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/

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/

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/