see a window like this:

In this situation you can draw in the graph paper some segments (using the mouse). Here is an example:

As soon as you draw the first segment, the button erase last segment allows to erase it, (in order to allow corrections). When you have completed the picture, you select the button Step 2. Now, if you click in the graph paper, you see that a small square appears (it can be erased with the button erase last square). These squares represent the points in which the branches of the fractal will start (the number of squares you can draw is limited by 9). Here is an example in which we put three squares on the picture drawn in Step 1:

Now you select Step 3. You will see that in the picture appears other smaller copies of the original sketch. Each of them has a small circle. If you move the mouse on that circle, then you can modify the scale factor and the orientation of each figure. Here is what you get at the beginning of Step 3 and after some modification:

Note that in Step 3 you have a panel which allows to choose between several kind of way in which you can modify the picture: rotations, scaling, etc. By default, the button rotation and scaling is selected. As soon as you have completed Step 3, you have given enough information to the software in order to draw the fractal: select Step 4. At this point you will see the same figure you have constructed in Step 1 but two new panels appear: Recursion level and fractal shift. The second one allows to move the fractal, the first one has several buttons. If you press the botton +1 several times you see the construction of the fractal. Here is an example:

And this is the final result:

You can go back and forth between Step 4 and Step 3, so you can slightly modify the data in Step 3 in order to get a better result in Step 4. You can also skip from Step 4 to Step 2 or Step 1, but in these cases the information given in Step 3 is lost.
The menu bar has two items: Settings and Examples. One of the items of Settings is Open I/O window. If you select this item (in Step 3 or Step 4), you see a new window in which you see:

• the coordinates of each segment drawn in Step 1 (they appear in the text area denoted by Segment coordinates. A sequence of 4 numbers like: 285, 80, 253, 162 represents a segment whose end points coordinates are (285, 80) and (253, 162).
• The coordinates of the branches origin;
• The angles of rotation of each branch,
• The scaling factor of each branch.