GENERATING A FRACTAL SQUARE

Transcription

1 GENERATING A FRACTAL SQUARE I 194 the Swedish mathematicia Helge vo Koch( itroduced oe o the earliest ow ractals, amely, the Koch Sowlae. It is a closed cotiuous curve with discotiuities i its derivative at discrete poits. The simplest way to costruct the curve is to start with a equilateral triagle o uit side-legth ad the brea each o its sides ito three equal parts o legth 1/ ad ext add a ourth elemet o legth 1/ to costruct a smaller equilateral triagle plus two boudig straight lies. Cotiuig this type o breaup ideiitely leads to the Koch Sowlae. We ca thi o the process as oe i which successive geeratios orm smaller similar triagles ad coectig lies orm a closed curve. The developmet through geeratios, 1, ad loo as ollows- There is o course o reaso that oly triagles ca be used to costruct a Koch curve. Oe such costructio which we have come up with ad which diers rom other extat costructios is oe based o a uit side-legth square oto whose edges are placed smaller ad smaller sel-similar smaller squares. Drawig such a curve we id that that geeratios -1- produce the ollowig igure-

2 The scalig betwee geeratios is ixed at a costat value o lyig i the rage < <1 such that the +1 geeratio has a square side-legth smaller tha the th geeratio by a actor o. This reductio i size betwee geeratios allows us to maitai sel similarity. You will otice that the costructio here allows attachmet o +1 geeratio squares oly to exposed edges o the th geeratio elemets. This procedure diers rom a stadard Koch curve. I we ow examie the total area cotaied withi the curve termiatig with the th geeratio, we id- A ( It is recogized that this iite series represets a icomplete geometric series whose closed orm solutio is ow. A little maipulatio produces the result- A {1 ( } 4 (1 Thus-

3 A 4 + (1 1 provided that <1/sqrt( ad overlap is allowed. Notice that the ratio o the area o the +1 geeratio compared to the th geeratio or 1 is- Area Ratio Let us ow loo at the special case o 1/. This produces the ollowig attractive coiguratio - I call the igure the Blac Sowlae. Notice that it has the iterestig property that taget diagoal lies with slope ±1 touch each geeratio. For this case the ratio o side-legth o the squares rom oe geeratio to the ext equals exactly three. We id A 1, A 1 1/9, ad A 4/7. The total area whe addig together all geeratios produces the iite value A 5/. This result maes sese sice the rotated square composed o the our taget lies metioed has a area o two. We ca also establish the A value by otig that the area ratio betwee geeratios is 1/ or this case ad hece-

4 A 1 ( 5 Whe looig at the Blac Sowlae the shortest distace betwee geeratio +1 ad -1 remais iite so there is o possibility o overlap. As we icrease above 1/ a poit will be reached whe a overlap ca occur. Let us see what the limitatios o are to prevet such a overlap. Clearly to avoid overlap it is ecessary or the smallest gap betwee the d (greead zeroth(blue geeratio i the above colorized igure be more tha This meas- O solvig we id Thus the restrictio o will be- < < 1 I order to have a ractal square with a iiite umber o geeratios. I oe is iterested i closed curves showig oly the irst geeratios tha this restrictio ca be relaxed. Let us ext calculate the perimeter P o the ractal square uder cosideratio. For the zeroth geeratio we have P 4(1 Whe the irst geeratio is icluded we id- P 1 4(1 + 4 (1 ( 4(1 [1+ ] ad the iclusio o the secod geeratio produces- P 4(1 [ } Thus we have that the total perimeter o the ractal square through the th geeratio becomes-

5 1 P 4(1 ( + 4( Notice the last term i this equality arises rom the act that the parts o the perimeter bloced by the +1 geeratio o loger exist. We thus see that lie or a stadard Koch curve the perimeter becomes iiite at whe >1/. However, you will also ote the uexpected result that the perimeter remais iite or ad <1/. It shows, or example, that P 1 ad A 17/1 or 1/4 while 1/ produces P ad A 5/. As we have doe or the Blac Sowlae, oe ca use computer graphics to quicly draw the ractal square through ay desired umber o geeratios or a ixed. Cosider drawig the igure usig oly the zeroth ad irst geeratio whe the startig poit is a uit square ad 1/. Our computer program usig MAPLE reads- with(plots; listplot([[1,1],[1/,1],[1/,],[-1/,],[-1/,1],[-1,1],[-1,1/],[-,1/],[-,-1/],[-1,-1/],[- 1,-1],[-1/,-1],[-1/,-],[1/,-],[1/,-1],[1,-1],[1,-1/],[,-1/],[,1/],[1,1/],[1,1]], colorred, scaligcostraied, axesoe; The graphic output is-

6 Note the total area is as give by the above ormula or A 1. The perimeter becomes P 1 8. Fially we briely metio a alterative way to geerate these ractal square coiguratios by a geetic algorithm approach as discussed by us i a earlier ote( ad related to the Lidemayer system. I this procedure ay closed curve ca be costructed rom straight lie segmets deied oly by their legth L ad the agle θ they mae with regard to the ext lie segmet. The desigatio o a lie will be [L,θ]. For the ractal square the agles are restricted to either θ π/ or couterclocwise or θ-π/ or clocwise. Thus or the square ractal with 1/ we have the basic buildig bloc- 1 π 1 π 1 π 1 π 1 π [, ],[, + ],[, + ],[, ],[, + ],[ A graph o this ive elemet bloc loo lie this or geeratios ad +1- I we coect the buildig bloc or the 1 geeratio our times the ollowig igure results-

7 As is see, the code geerates three staircase uctios which are hooed together to orm the ractal square. I oe is iterested i deiig a square ractal cotaiig the ext higher geeratio it is oly ecessary to icrease to +1 i the L1/ legths o the basic ive elemet code ad superimpose the result uto the exposed lie edges o the th geeratio squares.