The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
Modeling Spiral Growth in a GSP Environment, Dogan-Dunlap and Jordan

## Construction of GSP Program

1. Open a new sketchpad document.
2. Setting the background grid to a polar grid:
1. From the drop down Graph menu, select Grid Form as Polar Grid (see Figure 1):

3. Creating the base parameters (k and n) that are used in the production of spirals/rays.
1. Creating the seed incrementing parameter k. This parameter takes natural numbers as its values. It determines both the location of each seed (point), and the number of seeds (points) plotted.
1. Create a new parameter by selecting New Parameter from the Graph menu. Name the parameter k. Click OK.
2. The k object will appear in the sketchpad document in the upper left corner.
3. Right-click on the k object and select Properties. The Properties window will appear.
4. In the Properties window, select the Value tab. Set the precision to units and the value to 1. Once configured, the Value tab should appear as in Figure 2.

Figure 2. Seed number parameter configuration.

Figure 3. Seed number parameter configuration.

5. Select the Parameter tab by setting the parameter to Change Discretely, by 1.0 unit, and domain from 1 to 10000 (You can enter any natural number as you wish for the upper bound). Once configured, the Parameter tab should appear as in Figure 3. Click OK. If the Parameter tab is not on discrete setting, GSP may consider all real numbers as the values of k.
2. Creating seed placement parameter n with real values. This parameter determines the number of seeds (points) placed for each rotation (a 360° rotation). For instance, for n = 1/4, GSP plots four points (360/4) in the positions corresponding to the four angles (0, 90, 180, and 270) with 90 degrees between any two consecutive points. The values of n determine whether a number forms spirals or rays!
1. Create a new parameter by selecting New Parameter from the Graph menu. Make sure that no other objects are highlighted on the page. Name the parameter n.Click OK.
2. The n object will appear in the sketchpad document in the upper left corner.
3. Right-click on the object and select Properties. The Properties window will appear.
4. In the Properties window, select the Value tab. Set the precision to hundred thousandths. The highest decimal place our version of GSP provides for decimal approximations: if desired, one can set for fewer decimal places. Once configured, the Value tab should appear as in Figure 4. Click OK.

Figure 4. Seed placement parameter configuration.

4. Generating calculations based upon predefined parameters. Make sure no other objects are highlighted on the page.
1. Creating √k . It is the seed radius from the origin.
1. Create a new calculation by selecting Calculate from the Measure menu.
2. Once the New Calculation window appears, select sqrt from the Functions pull-down menu.
3. Now select the parameter k by clicking on the k object outside the Calculation window.
4. Once configured, the Calculation window should appear as in Figure 5 (note: if the parameter is properly selected, k will be highlighted). Click OK.

2. Creating a parameter called a. This is to determine the angle between any two consecutive points/seeds.
1. Create a new Calculation by selecting Calculate from the Measure menu.
2. Once the New Calculation window appears, select the n object located outside the New Calculation window.
3. Multiply this object by 2 and p. The number p can be selected from the Values pull-down menu.
5. Once configured, the Calculation window should appear as in Figure 6 (notice, if the parameter is properly selected, n will be highlighted). Click OK.

Figure 6. Parameter a creation.

6. Right-click on the calculation created in step (v), located on the GSP page, upper left corner, and select Properties.
7. Select the Label tab and set the label to a.
8. Once configured, the Label tab should appear as in Figure 7. Click OK.

Figure 7. Parameter a.

Figure 8. Seed angle configuration.

5. Seed angle configuration (Angle between a point/seed and x-axis):
1. Create a new Calculation by selecting Calculate from the Measure menu.
2. Once the New Calculation window appears, select the k object on the GSP page.
3. Multiply this object by a parameter by selecting the a object.
4. Once configured, the Calculation window should appear as in Figure 8 (if the parameter is properly selected, the objects k and a will be highlighted). Click OK.
6. Plotting (r, q) coordinates:
1. Select first the k object, and next the newly created equation, k.a = 360 as seen in Figure 8. The order of the selection is important!
2. Select Plot as (r, theta) from the pull-down Graph menu.
3. A point should appear on the graph. This point may coincide with the unit point. To eliminate an accidental activation of the unit point, right click on the point and select hide the unit point option. When you need the unit point to appear on the page, select the show all hidden option from the pull down Display menu.
7. Setting plotted point properties:
1. Deselect all the highlighted objects by clicking anywhere blank on the GSP page then select the newly plotted point. Make sure no other highlight appears except the newly plotted point.
2. Right-click on the point and select Trace Plotted Point from the menu.
3. The point will now leave a trace as it moves around the grid when running the Animation option. These traces will be the "seeds" of the spirals or rays. The Animation option will be introduced in the next section.
8. Setting the Grid zoom:
1. Deselect all objects then select the unit point in the document. The unit point is located at (1, 0) (Make sure it is not the newly plotted point).
2. Drag the unit point toward the center of the grid to zoom out until a view close to the one in Figure 9 appears. This can be adjusted to a desired zoom in order to observe more spirals and seed numbers.

Figure 9. Polar grid set at desired zoom.

9. Saving the GSP document:
1. Select Save As from the pull down File menu.
2. Choose a name for the file. Spirals.gsp will be used as the name of the file for our purposes. You can find the active GSP file at Spirals.gsp (The file requires a GSP copy on your computer).