Think: Spin Shrink!Author(s): Martin HansenSource: Mathematics in School, Vol. 23, No. 2 (Mar., 1994), pp. 13-16Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215090 .

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Think: Spin Shrink !

by Martin Hansen, The Schools, Shrewsbury

Transformation geometry is a topic that I have always enjoyed teaching. Its visual nature I find inspiring; and there is a satisfaction in knowing that a mathematical process has been correctly carried out because the resulting picture "looks right".

A transformation which, I think, ought to be better known to schoolchildren is the spiral similarity transform- ation. A spiral similarity is simply the combination of an enlargement and a rotation. Mixing these two transform- ations into one that is new, however, gives rise to some lovely diagrams, as I shall show in this article. The diagrams are of particular interest as they are, in fact, simple fractals and they are generated in the same way that all fractals are; by iteration.

The iteration can be of an intuitive nature as I showed in an article titled "Let's Twist" in the May 1992 edition of "Mathematics In School". However, as the diagrams that one strives to produce either involve a more complex shape, as in Figure 1, or become more spaced out as in Figure 2, getting it right by intuition becomes somewhat tricky.

Thus, one quite naturally wants a more rigourous, mathematical method for getting the pictures right. Different tools suggest themselves at GCSE level and A-level.

At GCSE level a multiplication by a suitable matrix will do the job of both spinning and shrinking the coordinates of the start shape. The matrix needed must combine an enlargement of centre the origin and scale factor square root 0.5, with a rotation about the origin of 45', this being the spiral similarity at work in "Let's Twist" and Figures 1 and 2. The matrix that will do this is;

0.5 -0.5

0.5 0.5-

This is derived by multiplying the general matrix for an enlargement centre the origin and scale factor k;

[k 01 Ok

with the general matrix for a rotation about the origin of angle 0';

cos -sin1 -sin cos

k being square root 0.5 and 0 being 45'. In Figure 2, A is the triangle with vertices P(4, 0),

Q(4, -4) and R(8, -4). These are written as a multi- point matrix;

P Q R

xr4 4 81

y 0 -4 -4

Multiplying the multipoint matrix for A by the spin shrink matrix gives the multipoint matrix of triangle B;

[0.5 -0.5r4 4 8 2 4 6

0.5 0.5-0 -4 - 4 2 0 2

Fig. 1

Fig. 2

B

A'

Mathematics in School, March 1994 13

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14 Mathematics in School, March 1994

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This new triangle, triangle B, is now multiplied by the spin shrink matrix to give the next triangle in the spiral. This is the iterative process that is repeatedly applied to generate the Figure 2, or at least, one quarter of it. However, to make an obvious observation, this repetition of a simple process, whilst being a reason for the diagram's fascination, soon becomes somewhat tedious to do by hand. The challenge, once the first couple of diagrams have been plotted, is in finding shapes that iterate into a picture pleasing to the eye and the task of the actual iteration is something better delegated to a computer.

One of the nice improvements made to BBC Basic for Acorn's Archimedes range of computers was the addition of some powerful matrix algebra commands. Box 1 contains a short program which will take a set of start shape coordinates and generate a picture in the style of Figures 1 and 2. I have kept the program short knowing that some readers will enjoy making modifications to it but even as it stands it does the job. To make it iterate a shape of your

Mathematics in School, March 1994 15

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10 REM>SpinShrink 20 REM For the Acorn Archimedes 30 REM by Martin Hansen 40 REM (c) Mathematics In Schools 50: 60 MODE 12:ORIGIN 640,512 70 READ numpts 80 DIM mtx (2,numpts) 90 DIM s45(2,2),r90(2,2)

100 s45(1,1) =0.5: s45(1,2) = -0.5 110 s45(2,1)=0.5:s45(2,2)= 0.5 120r90(1,1)=0 :r90(1,2)=-1 130r90(2,1)=1 :r90(2,2)= 0 140 FOR pt=0 TO numpts 150 READ x,y 160 mtx(1 ,pt) = x*200 170 mtx(2,pt) = y*200 180 NEXT pt 190 FOR iteration= 1 TO 20 200 FOR quarter=l1 TO 4 210 MOVE mtx(1,1),mtx(2,1) 220 FOR pt=2 TO numpts 230 DRAW mtx(1,pt),mtx(2,pt) 240 NEXT pt 250 DRAW mtx(1,1),mtx(2,1) 260 MOVE mtx(1,0),mtx(2,0) 270 GCOL quarter 280 FILL BY 0,0 290 GCOL 7 300 mtx( ) = r90( ).mtx( ) 310 NEXT quarter 320 mtx( ) = s45(). mtx( ) 330 NEXT iteration 340 : 350 DATA 6, 3,- 3 360 DATA 2,- 2, 2,- 6, 4,- 8 370 DATA 4,- 4, 8,- 4, 6,- 2

Box 1

Shape A 350 DATA 3, 0,6 360 DATA - 4,8, 4,8 370 DATA 0,4

Shape B 350 DATA 5, 6,0 360 DATA 8,4, 8,0 370 DATA 6, - 2, 4,0 4,4

Shape C 350 DATA 8, 6,-6 360 DATA 2, - 6, 2, - 4, 4, - 4, 4, - 2 370 DATA 6,- 2, 8, - 4, 8, - 8, 4, - 8

Shape D 350 DATA 3, -4, -6 360 DATA -4,-4, 0,- 8 370 DATA - 8,- 8

Shape E 350 DATA 7, -6,0 360 DATA - 8, - 4, -8,4, - 4,4, - 6,2 370 DATA -4,0, - 6,- 2, -4,- 4

Box 2

A

E B

D C

Fig. 3

choice the three DATA statements at the end of the program need changing. This is where the program looks to find the coordinates of a start shape. On the line numbered 350 the first number following the word DATA states how many coordinates make up the start shape and the next two numbers on that line identify a point somewhere in the interior of the shape. On lines 360 and 370 are the coordinates themselves. I have inserted spaces to format them into pairs. Such spacing is ignored by the computer. Box 2 and Figure 3 give a variety of interesting start shapes to take for a spin shrink.

An A-level group might be aware that complex number multiplications give rise to spiral similarities, and the hunt is then on to find the relevant complex number. Wanted, you may recall, is a transformation which combines an enlargement of centre origin and scale factor square root 0.5, with a rotation about the origin of 45'. A multiplication by the general complex number a+bj gives a rotation about the origin of tan(b/a) and an enlargement centre origin and scale factor (a2 + b2)1/2. Thus, tan-1 45' being 1 implies that a=b. Matching (2a2)1/2 with 0.51/2 gives two suitable complex numbers; 0.5 +0.5j and -0.5-0.5j. The start shape is now considered to lie in the complex plane. Returning to triangle A in Figure 2;

(0.5 + 0.5j)(4 + 0j)= (2 + 2j)

(0.5 + 0.5j)(4 - 4j)= (4 + Oj) (0.5 + 0.5j)(8 - 4j)= (6 + 2j)

Triangle B thus found and plotted, the process is repeated by multiplying again by 0.5+0.5j. Several calculators have a complex number multiplication facility and, indeed, I have also found that A-level students enjoy using the matrix algebra facility of their graphics calculators to see how complex numbers and matrices can both produce spiral similarities. [-

References Hansen, M. (1992) Let's Twist. Mathematics in School, 21, 3. Hansen, M. (1993) The Theorem of Pyfudgerous. Mathematics Review,

4, 2.

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