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.

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.

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.

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