A New Recursio for Space-Fillig Geometric Fractals Joh Shier Abstract. A recursive two-dimesioal geometric fractal costructio based upo area ad perimeter is described. For circles the radius of the ext circle is a costat times (gasket 1 area)/(gasket perimeter). Placemet of the circles withi a boudig circle is doe by ooverlappig radom search. The th circle area A goes over to a power law A ~ -c for large, where is the umber of placed circles ad c is a costat. For circles it is foud that if is a quotiet of itegers, c is also a quotiet of itegers. Computatioal evidece idicates that this algorithm is space-fillig, ad that it is ohaltig over a substatial rage of values. A study of the three-dimesioal case of spheres shows similar behavior. imited studies of shapes other tha circles show that the algorithm works for a variety of shapes. 1. Itroductio I previous work [2] a space-fillig geometric algorithm was described, based o (a) a sequece of shapes whose areas obey a egative-expoet power law versus shape umber, ad (b) placemet of successive eversmaller shapes by ooverlappig radom search. Quite early i the study of these fractals it was cocluded that the quatity gasket area w (1) gasket perimeter ca be viewed as a measure 2 of the average gasket width. For circles oe ca defie the dimesioless average gasket width as gasket area after placemets b( c, N, ) (2) ( gasket perimeter after placemets)( circle 1diameter ) Numerical studies for the statistical geometry algorithm [2] showed that b is early idepedet of for large. This was viewed as evidece that the available space i the gasket falls i direct proportio to the circle diameter, so that the algorithm does ot halt. The preset study builds upo this idea. If w is a valid measure of average gasket width, the it should be possible to fill a spatial regio with a sequece of ever-smaller circles accordig to a recursive rule for the circle radius r +1 : r gasket area after placemets 1 (3) gasket perimeter after placemets where is a dimesioless parameter. The gasket area is always positive ad is a dimiishig fuctio of, while the gasket perimeter grows without limit as icreases. Thus the circle radius r rapidly decreases. 2. Mathematical Properties of the Recursio. The reader should keep i mid that the coclusios preseted here are based o computatioal experimets. Proofs (or refutatios) of the claims will require further study. We first cosider the behavior of the recursio of Eq. (3) before cosiderig whether ooverlappig radom search ca fill a regio with circles havig radii geerated this way. Recursive r values were computed 1 The gasket is the part of the boudig regio ot withi a placed circle, e.g. the white regio betwee circles i Fig. 3(a). Study of the Shier-Bourke paper [2] will make it much easier to uderstad the results described here. 2 The ratio w as a measure of gasket width is a difficult idea for may people. Much metal sumo wrestlig may be eeded before it is comfortable. Its usefuless is supported by the results reported here ad i [2]. Shier -- Gasket Area/Perimeter Algorithm -- v. 11 -- Jue 2015 -- p. 1
for circles usig double-precisio arithmetic, with the results show i Fig. 1. It ca be see that i log coordiates both of the plotted quatities go very rapidly to a straight lie as icreases, idicatig that the data follows a power law whose expoet is the slope 3 of the lie. With = 2.5 (Fig. 1) the circle area data (red) has a slope 1.384613 for large (with accuracy of 6 sigificat digits). The gasket area (blue) has a slope of.384613. Such expoets differig by 1 have bee cited previously [3-5] i studies of radom Apolloia fractals. We let c be the (positive) expoet 4 for circle area, which is thus c = 1.384613. It is foud that 1.384613 = 18/13 (to 6 sigificat digits) whe = 2.5. The large- slope of the data i Fig. 1 is foud to be idepedet of the details of the first few circle radii, so that the first few recursios ca have differet rules to aid i lauchig (see sec. 4). The sizes of the first few circles have a sigificat effect o the ability of the algorithm to ru at high values. Fig. 1. og-log plot of th circle area (red) ad the gasket area (blue) versus the circle placemet umber with = 2.5. I geeral if a quotiet of itegers is chose for, c is also a quotiet of itegers, with the c value give by 3 If y = x c, the log(y) = c log(x), so that a plot of log(x) versus log(y) is a straight lie with slope c. 4 Thus the symbol c follows the usage for statistical geometry fractals [2]. Shier -- Gasket Area/Perimeter Algorithm -- v. 11 -- Jue 2015 -- p. 2
4 2 c 4 a c 1 4 c 2 b Equatio (4a) was foud by trial-ad-error ad implies that 1 < c < 2. Thus the fractal dimesio D = 2/c must obey D > 1. Equatio (4a) agrees with computed c values to 6 sigificat decimal places. It should be possible to derive Eq. (4a) from Eq. (3) i the limit of large. The power law i obeyed by the gasket area (blue poits i Fig. 1) idicates that the gasket area ca be made smaller tha ay give value if is sufficietly large. This is the basis for the claim that the algorithm is space-fillig if the placed circles do ot overlap. 3. Circle Placemet by Nooverlappig Radom Search. Nooverlappig radom search is used i the statistical geometry algorithm [2]. Figure 2 shows the flow chart for this algorithm. Fig. 2. Flow diagram for ooverlappig radom search. The shapes (here circles) get smaller as is icremeted followig Eq. (3). The diagram has o STOP because available evidece idicates that it does ot halt for a sizable rage of values. Successive shapes (here circles or spheres) have radii calculated usig Eq. (3) or Eqs. (5-7) or Eqs. (8-9). Ca ooverlappig radom search proceed without haltig with the recursive rule of Eq. (3)? Figure 3 shows a example. The algorithm does sometimes halt, subject to a haltig probability (like statistical geometry [2]). A study of the haltig probability for circles i a circle was made such that ohaltig rus stop at 50 circles ( =.4 ad tra = 6 as defied i sec. 4.). A ru was viewed as haltig if 6,000,000 trials were made without a placemet. The results were as follows: With = 1.60 0 of 100 rus halted With = 1.80 0 of 100 rus halted With = 2.00 2 of 100 rus halted at placemets 6 6 With = 2.20 18 of 100 rus halted at placemets 6 7 7 7 7 6 7 6 6 4 4 6 8 6 7 8 3 4 With = 2.40 45 of 100 rus halted at placemets 6 3 6 8 6 6 6 3 6 7 5 6 6 5 6 8 8 5 6 4 11 6 3 7 6 6 7 7 6 4 6 6 5 6 5 4 6 6 5 6 6 4 6 7 5 It is cocluded that for low values the haltig probability is close to zero. For higher values some fractio of the rus halt, ad whe they halt they do so at a small umber of placemets. This patter of behavior is similar to that for statistical geometry [2]. Shier -- Gasket Area/Perimeter Algorithm -- v. 11 -- Jue 2015 -- p. 3
The use of ooverlappig radom search with Eq. (3) requires special cosideratio for the first circle size. What should the iitial area ad perimeter be? It is reasoable (but somewhat arbitrary) to take the startig area ad perimeter to be their values for the boudig circle. With the assumptio of costat we must have < 2 for the first placed circle sice otherwise the first circle would be larger tha the boudig circle. With costat i most cases the first circle occupies a domiat part of the boudig area. More iterestig results are see if (oly) the first few circles depart from Eq. (3) as i Eqs. (5-7). Figure 3(a) shows a example where = 0.5 for the first circle ad = 5/3 otherwise. Fig. 3. (a) A space-fillig patter of 400 circles i a circle, = 5/3, c = 22/17, 87% fill. The first value was 1/2. (b) A log-log plot of the cumulative umber of radom trials eeded to place circles. Figure 3(a) looks much like patters made with the statistical geometry algorithm. I Fig. 3(b) the data approaches a straight lie for large, idicatig that the cumulative umber of trials versus ca be satisfactorily approximated by a positive-expoet power law. Such a power law ever becomes ifiite; thus the coclusio that the algorithm does ot halt i this example. A formal proof of ohaltig for a rage of values would be of much iterest. 4. auchig. It is foud that oe ca defie the first few circles i a variety of ways while the algorithm cotiues to ru ad coverge to a power law (with the same expoet) for circle area after a large umber of placemets. The treatmet of the first few circles will be called "lauchig" (some might call it "seedig"). Alterative radius rules are defied here. I Eqs. (5-7) we have parameters ad tra i additio to. The parameters ad tra oly affect the first few circles; after tha Eq. (3) holds for all. Shier -- Gasket Area/Perimeter Algorithm -- v. 11 -- Jue 2015 -- p. 4
Whe = 1: ( area of boudary) r1 (5) ( perimeter of boudary) 2 1 (6) tra 1 gasket perimeter after placemets Whe 1 < < tra : tra gasket area after placemets r 1 Whe tra : r gasket area after placemets 1 (7) (same as (3)) gasket perimeter after placemets Equatio (6) causes the "effective" value to icrease smoothly (as a parabola) from to over several steps. This flexibility i lauchig allows relatively high values to be used without early haltig. Figure 4 shows a example of the high values possible with the scheme of Eqs. (5-7). Fig. 4. (a) A space-fillig patter of 400 circles i a circle, = 8/3, c = 7/5, = 0.3, tra = 6. 94% fill. (b) A log-log plot of the cumulative umber of radom trials eeded to place circles. og-periodic color with oe full cycle. For statistical geometry a cumulative trials plot such as Fig. 4(b) is a straight lie with oise. Here we have a "stair step" behavior i which may trials are eeded to place a few of the circles. For > 100 the data i Fig. 4(b) ca be approximated by a straight lie, which is evidece for ohaltig. Aother lauchig scheme calls for computig two ext-radius values: r A 1 R (8) Shier -- Gasket Area/Perimeter Algorithm -- v. 11 -- Jue 2015 -- p. 5
r B gasket area after placemets 1 (9) gasket perimeter after placemets ad choosig the smaller of these values. Here is a dimesioless parameter ad R is the radius of the boudig circle. As the process begis r B is quite large, but it decreases steadily, evetually becomig smaller tha the fixed value r A. Fig. 5. (a) A space-fillig patter of 800 circles i a circle with the lauchig scheme of Eqs. (8) ad (9). = 8/3, =.26. 94% fill. (b) A log-log plot of the cumulative umber of radom trials eeded to place circles. og-periodic color with oe full cycle. The seve largest circles have the same radius. Figure 5 shows a high- patter with the lauch scheme of Eqs. (8) ad (9). The area occupied by the fixed-radius circles is typically aroud 50% of the boudary area. Figure 5(b) shows the same kid of stair-step behavior see for other examples. 5. Three Dimesios: Spheres. The obvious geeralizatio of Eq. (3) to spheres i three dimesios is r gasket volume after placemets 1 (10) gasket surface area after placemets Figure 5 shows a typical example of iteratig Eq. (10) for spheres (with the lauch scheme of Eqs. (5-7)). The value = 2.25 is about as high as oe ca go for spheres. For large the slope of the sphere volume data is Shier -- Gasket Area/Perimeter Algorithm -- v. 11 -- Jue 2015 -- p. 6
c = 1.05555 (= 19/18) while the slope of the gasket volume is.05555. As for circles, if is a quotiet of itegers, c is also a quotiet of itegers (accurate to 6 sigificat digits). Data for c versus is give i Table 1. Fig. 6. og-log plot of the volume of sphere (red) ad of the gasket volume (blue) versus the sphere umber. The lauchig sequece for the spheres i Fig. 6 follows Eqs. (5-7) with the obvious adjustmets for spheres. The egative-expoet power law see for the gasket volume i Fig. 6 idicates that a ooverlappig fractal based o Eq. (10) will be space-fillig, but it will fill very slowly. Table 1. Power-law slope c ad fractal dimesio D for sphere volume versus. The fractal dimesio is 3/c [2]. c fractal D The data of Table 1 has bee foud by trial ad error to agree with the equatio 36 3 c 36 2 It is see from Eq. (11) that 1 < c < 3/2. The fractal dimesio D must be > 2. It is cojectured that the coefficiets i Eqs. (4) ad (11) are itegers, ad that the 3 ad 2 i Eq. (11) are Euclidea dimesios. Shier -- Gasket Area/Perimeter Algorithm -- v. 11 -- Jue 2015 -- p. 7
(a) (b) Fig. 7. (a) Cross sectio of a space-fillig patter of 2500 spheres i a cube with = 2.4, = 0.4, tra = 6, c = 18/17 = 1.0588, 75% volume fill. The sectio rus through the cube ceter i the xy plae. og-periodic color with oe full cycle, with the largest ad smallest spheres havig the same color. The smallest shapes are spheres which have oly a small itersectio with the plae of the cross sectio (b) A log-log plot of the cumulative umber of radom trials eeded to place spheres for the same computer ru. The recursio (10) ca produce a space-fillig fractalizatio of spheres for values up to about 2.5. The tred i Fig. 6(b) is evidece that the algorithm does ot halt. A formal proof of ohaltig is a iterestig challege. Figure 8 shows aother sphere-i-cube example, this time usig the lauch scheme of Eqs. (8) ad (9). With these parameters the haltig probability was high ad this is a "lucky" survivor ru. The first 25 spheres all have the same radius ad have a mageta color. The rus which halted all did so at a placemet fairly close to 25. Sice this is a cross sectio, some mageta circles are small, which reflects the fact that the plae of the cross sectio cuts through oly a small part of the correspodig sphere. The curve of Fig. 8(b) divides ito two regimes. For low values the umber of trials eeded to place the fixed-size spheres rapidly icreases. After the area/perimeter rule takes over the curve flattes ad we see a fairly eve icrease of trials i log-log coordiates. The tred i Fig. 8(b) shows that for large the umber of trials eeded to achieve a give umber of placemets is quite well-behaved ad is accurately described by a power law. Such a power law is evidece for ohaltig. The expoet of this power law was foud to be 1.166 based o least-square fittig of the last 15000 poits i the log-log data. The lauch sequeces which have bee explored here hardly exhaust the possibilities. Is there a lauch sequece which is "optimum" or "best" i some sese? Ca oe achieve iterestig results by pre-placemet of the first few shapes? The use of periodic boudaries resultig i patters which ca be tiled (see [2]) is oe of the iterestig studies which remai to be doe. Shier -- Gasket Area/Perimeter Algorithm -- v. 11 -- Jue 2015 -- p. 8
Fig. 8. (a) Cross sectio of a space-fillig patter of 30000 spheres i a cube with = 26/11, =.28, c = 237/224 = 1.058036, 82% volume fill. The sectio rus through the cube ceter i the xy plae. og-periodic color with oe full cycle. (b) A log-log plot of the cumulative umber of radom trials eeded to place spheres for the same computer ru. 6. Discussio. Oe of the iterestig ope questios about the statistical geometry algorithm [2] is whether the ooverlappig radom search method ca be used for a space-fillig 5 costructio with a differet area rule. Such a differet rule is preseted here. A power-law rule for the areas of the circles is ot assumed a priori, but it is foud that such a power law describes the data quite accurately i the limit of a large umber of shapes. The algorithm is quite simple. Oce lauched, it requires oly the ideas of perimeter, area, ad volume, plus a sigle dimesioless parameter. It reveals a iterestig property of area ad perimeter (or volume ad area). It is ot obvious how oe would use this idea i oe dimesio. May recursive fractals are kow (e.g., Sierpiski) i which each recursio creates k times more (smaller) shapes tha the previous recursio, i.e., the recursio is betwee "geeratios". This recursio is ot geeratioto-geeratio, ad each shape has a differet area. The algorithm has some uique properties. It ca create a space-fillig power-law fractal based solely o perimeter-area-volume. The defiitio ad calculatio of power laws with expoets other tha itegers or halfitegers is ordiarily doe with logarithms. Iteratio of Eq. (3) ca geerate such a egative-expoet power law without referece to logarithms. The observatio that if is a quotiet of itegers, c is also a quotiet of itegers is see i Eq. (4a) ad Eq. (11). The coefficiets i Eqs. (4a) ad (11) are cojectured to be itegers. Derivatio of Eq. (4a) from Eq. (3) would be iterestig ad should be possible. Formally this requires demostratio of the limit: 5 I have avoided use of the word "packig" ad used "fillig". Packig usually refers to geometric costructios where the elemets touch each other, which is ot the case here. Shier -- Gasket Area/Perimeter Algorithm -- v. 11 -- Jue 2015 -- p. 9
log r 2 2 log r 4 2 1 log( 1) log( ) 4 as While circles ad spheres are studied here, limited work with other shapes has bee successful. It is cojectured that the algorithm rus without haltig for ay placed shape 6 over some rage 0 < < max. Oe descriptio for this work ad that of [2] is "o-apolloia geometry". 7. A Almost-Idetity. Why should the computed expoets i Fig. 1 differ by exactly 1? Would't it be iterestig if oe could fid a value such that 1 ( c) i1 1 1 1 (13) c i where ( ) is the Riema zeta fuctio. Oe ca come quite close by takig = c 1. This relatioship for versus c is used i certai published papers (e.g., [3-5]). If you compute the left-had side of Eq. (13) you fid 1st term 2d term sum 100 0.787372 0.251189 1.038561 1000 0.893290 0.125893 1.019182 10000 0.946511 0.063096 1.009607 100000 0.973192 0.031623 1.004814 1000000 0.986564 0.015849 1.002413 The sum differs from 1 by oly a small amout, ad it appears (without proof) that if you take big eough the differece from 1 goes asymptotically to zero. The relatio = c 1 is valid for the data of Fig. 1 ad Fig. 6 if is the expoet for the gasket area. 8. Refereces [1] J. Shier, Hyperseeig (Proceedigs of the 2011 ISAMA Coferece), Jue, p. 131 (2011). [2] J. Shier ad P. Bourke, Computer Graphics Forum, 32, Issue 8, 89 (2013). This is the first publicatio i a refereed joural. The last versio set to the editor ca be dowloaded at [6] or [7]. [3] P. Dodds ad J. Weitz, Phys. Rev. E, 65, 056108-1 (2002) [4] P. Dodds ad J. Weitz, Phys. Rev. E, 67, 016117-1 (2003) [5] G. Delaey, S. Hutzler, ad T. Aste, Phys. Rev. etters, 101, 120602 (2008). [6] P. Bourke, web site http://paulbourke.et/fractals/ [7] J. Shier, web site http://joh-art.com/stat_geom_likpage.html 6 Equatio (3) ca be formulated for ay shape, but requires that a somewhat arbitrary legth must be specified o the lefthad side of the equatio. For example, for equilateral triagles oe could choose the edge legth or the ceter-to-vertex legth. The choice of this legth affects the scale of values (see Appedix A). Shier -- Gasket Area/Perimeter Algorithm -- v. 11 -- Jue 2015 -- p. 10
Appedix A -- Scalig It is possible to put Eq. (3) ito a more uiversal form. et be a legth parameter such as the edge legth of a square. We the defie the area for the shape to be 2 ad the perimeter to be, where ad are dimesioless parameters ( = for a circle if is the radius). If we have a square ad take to be the edge legth we fid = 1 ad = 4. If we istead take the legth to be the half-edge legth, the = 4 ad = 8. The combiatio / 2 will be the same regardless of the choice made for the legth parameter. Usig ad Eq. (3) becomes 1 2 2 2 2 0 i 0 i i1 i1 (A1) 0 i1 i 0 i1 where 0 is the legth parameter for the boudary shape. If we defie a ew parameter g by the g / i. e. g / (A2) 2 0 i1 1 g (A3) 0 i1 2 i i i Whe expressed i terms of ad g oe ca calculate uiversal results, ad get the correspodig values for a particular case usig Eq. (A2). The combiatio / 2 chages if we cosider a differet shape. For example if we have a rectagle with legth 3 times its width ad use the short dimesio as the legth uit, the = 3 ad = 8, so that / 2 is 3/64 rather tha 4/64 as it was for the square. If we use the log dimesio, = 1/3 ad = 8/3. This descriptio assumes that the boudary shape ad the placed shape are the same (e.g., circle). The formulatio is easily adjusted for a boudary shape which differs from the placed shape, or for a lauchig scheme (sec. 4). Shier -- Gasket Area/Perimeter Algorithm -- v. 11 -- Jue 2015 -- p. 11