Introducing Fractal Penrose Tilings

Introducing Penrose’s Original Pentaplexity Tiling

Comparing Fractal Tiling and Pentaplexity Tiling

Pentaplexity: Comparing fractal tilings andPenrose tilings

Adam Brunell and Daniel Sherwood

Vassar College

adbrunell@vassar.edu | dasherwood@vassar.edu

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 andDarts?

We investigate the origins of Penrose’s tiling and then makeconnections.

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 DerivableEach 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