We show in this clip that aperiodic Penrose tilings can be viewed as projections of a five dimensional cubic lattice. We focus on the geometric properties of this projection and try to motivate the choices that are required for a Penrose tiling. To cope with the five-dimensional geometry, lots of toy models and analogies are exploited.
content:
0:00 Introduction
0:45 Branding
1:35 Toy model 2D
3:55 Algebra
5:15 Toy model 3D
6:50 Cubes in various dimensions
12:25 A particular rotation in 3D
14:20 A particular rotation in 5D
16:00 The convex hull for 2 co-dimensions
16:55 The convex hull for 3 co-dimensions
17:45 Towards Penrose tilings
18:15 A flatlander's point of view
20:28 The full picture
21:50 Transition to the other Penrose plane
22:16 Sneak preview
references:
Here is a link to Dugan Hammock's community post:
https://community.wolfram.com/groups/-/m/t/2992310
and you might also want to have a look at his higher dimensional excursions on youtube:
https://www.youtube.com/@VJDugan
For a more rigorous mathematical approach we recommend the seminal paper by de Bruijn:
Algebraic theory of Penrose's non-periodic tilings of the plane.
N.G. de Bruijn
Proceedings A 84 (1), 1981
A link to the mathematica notebook that was shown in the video:
https://www.dropbox.com/scl/fi/sabmv2omlltw10rv6lexk/PenrosePresentation.nb?rlkey=0k68ntxeeh2gbjpdoiq2wob7d&st=bkispvcz&dl=0
