Percolation Properties of Triangles with Variable Aspect Ratios Gabriela Calinao Correa University of Massachusetts at Amherst Department of Electrical & Computer Engineering Dr. Alan Feinerman University of Illinois at Chicago Department of Electrical & Computer Engineering

Outline Percolation Overview Methods & Analysis Matlab AutoCAD Data Conclusions Acknowledgements & Citations

Percolation Theory Overview Percolation: study of random networks & their nature of connectivity Percolation Threshold ( p c ): Below p c no giant connected component exists Above p c a giant connected component exists

Methods & Analysis Data Percolated Aluminum Mylar CO 2 LASER AutoCAD MATLAB Resistance Measured pcpc

Matlab Define Parameters Percolation Area Location in AutoCAD file Dimensions Values Kerf Node Size Triangle Aspect Ratio Area Location Pseudorandom Within Boundary Orientation Pseudorandom Angle Rotation Matrix Generate Triangles AutoCAD CAD Commands Define Shapes Determine Area Loss Binary Matrix 1 - indicates Node removed 0 - indicates untouched Node Threshol d Label Clusters Find if Path Exists Percent Area Left Percolated

Logic for Triangle Generation if intersection exists if i in box edge on box add 1 st intersect add point i if i+1 in box add point i+1 if intersection DNE add 1 st intersect two points in box YesNo add point i+2 add 2 nd intersect if triangle on corner add corner point Draw Whole Triangle i+1 i i+2 intersection point

AutoCAD Loads file made by Matlab Defines Laser Settings Creates Parts o Guides o Acrylic Fasteners

Aluminum Coated Mylar Film Paper Sandblasted Aluminum Base

Exponential Relationship Percolation Properties of Random Ellipses B. Xia and M. Thorpe, Physical Review A 38, 2650 (1988). p c = exp(-πabn c ) p c is the percolation threshold n c is the hole density per unit volume at p c b/a is the aspect ratio of an ellipse x = Aspect Ratio f(x) = Percolation Threshold f(x) = a*exp(b*x) + c*exp(d*x) Coefficients (with 95% confidence bounds): a = ( , ) b = ( , ) c = ( , ) d = ( , )

Conclusions Generated Data from o Matlab o Experimental Setup Exponential Relationship o Triangular relationship ~ Elliptical relationship Additional Experiments o Refine Data o Percolate over Larger Area Further Geometric Exploration: o Double Triangles  e.g. equilateral triangle to equilateral rhombus

Nils Burglund B. Xia and M. Thorpe, Physical Review A 38, 2650 (1988). EEC-NSF Grant # Dr. Takoudis, Dr. Jursich, & the REU staff Tatjana Dankovic, Prateek Gupta, Michael Walsh The University of Illinois at Chicago Citations Acknowledgements