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