Volume 8. February 2008. Article ID 1681

This article introduces a mathematical model of spiral growth in a Geometer’s Sketchpad (GSP) environment. The model makes a graphical display of the behaviors observed in nature among many plants such as pinecones, cauliflowers, and sunflowers. The GSP program can be used in early college mathematics courses to motivate activities on rational and irrational numbers as well as the decimal approximation of irrational numbers.

- spiral growth
- Geometer's Sketchpad

The basic exposition in this article is accessible with a modern web browser. However, to run the GSP program, you will need The Geometer's Sketchpad software.

We introduce the construction of a Geometer’s Sketchpad (GSP) program that provides the graphical display of spiral growths observed in nature among many plants such as pinecones, cauliflowers, and sunflowers. For more background information, please visit some of the following external sites:

- Fibonacci Numbers and Nature
- Phyllotaxis: Helical arrangement of leaves and staggered dots on shells-two corresponding patterns
- The Golden ratio and Beauty in Nature
- Phyllotaxis form Wolfram MathWorld.

Documents such as the National Council of Teachers of Mathematics’ Technology Principle and the International Society for Technology in Education’s Standards for Students portray a unified vision of technology, pointing out its importance in meeting the needs of students. Furthermore, an effective implementation can help students of varying abilities learn mathematics at a deeper level by helping students externalize both procedures and connections. Instructional technology can be helpful for students’ performance in a society saturated with opportunities for problem solving, communicating, and analyzing [Barron, et al.], [Solomon]. Therefore, "it is in the best interest of both today’s young people and the nation as a whole that all students have an opportunity to master the elements of technology they will need to have a productive future" [Milone & Salpeter, p. 39]. Educators understand that technology, especially in a mathematics classroom, provides students with opportunities to visualize concepts, and allow more time for inquiry by reducing the time spent in complex computational tasks [Office of Technology Assessment]; [Mathematical Sciences Education Board], [Perks, et al.].

The Geometer’s Sketchpad (GSP) along with technologies such as graphing calculators and computer algebra systems are used in mathematics lessons for the purposes of visualization and investigation [Dogan-Dunlap, 2003a], [Dogan-Dunlop 2003b]. (See also Quick Interactive Web Pages with Java Sketchpad). This article introduces [Naylor’s] mathematical model in a Geometer’s Sketchpad (GSP) environment in order to obtain the graphical display of various aspects of rational and irrational numbers. There have been other technologies such as *Mathematica* used to simulate Nalor's model (for example, Spiral Distribution of Seeds on a Flower). The intuitive nature and the widespread use of GSP however may make the model accessible to more educators. We hope that by learning to construct the particular GSP program, mathematics teachers advance their knowledge of the basic GSP functions.

When a plant such as a sunflower grows, it produces seeds at the center of the flower and these push the other seeds outward. Each seed settles into a location that turns out to have a specific angle of rotation relative to the previous seed. It is this rotating seed placement that creates the spiraling patterns in the seedpod. [Naylor, p. 163].

[Naylor] introduces his mathematical model considering each seed as a unit measure. The area of a circular surface is then expressed in terms of the number of seeds (denoted by `k`), and consequently, the radius of the circle is computed as a function of the number of seeds: √`k` / π. That is, the distance from the center of the flower to each seed varies proportional to the square root of the seed number. Considering that each new seed in a flower settles into a location at a specific angle, one can choose an angle value (denoted by `α`) that determines the rotation of each seed from the 0° line (`x`-axis). Here, a angle mimics the angle, and the square root of the seed number mimics the outward notion that flowers use in placing new seeds. Then, the approximate location of a seed can be described by the polar coordinates: `r` = √`k`, `θ` = `k``α`. We should point that for simplicity, the effect of the irrational number is ignored in our calculation of the radius. If one wishes, it can be included back in. This however does not change the graphical behavior but leaves shorter distance between points resulting in a more condensed graph. Remember that since π is an irrational number, one can only use its approximate rational or decimal values.

Before constructing the GSP program, let us look at a few examples to further explain the model. If the `k`th seed is to be placed on a flower, it is plotted at the `k``α` angle at a distance of √`k` from the origin. For instance, if we choose the angle to be `α` = 90°, (or 1/4 of a complete rotation) then the 1st seed will be on the `y`-axis at the distance of √1and the angle of 90° (See Figure 1). The second seed will be at the distance of √2 with the angle 90° from the first seed (located on the `x`-axis). Furthermore, the 3rd seed will be on the `y`-axis at the distance of √3 with the angle of 3 * 90° = 270° counterclockwise from the 0° line. Continuing with this process, one can observe that the 4th seed will be located at the 0° line with the radius of √4. Notice that 90° angle results in exactly four lines forming four rays with wide space between each. If an appropriate angle value is chosen between any two consecutive seeds, then one may get a better distribution of seeds. An appropriate angle may keep the seeds from lining up very early in the process. In our GSP model, we enter rational and decimal numbers as well as the approximate values of irrational numbers to determine the portion of a full rotation as an angle. For instance, for our GSP model to consider 90° as the angle between any two consecutive seeds, we enter either the rational number 1/4 or its decimal equivalent 0.25 to compute the portion of a full rotation, 360°.

You may notice that the choice of the rational or the decimal number determines the type of spirals or rays the seed placement process forms. For instance, if one uses the rational number 1/6 to compute the angle, then the mathematical model determines the position of the first seed at the radius of √1 with 60° angle from the 0° line. Moreover, after the sixth seed is placed on the 0° line, the 7th seed is plotted at the line where the first seed is located. If the angle, on the other hand, is chosen to be 2/7 of a complete rotation, then the 7th seed is rotated by 7 * 2/7 of 360° (720° angle). That is to say, not one but two complete rotations bring the seed back to the 0° line. Then, the formation of a small spiral can be observed among the first seven seeds. After the 7th seed however the process repeats itself forming seven rays. Observe the process for the first 34 seeds in Figure 2. Here, blue dots indicate the first seven seeds forming a spiral. For more seed placement, click on Figure 2, and run the Animation option from the Display menu. Note that the model for 2/7 forms seven rays since gcd(2,7) = 1 (see the discussion of relatively Prime numbers at Wolfram MathWorld). Consider the following relatively prime numbers, 2 and 21. For the rational number 2/21, the process plots 21 points in two full rotation and starts forming 21 rays after the 21st point. The first 21 points form spirals that can be observed easily. For the rational number 2/6, on the other hand, the process forms three rays since gcd(2, 6) = 2. That is, 2/6 is equivalent to the rational number 1/3. The sixth seed still falls on the 0° line but it is not the first to be placed on. The third seed is the first to be placed on the 0o line. Another word, after the third seed, the process starts forming rays but no spirals. In summary, *if the angle is any rational of one full rotation, say a / b, seed b will fall on the 0 ° line...Therefore the pattern will repeat* after the

The rest of the paper offers the construction of the GSP program, the procedure for running the program, and a set of suggested activities with a worksheet. The construction does not assume an advanced GSP knowledge but an ability to recognize the GSP software on one’s computer. It should be noted that the demonstrations included in this paper are created by GSP version 4.01. Now, we are ready to construct the program.

- Construction of the GSP program (those who are familiar with the GSP functions can skip this section).
- Instructions on how to run the GSP program Spirals.gsp.
- Suggested activities.

The paper provided the construction of a GSP program modeling spiral growth occurring in nature, a procedure for running the program, and a set of suggested activities. The activity can be used in early college mathematics classrooms especially in mathematics courses for prospective teachers to motivate activities on rational and irrational numbers, and the rational and decimal approximations of irrational numbers. The teachers of mathematics might either want to create the sketchpad file, Spirals.gsp, and have the class input different values, or allow students to construct the program on their own if time is allowed. The objective is to have students investigate the differences in behavior between the rational and irrational numbers (their approximations) that are used to determine the angle between points, learn about the rational and decimal approximations of irrational numbers, and to provide opportunities for students to use both the rational and decimal representations interchangeably. At the most basic level, students learn that rational numbers eventually form rays, and that the formation of rays and spirals depends on the greatest common divisor (gcd) of the numerator and the denominator of the rational numbers.

In this paper, we mentioned a few mathematics topics that can be covered through the GSP model but one can also use the same file to visit many other topics. Furthermore, the instruction of the GSP program may provide teachers an opportunity to learn more about the basic GSP functions. As a result, having advanced their knowledge of GSP functions, the teachers of mathematics may construct new GSP-based activities that can support and enhance lessons on many other mathematical concepts.

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- Phyllotaxis from Wolfram MathWorld.
- Relatively Prime from Wolfram MathWorld
- Beauty of the Golden Ratio: Golden ration and Beauty in Nature from Oracle ThinkQuest.
- Phyllotaxis from Wikipedia
- Technology Foundation Standards for Students from International Society for Technology in Education
- Fibonacci Numbers and Nature from Fibonacci Numbers and the Golden Section
- Spiral Distribution of Seeds on a Flower.
- Quick Interactive Web Pages with Java Sketchpad from Journal of Online Mathematics and its Applications.
- Phyllotaxis: Helical arrangement of leaves and staggered dots on shells-two corresponding patterns. from Max Planck Institute for Developmental Biology.
- The Technology Principle from National Council of Teachers of Mathematics