# Geometry page

"Lessons in calculation have been devised for tiny tots to learn while they are enjoying themselves at play..." wrote the ancient Greek philosopher Plato, explaining why the Egyptians were so good at mathematics. "Next, the teacher puts the children on to measuring lengths, surfaces and solids, a study which rescues them from the deep rooted ignorance, at once comic and shocking, that all men display in this field."^{ note 1}

The aim of this page is to provide a key to the door of mathematics, which many people will be surprised to discover is a world of considerable wonder and beauty. It is not necessary to understand pages of mind-numbing symbols in order to enter this world and begin to explore it.

## Introduction to geometry through Islamic patterns

If you follow the instructions given here you should be able to create something beautiful and at the same time learn one of the fundamental constructions of geometry.

Historically, the ideas on which this presentation are based derive from the Islamic world. Islam preserved and enhanced the sciences inherited from the Greeks, who in turn had learned much of their culture from the Egyptians. Islamic learning began to filter back into Christian Europe around the 11th century AD.

The following material has been tried out on a mixed ability class of 10 year olds with some success. If you try it out with children then please give as much help as necessary without actually taking the compasses out of their hands, and give plenty of praise. They are bound to produce something worth looking at if you have eyes to understand it correctly! Look not only at the finished product but also at the effort that went into making it, which you will see in their faces. If praised, they will understand that, with a little effort, they have been given a key to a new world. If they don't want to do it and are not in school, let it go for another time. I strongly suggest that adults do the construction themselves before presenting it to children.

You will need a pair of compasses, a straight edge or ruler, a sharp pencil, a large sheet of paper (at least A3, ie: twice the area of ordinary exercise book paper), some scrap paper for practice, and some coloured pencils or felt tip pens. At the outset it must be understood that the aim of this is to have some fun and to feel positive about geometry, so if your pattern ends up not as precise as the ones presented here, remember that Plato taught that the things we see here on earth are only at best approximations to a finer reality, and that even the most beautiful creations of nature have flaws and imperfections. In the class of 10-year olds, some produced wonderfully precise versions with added intricacies of their own invention, while others produced pages of more loosely scattered stars that danced on the page, like real stars. They all looked beautiful when coloured in. Whatever you end up with, colour it in!

To start with, practice drawing a whole circle on one of the spare pieces of paper. This is the tricky part. Make sure that the compasses are tight so that the radius doesn't change as you are using them. Preferably hold them at the top, so that you can complete as much of the circle as possible in one movement. Pay attention to where the pencil point touches the paper. Relax: you don't need to be tense!

When you are ready to start on the pattern, set the compasses so that you will be able to fit many circles onto your paper (about one and a half inches or 4cm between the points of the compasses if your paper is A3). Place the point of the compasses in the centre of the paper.

This represents the beginning of creation. Modern physicists believe that the whole of creation arose from a single point, from which in an inconceivable explosion emerged not only the unformed substance from which matter would eventually condense, but also time and space themselves. Here is your first circle: the as yet unformed substance of creation expanding outwards (figure 1).

Next, place the point of the compasses anywhere on the edge of the first circle, and draw another circle. In this and all subsequent steps be as accurate as possible, taking particular care to keep the radius (the distance between the compasses points) the same: do not press too hard. You should find that the new circle passes through the centre of the first circle.

Now move the point of the compasses to one of the two places where your second circle has cut the edge (circumference) of the first circle, and draw a third circle. This in turn will cut your first circle at another two points: one is the centre of the second circle, and the other is new. Move the point of the compasses to this new point on the circumference of the first circle and draw yet another circle, keeping the radius always the same. Repeat making circles round the edge of the first circle until you get back to where you started (fig. 2).

This is perhaps the first miracle of geometry: if your drawing has been accurate you will see that the last circle passes through not only the centre of the original circle but also through the centre of the second circle you drew, so that now you have exactly six circles around the edge and the original circle in the middle. Why exactly six (and not, say, six and a half)?

The expanding matter of the universe has begun to differentiate, to make new forms; it is opening like a flower. Look at the middle of your drawing and you will see a flower.

Now each of these little universes will begin to make universes of their own, so you will need to continue creating new circles, still the same radius, around each of the circles you have drawn so far, until you run out of paper (fig. 3).

From here, endless possibilities arise, of which what follows is only one. In your first circle, find one of the points where the other circles cut the edge, and draw a straight line from there, not to the next point on the edge but to the next-but-one. Continue until every point on the edge of the circle is joined to its next-but-one. You should now have a star made of two interlocking triangles (a Jewish girl in the class exclaimed: "that's a Jewish star!"). Now you have to find one of the circles whose star will touch the first one: this is not so easy now as here are so many circles! When you have found one, draw its star too, and continue in this way drawing as many stars as you can (fig. 4 below). I have left out the flowers in the middles of the circles for clarity, but there is no need to rub them out in your drawing. You can colour over them later or make them part of your design.

In fact there are at least two ways of doing this: one leads to stars with hexagons between them and the other, as here, leads to diamonds (fig. 5). Both variants were discovered by the children.

Islamic artists used often to relieve the severity of their geometric designs by depicting flowers or other organic forms in the middles of the larger shapes, which you may wish to do. Now for colouring in!

Note 1. Plato, "The Laws", trans. Trevor Saunders, Penguin Classics 1970, 819 p.313.

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