THE CONTINUING ADVENTURES OF MYRTLE THE TURTLE

Myrtle the Turtle has been travelling in time inventing geometry. Having shown Pythagoras how to draw polygons Myrtle climbed back into the time machine happy to have been able to help. She pressed another switch. "Here goes. I wonder where I'll end up this time."

The door opened. The time machine had landed in an ancient city. The day was hot and crowds were hurrying about their business. "I'd better find out where I am," Myrtle thought. She climbed out of the machine and went up to a stranger. "Excuse me, could you tell me where ..." Myrtle was interrupted by a naked man, running down the street. He ran between Myrtle and the stranger, yelling "Eureka! Eureka!" and disappeared around a corner. Myrtle was shocked, but nobody else took any notice. "Oh, don't mind him," the stranger said, "That's our professor. He's a bit crazy but very clever. His name's Archy Medes, he gets his best ideas when he's in the bath. You often see him, running wet and naked to his laboratory, to try them out before he forgets. We're used to him. You must be a stranger in Syracuse."

"Yes, I'm Myrtle the Turtle. I come from ... quite far away."

"Well I'm Ray Dius, a wheelwright. I make wheels for chariots. Archy has been helping me with a problem. When I've finished a wooden wheel, I must bend a metal tyre around the outside. I never know how long to cut the metal before I bend it. Would you like to meet Archy? His laboratory's round the corner. I'd like to know if he's solved my problem yet."

"Yes, I'd like to meet him. I hope he's got some clothes on by now."

Myrtle and Ray went to the laboratory. Archy was pleased to see them, and he'd put some clothes on. The floor was covered in apple pies, rulers, string and paper.

"Mind the pies," Archy said, "I've been using them to help me solve your problem. I've measured them round the edges, that's called the "circumference", and measured across the middle, that's the "diameter". I've discovered that if you multiply the diameter by a special number I've calculated, you'll get the size of the circumference."

Ray was excited. "So now if I know the diameter of the wheel, I can find the length of the metal tyre by multiplying it by your special number?"

Archy nodded. Myrtle was astounded at meeting someone so clever. "Have you a name for your special number?" Myrtle asked.

Archy looked at the pies on the floor. "Well it worked with the pies, so I think I'll call it pi," he said. He was happy to find someone who was interested. He showed Myrtle how to draw circles and they spent a long time testing his special number. It worked every time. That evening Ray, Archy and Myrtle feasted on Archy's apple pies. Then Myrtle thanked her friends and returned to the time machine.

We saw in Issue 7 how we could build a whole sequence of procedures for drawing regular polygons:

It was suggested that similar procedures could be written for drawing regular polygons with larger numbers of sides. There is certainly a strong pattern emerging from these that should be of help!

We are now possibly on the verge of making a large generalisation - a procedure, with a variable length of side ("edge" might be a preferable term), with which we can construct a regular polygon with any number of edges.

• Creating POLY
When children seem ready to take this step, a fair amount of experimentation and discussion will inevitably be involved.
Two variables will certainly be required:

• the number of edges that the regular polygon will have (NUM)
• the length of each edge, as before (SIDE)

The problem in all this is deciding, in advance, what the turn must be for any given number of edges.
ie.

Previous discussion of the "total turtle trip theorem" will need to be recalled here. If the total turn required is 360 degrees, then we will need to determine a turn which, when repeated NUM times, will give us this total.
So:

(The symbol "/" for division may not be familiar and may demand discussion.)

• Activity
Try out this new procedure POLY with dififerent numbers of edges.
eg. POLY 12 10
    POLY 18 12

You will have to experiment with values of SIDE to make sure that your drawing is of a manageable size.

Try some large values of NUM. What happens to your drawing?

• Drawing circles
We have here the secret of making the Turtle draw a circle, and eventually other curves. Embedded in the last activity is an interesting and powerful idea - that a circle can be regarded as a regular polygon with a "sufficiently large" number of edges. In fact, the value of NUM does not have to be very large for an acceptably smooth circle to be obtained.

Using this knowledge, try the following problems:

• Problem 1
Write a procedure SEMI which will make the Turtle draw a semi-circle.

• Problem 2
Write a procedure QUAD to draw a quadrant (quarter) of a circle.

• Problem 3
Use what you already know to construct a procedure WIGGLE which will produce a wavy line like this:

Can you now make this longer?
Can you make the line less "wavy"?