PROPERTIES OF FRACTAL SQUARES Several years ago while studying ractals, including the Koch Fractal Curve and the Sierapinski Triangle, we ound an interesting new igure based on a zeroth generation square. We termed the higher generation ractal arising rom this square the Black Snowlake. We wish here to look more at the properties o this snowlake and also another related ractal. Our starting point is a igure o the our-old symmetric Black Snowlake shown- It is constructed by use o the concept o generations. Here the zeroth generation is a square o area A 0 = L. Attached to this square are our irst generation squares o area A 1 =L /9 each. Next one orms the second generation by attaching 6 even smaller squares o area A =L /81 each to the irst generation. Note that, unlike or a Koch Curve, the nth generation may only touch the (n-1) generation and never the (n-), (n-), etc generations. Continuing the addition o generations out to ininity, we get a total inite area o- A L 1 6 1 1 5 1... L 1 [1...] L 4 9 81 79 by use o the geometric series. We note that the distance rom the center o the zeroth generation out to the ininite generation is just-
1 1 1 1 L... 1L 9 7 This means that i we enclose the snowlake in the smallest possible square, the square will have an area o L. So that the ratio R between Snowlake area and to the circumscribing square area, rotated by 45deg, will be- 5 R 8. percent 6 In terms o a Hausdor dimension, the Black Snowlake has the dimension- d ln(5) ln() 1.46497... We can also calculate the perimeter o the snowlake by noting that the zeroth generation has the perimeter- P 0 =L{4-(4/)}=8L/ since part o its perimeter is blocked by the next generation. The contributions to the perimeter o all subsequent generations is also ound to be 8L/ or each. Thus the perimeter o the entire Black Snowlake, going out to the ininite generation, is ininite. Such behavior is to be expected when dealing with ractals having non-integer dimensions. Since a condition or the Black Snowlake is that only the (n+1) and (n-1) may touch the nth generation, it is clear that there must be a restriction on the side length ratio o the nth generation square. To ind what this restriction is, we look at the ollowing diagram-
We see that each generation is represented by squares o side-length n L. So looking at the part going down rom the L square, we must have- L L L 4 L 5 L,,, Cancelling out the L/ term and making use o the geometric series since <1, we ind- This solves as 1 1 0.414156... The Black Snowlake which has =1/ alls into the non-overlap case, but taking =1/ would produce a deinite overlap ater the third generation. To see how many generations it would take to produce an overlap when >sqrt()-1, we must have 4... n Ater application o the inite geometric series, this is equivalent to saying n 1 or the beginning o overlap. Thus when =1/ overlap begins with the third generation(n=). One can produce many variations o the Black Snowlake ractal. One o the most interesting new conigurations which we have just ound starts with a square o side-length 4L or the zeroth generation. We break the sides into our equal length segments o L each and draw on each o the sides an outward square ollowed by an inward square o area L representing the irst generation. Since there is no change in eective area, the total area o the irst generation remains at A=16 L.. However, the perimeter increases by a actor o two to L. The irst generation looks like this-
Following the same procedure or the second generation, we ind the intricate ractal pattern It took some eort to draw this second generation ollowing the same double pulse procedure. When I irst saw this pattern it reminded me o a lion, a dancing clown, or a swastika. What struck me on irst seeing the igure was that its area
remains the same as the original 4Lx4L square while the perimeter has increased by a actor o our. Clearly the ratio o perimeter to area will continue to increase with ever higher generations. This act is something to keep in mind or those individuals involved with convective heat transer or the design o improved air and water ilters. The our old symmetry o the ractal square is shown more clearly in the ollowing graph or the second generation- We see that each quadrant contains 64 squares so that the total area equals- 64x4xL /16=16L and so remains unchanged rom earlier generations. August 015