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Percolation Properties of Triangles with Variable Aspect Ratios

This research investigates the percolation properties of triangular shapes with variable aspect ratios. Using MATLAB and AutoCAD, the study explores the connectivity of random networks by defining the percolation threshold and clustering logic for triangle generation. Experimental data, including resistance measurements of percolated aluminum and Mylar, reveal an exponential relationship between aspect ratio and percolation threshold. The findings contribute to understanding geometric structures in random networks and suggest avenues for further exploration in percolation theory.

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Percolation Properties of Triangles with Variable Aspect Ratios

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  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

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

  4. Methods & Analysis MATLAB pc AutoCAD Data CO2 LASER Resistance Measured Percolated Aluminum Mylar

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

  6. Logic for Triangle Generation Draw Whole Triangle if intersection DNE Draw Whole Triangle if intersection exists edge on box if i in box if i+1in box add 1st intersect add 1st intersect add point i+1 add point i i+1 intersection point Yes No two points in box i add point i+2 add 2nd intersect i+2 if triangle on corner add corner point

  7. AutoCAD • Loads file made by Matlab • Defines Laser Settings • Creates Parts • Guides • Acrylic Fasteners

  8. Sandblasted Aluminum Base Aluminum Coated Mylar Film Paper

  9. Exponential Relationship Percolation Properties of Random Ellipses B. Xia and M. Thorpe, Physical Review A 38, 2650 (1988). pc = exp(-πabnc ) pc is the percolation threshold nc is the hole density per unit volume at pc 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)

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

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

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