Let's Twist

Let's TwistAuthor(s): Martin HansenSource: Mathematics in School, Vol. 21, No. 3 (May, 1992), pp. 16-18Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214877 .

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by Martin Hansen, The Schools, Shrewsbury

LET'S

, An idea that has gained in importance in recent years is that of iteration for it lies at the heart of both Chaos theory and Fractal geometry. However, whilst repeatedly applying the same set of rules to a number is an almost natural thing for a computer to do, it is hard and tedious work for us humans.

Many GCSE courses now include some simple solving of equations using iteration techniques and so, clearly, it is felt that pupils should be given a taste of what iteration is about. Having taught this work recently, it struck me as being a little sad that the visually stunning diagrams of fractal geometry are out of reach. For them, one must iterate complex numbers and this delight is a province of undergraduate level mathematics.

The beautiful diagrams of fractal geometry arise because complex numbers are composed of two parts, a real bit and an imaginary bit. These are plotted as x and y coordinates. With this idea in mind I started thinking about how one might "jump in" by starting with a simple geometric shape such as a square and a triangle and then iterate it in some way. What follows is how I now introduce to my classes the ideas of iteration.

e Upon a standard A4 page of 1 cm square graph paper mark out a large boundary square of side 16 cm.

Fig. 1 (a-d) Fig. 2 Fig. 3 (a-d)

16 Mathematics in School, May 1992



.

Fig. 4

* Inside this draw a shaded in square of side 4 cm as shown in figure la.

s A triangle is to be added on top of the 4 cm square. It is isosceles and right angled with it's hypotenuse along the top of the square. See figure lb.

* This combination of square with triangle on top is the shape to be iterated. The iteration is done by drawing a second square and it's associated triangle on the side of the first triangle as shown in figure 1c.

Fig. 5

* The 16 cm boundary was chosen as this removes the need for extensive calculations or excessively accurate drawing of lengths and angles. Figure 1d shows the process taken to it's limit.

The entire area of the 16 cm boundary square can now be tesselated with four of these spirals as shown in figure 2. For added interest figures 3 and 4 show how to add in a variation that iterates a triangle in a similar manner. Other variations are not hard to find.

If some pictures of fractals are available it will be noted that they too often exhibit a spiralling component. One can also talk at this stage of being able to zoom in indefinitely upon the centre without the detail of the structure deminishing which is a property of both fractals and of our shape.

The square and triangle chosen were special in that they result in a spiral shape which tesselates the plane. (Inciden-

tally, a fractal that tiles the plane is known as a fractile.) As shown in the introductory "rocket", however, one can swap the square for a rectangle and by keeping the ratio of length to breadth constant develop further spiral designs.

Figure 5 was the background grid used and the "rocket" designed in it's first rectangle. Successive rectangles contain similar grids to scale down the "rocket" correctly. The triangle too can be varied but the original one has the property of halving the rectangle's area each time. This is the ratio by which a photocopier reduces diagrams when going A4 to A5. Thus, any A4 picture could be reduced in stages and by inserting appropriate triangles be made to spiral across a wall.

A little study of the diagrams reveals that the spiral generated in figures 1 to 4 was done so by turning through 45. each time one of eight spokes radiating from the centre was encountered. Such a spiral is called an equiangular spiral and, in polar coordinates, has a very simple equation of the form r = aek*. In fact, the spiral that the square with triangle atop follows is r= 24/90 for 45 in degrees. Figure 6

Fig. 6

shows how the curve with this equation relates to the earlier diagrams.

Rotating the curve in 11.. steps and adding in spokes spaced at 5. intervals leads to spiral graph paper. By first designing a shape by shading in squares on, say, an 8 by 8 grid and then transferring it onto the spiral grid some interesting distortion effects result.

a-.r: $~:

/~c

I'C/c, ic

Fig. 7

Mathematics in School, May 1992 17



I recently saw Benoit Mandelbrot

being interviewed on television about

the excitement of discovering fractals,

of sitting up night after night

watching wierd and wonderful shapes

"blowing over" his computer screen.

It thus seemed appropriate to

conclude this short article with a

page of what I hope are thought

provoking and inspiring designs.

Two of them are explained and two

are left for readers to puzzle over.

18 Mathematics in School, May 1992



Let's Twist

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