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Pentaplexity: Comparing Fractal Tilings and Penrose Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Pentaplexity: Comparing fractal tilings and Penrose tilings Adam Brunell and Daniel Sherwood Vassar College [email protected] [email protected] | April 6, 2013 Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Overview 1 Introducing Fractal Penrose Tilings 2 Introducing Penrose’s Original Pentaplexity Tiling 3 Comparing Fractal Tiling and Pentaplexity Tiling Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Penrose’s Kites and Darts Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Drawing the Aorta in Kites and Darts Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Fractal Penrose Tiling Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Fractal Tile Set Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Connections How does this Fractal Penrose Tiling relate to the Kites and Darts? We investigate the origins of Penrose’s tiling and then make connections. Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Tiling the Plane with Pentagons [Penrose, 1979] Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Hole Tiles [Penrose, 1979] Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Pentaplexity Tiles [Penrose, 1979] Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Pentaplexity Tiling [Penrose, 1979] Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Claims Mutually Locally Derivable Each tiling is obtained from the other by a simple local rule. Correspondence Between Tiles Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling 5-in Pentagon / Big Star Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling 4-in, 1-out Pentagon / Big Ghost Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling 3-in, 2-out Pentagon / Bendy Bit Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling What’s Left? Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Rhombus Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Boat Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Star Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Corresponding Tiles Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Corresponding Tiles Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling Further Questions Adam Brunell and Daniel Sherwood Penrose Fractal Tilings Introducing Fractal Penrose Tilings Introducing Penrose’s Original Pentaplexity Tiling Comparing Fractal Tiling and Pentaplexity Tiling References Roger Penrose (1979) Pentaplexity: A Class of Non-Periodic Tilings of the Plane The Mathematical Intelligencer 2(1), 32 – 37. Natalie Frank (2013) Personal communication with the authors Adam Brunell and Daniel Sherwood Penrose Fractal Tilings.
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