13  Fractals in crumpled paper 



A Mexican team has uncovered some surprising geometric properties of crumpled paper balls .
The next time you angrily snatch the copy of your love letter or complaint to the crease for a ball to the basket , think before moving to act: a ball of crumpled paper has enough intriguing geometric properties for interest to you and calm your mood .
At the National Polytechnic Institute of Mexico , Alexander Balankin and four colleagues were so interested in how , statistically , the two sides of the sheet is spread over the surface of the ball of paper, and the fragment distribution paper obtained when cutting the ball into two.
For this they have in their hands, crumpled in a ball 100 square sheets of paper and end of five sizes (4, 8 16, 32 and 64 inches square) . After waiting two weeks for the size of the crumpled balls stabilized, they measured the mean radius of each ball . They then performed on these objects two types of experiments and measurements .
A first type of experiments was to paint it black , brush the surface of ten balls of each size and each side of the sheet défroissée then has black spots. Both sides of each leaf were recorded scanner to facilitate measurements and analysis .
A. Balankin and his colleagues have found the results already obtained, including the mass M a ball of crumpled paper is proportional to R^{D}Where R medium is its radius D is its fractal dimension , which measures the number density with which the object fills the space . They are D = 2.27 ± 0.05 ( for a ball of solid material , the dimension D is equal to 3) . Mexican team also obtained the total area painted S is proportional to R^{OF} with D‘ = 2.11 ± 0.05 (for a solid ball that number OF is equal to 2 ). Both satisfy fractal dimensions although the relationship OF = 3 – 2/DWhich was obtained by theoretical arguments and simulations.
In the second type of experiment, for each sheet size , the researchers divided into two ten balls crushed , painted fragments derived from one half black, and then reconstituted leaf by assembling the puzzle pieces .
The analysis of black and white mosaics obtained reveals that the distribution of fragments (black or white ) according to their size T (Measured in pixels) is characterized by a fractal exponent equal to 1.68 ± 0.04 and independent of the size of the sheet . In other words , the number N(T) fragment size black T is proportional to T^{–1,68}. The fractal dimension seems equal to that ( 1.64 ± 0.05), which characterizes the intersection of a ball of crumpled paper with a plan . Is this a coincidence or the translation of properties deeper ? We do not know .
But this work should stimulate further research to better understand the values found and clarify their universality . Beyond paper balls , these are all phenomena of creasing and folding (folding of proteins , membranes, nanoparticles , geological folds , etc. . ) that are potentially affected . This does not mean that these phenomena could be understood and controlled only from descriptions fractal …
Category: physics  Tags: crumpled balls, geometric properties, puzzle pieces 