Way back in the early 1980s, the first Apple II (Integer
BASIC) program I ever saw programmed a pencil and graph paper exploration called
*Spirolaterals*. I had a lot of success using Spirolaterals with middle school and secondary students. I used them to
introduce the concept of mathematical *conjecture.*

The simplest spirolateral is defined as follows:

Start at some grid point on the Cartesian plane.

move forward *x* steps, turn
right 90,

move forward *2x* steps, turn
right 90,

move forward *3x* steps, turn
right 90.

move forward *4x* steps, turn
right 90

…

move forward *nx* steps, turn
right.

The coefficient of the last *x *is called the *order.*

This graphic shows the path of a 4x or order 4 spirolateral.

The order can be repeated and the number of repetitions is called the *cycle*. In the following graphic the
order 3 spirolateral has been repeated 4 times, starting with the red, then the
green, the blue, and finishing with the black order 3.

As is evident in the graphic, the *order 3 cycle
4* spirolateral closes. It’s the relationship between order and cycle that
can be explored and, by reviewing order-cycle data, a conjecture can be formed.

The following graphic is a screen shot of the Scratch
program that just computed the path followed by an order 9 cycle 4
spirolateral.

This project can be viewed and downloaded by clicking
on the following link.

http://scratch.mit.edu/projects/11260219/

The second Spirolateral Scratch project mixes left and
right turns. This greatly increases the number of patterns an order 5
spirolateral can compute. For example, there are 32 ways to order right and
left turns for Order 5. In the graphic below, the right-right-left-right-right
pattern for order 5 cycle 4 has been computed.

This project is also a simple exercise for learning
about how to store and retrieve data from a list.

If enough data is collected by experiment, patterns
in behavior do emerge. In other words, there is a relationship between order
and cycle in this project that also leads to a conjecture.

This project, Spirolateral Bot 2, can be viewed and
downloaded by clicking on this link.

http://scratch.mit.edu/projects/11267799/

Patterns really begin to get complicated (and even
more interesting) when other turns, like 60 degree turns, and both left

and
right turns are allowed.

The project Spirolateral Bot 3 implements 60º right
turns but can easily be modified to allow turns of any degree and both left and
right turns. Doing so would make a nice exercise.

In the graphic, an order 5 cycle 3 spirolateral has
been computed.

The project can also be viewed and downloaded by
clicking on the link below.

http://scratch.mit.edu/projects/11265861/

I do have
pdf files for Spirolateral Bots 1 and 2 that go into more detail about
programming spirolaterals in Scratch. To request a free copy, send an email to
grandadscience@gmail.com