1. 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

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

4. 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

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

7. 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

8. 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

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

11. Aluminum Coated Mylar Film Paper Sandblasted Aluminum Base

14. 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 = 0.3887 ( 0.2579, 0.5195) b = -3.637 (-5.2340, -2.0390) c = 0.4945 ( 0.3562, 0.6328) d = 0.08325 (-0.1832, 0.3497)

15. 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

16. Nils Burglund B. Xia and M. Thorpe, Physical Review A 38, 2650 (1988). EEC-NSF Grant # 1062943 Dr. Takoudis, Dr. Jursich, & the REU staff Tatjana Dankovic, Prateek Gupta, Michael Walsh The University of Illinois at Chicago Citations Acknowledgements