As is often the case when looking for something, you become distracted by other more exciting ideas. Such was the case recently when retrieving some old tiling patterns.

How could I have overlooked the Penrose tile? A very special tile named after its inventor - Sir Roger Penrose OM, until recently the Rouse Ball Professor of Mathematics at the University of Oxford. He is also a well-known populariser of mathematics.

The Penrose tile is a rhombus whose side length is f (phi - the Golden section) and whose interior angle is 72^° . The tile is cleverly dissected into what John Conway - another well-known populariser of mathematics, calls 'darts' and 'kites'.

These two shapes can be used to tile the plane in very interesting and stimulating ways, as illustrated below:

Can you, by first defining the kite and dart within the context of LOGO, devise elegant programs to replicate the tilings above?

See the website of Pentaplex Ltd. for details of some games and jigsaws based on these tilings. There is an article about these tilings by Bill Richardson in the January 2000 edition of Maths in School, published by the Mathematical Association. Bill has sent us this image showing much more of the second tiling pattern above. You might like to try to re-produce it using LOGO.

FIRST FORWARD

Beginners can start with FIRST FORWARD 1, 2 and 3 here or go back and follow through the series month by month starting from the Introduction to LOGO in July 1999 .

• FIRST FORWARD 1
• FIRST FORWARD 2
• FIRST FORWARD 3

FIRST FORWARD 4

Last month you investigated the instruction:

REPEAT 360 [FD 1 RT 1]

Hopefully you altered it as you saw fit and produced some exciting shapes.

So it is timely to introduce you to the idea of a variable. As the word implies, a variable is something that varies, something that changes.

You may find the following a quicker way of drawing circles and circular patterns. It is called a procedure and has to be written in a particular way.

TO CIRCLE :CH
REPEAT 360 [ FD :CH RT 1]
END

Notice how the procedure is set out:

N.B. the variable, :CH could have been any letter or letters that you wish!

However experiment with the procedure: CIRCLE 3 CIRCLE 6 CIRCLE -3 CIRCLE 9

Building on this idea of a procedure we could have written:

TO CIRC :CH :ANG
REPEAT 360 [ FD :CH RT :ANG]
END

This is a procedure with two variables!
Can you anticipate what will change and what will happen now?
Can you see what is now possible?

Go on experiment! Try:

CIRC  3 3 
CIRC  3 6
CIRC  3 30

What is happening?

Why not be bolder and try say: CIRC 30 90 CIRC 30 45 CIRC 30 18 Etc., etc.,

What do you notice?
What conclusions can you come to?