Fractals in Italy: a High School project

Transcription

1 1 Fractals in Italy: a High School project Gabriele Brambilla Istitute E. Breda; Liceo Scientifico Tecnologico, V^ A; Sesto San Giovanni, Milano In response to an e mail received last year I have been advising Gabriele on a project looking at the geometry of electrochemically generated fractals. The report L INFINITO NEL FINITO: Frattali e Diffusion Limited Aggregation (or less evocatively in English The Infinite Into The Finite Fractals And Diffusion Limited Aggregation ) is a scientific report of exceptionally high standard. It shows how a good idea can result in quality work, even if the scientist is still at school, does not have access to high tech resources, or is not working in a team of experts. The report also is a nice example of science progressing by careful observation (see Gabriele s comments on his anomalous point in figure 4). Gabriele worked with classmates Giacomo Brogin, Simone Mangano, Iacopo Pancotti and Jacopo Pellegatta, whom he rightly acknowledges as collaborators. The extracts below (slightly edited by DBH, with further comment in [square brackets], and renumbering of figures) give some details of the experiment, and show the results and interpretations made by Gabriele. Please contact him at: gabritenebra@gmail.com for a copy of the report The experiment Apparatus Materials Voltage generator (voltage range from 0 to 25 V, sensitivity of ± 0.2 V). Petri dish (radius 5.5 cm). Petri dish (radius 9 cm). A copper wire (diameter 1 mm). A copper annulus (thickness 1 mm, internal radius 5 cm and external radius 6 cm). Chromatographic paper with specific gravity of 150 g/cm 2 (recommended: Aldet 205). Supports. Electric cables for the generator. Aqueous solution of CuSO 4 (1 M)

2 2 Aqueous solution of H 2 SO 4 (1 M) H 2 O Experimental arrangement Figure 1: Experimental set up. Procedure 1. Cut the paper in circles with radius 6 cm. 2. Place the smaller Petri capsule upside down inside the bigger one. 3. Place the annulus on the smaller capsule and connect it with cables to the positive pole of the generator. 4. Saturate the paper with an equimolar solution of H 2 SO 4 and CuSO 4 and place it over the annulus. 5. Connect the copper wire to the negative pole of the generator. 6. Place the copper wire in the centre of the paper without breaking the paper. 7. Turn the voltage source to the desiderated voltage. 8. When the cluster has reach the desired dimensions turn off the generator and wash the paper in water to avoid that the acid and the oxidation that it causes damage the fractal structure. To obtain clusters with an easily observable structure select a voltage between 1 and 20 V. To keep the equipment fixed it is advisable to use supports. It is extremely important to often clean the electrodes to avoid a non uniform distribution of the potential (causing the deformation of the structure). For electrodeposition with a difference of potential (d.d.p) 10 V strips of paper (pescanti in figure 1) must be used that draw H 2 O in the electrodeposition paper from the bigger Petri capsule; this is necessary because the process of aggregation at these values of d.d.p. is very slow and the

3 3 evaporation of the solvent can dry out the paper causing the deposition to stop. [These paper strips act as wicks]. The experimental procedure can take from half an hour to several hours, depending on the chosen voltage. Results Figure 2: Cluster obtained with solution 1M and with a voltage of 2V. As it can be seen from the image every branch is different from the others. In spite of it I noticed that in the different clusters some structures of branches appear with a greater frequency, on which I based the mathematical model to describe the changing in the structure.

4 4 The model consists in a father branch (X) which give origin to three son branches (a,b,c) of which each one presents an omotetia factor and a rotation angle independent from the ones of the others. Figure 3 Illustration of the method of misuration on a branch of a cluster. X is the father branch, while a,b,c are the son branches. The model This kind of model is obtained combining rotations, omotetia and translations. [Thanks to Wikipedia and Google language translator, omotetia = dilation: A dilation is a specific geometric transformation of the plane or space, which expands or contracts the objects, while maintaining its shape and orientation.] Here we report the geometric transformations singled out with the Barnsley Theorem, that applied to the father branch compose my model: Son branch a Son branch b

5 5 Son branch c Where,, are the dilation factor of the respective branches and,,, are the coordinates of the terminal points of the segment X (father): max is the terminal point from which the son branches are originated (a, b, c) while min is the terminal point from which the fatherbranch was originated. The dilation factor and the rotation angles have to remain constant in the whole development of the structure (in which every son branch becomes a father branch for three new branches) to confirm the auto similarity that makes this structure a fractal. Selected Results The data indicate that the ratios between the son branches (a, b, c) and the father branch (X) increase with the increase of the applied voltage (the dilation factor increases), while the values of the rotation angles between the son branches and the father branch decrease. In the schedule of the mean of and this can be caused primarily by two things: (figure 4) we find an anomalous result on the fractal built at 10V 1. The wideness of the angles that must be considered is the addiction of and, and not the single one, because the structure of the cluster doesn t have reasons to prefer right or left (in fact the graph of figure 5 seems to confirm this hypothesis). 2. A human error in taking the photograph (but improbable because in that case a value of the dilation factor will appear anomalous). Figure 4 relation between the mean of the value of γ and the voltage.

6 6 Figure 5 Relation between the value of the addition of the means of α and of γ and the voltage. Simulation To verify the data obtained Iacopo (one of the members of the team) have compiled a program that can simulate the mathematical model hypothesized by me. [C++ and Matlab code are given in the report] Figure 6 building in Matlab of the first group of son-branches originated by the fatherbranch.

7 7 Figure 7 graphic comparison between branches obtained at 18V and their simulation. As can be seen in the figure 7 the graphic of the situation simulated reflects in an enough faithful way the reality of the structure of the cluster. This fact indicates that the mathematical model developed is a good approximation of the situation of the branches of the clusters obtained through the DLA and this signifies that it is adapt to describe this reality. ACKNOWLEDGEMENT I thank the Istitute E. Breda for the tools and the substance supplied, all my teachers of this scholastic year, in particular the Prof. Francesco Driglia, tutor of my project, for all that this has involved, the Prof.essa Elisa Brivio and the Prof.essa Cristina Perini for the great time they have dedicated to the project and the Prof. Fabrizio Favale for the advice; the Prof. David Brynn Hibbert of the University Of New South Wales of Sydney, the Ing. Giacomo Favini, the Ing. Matteo Ceserani and the Prof. Paolo Brambilla of the University of Milano Bicocca for the aid they gave to me and at last (but not least) the Prof. Francesco Prestipino for the encouragement provided during the development of the project.