pentaplexity: comparing fractal tilings and penrose...
TRANSCRIPT
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
[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 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