Stephen Wolfram's: A New Kind of Science | Online
Jump to Page
Look Up in Index

Chapter 5 Notes > Section 4 > Page 932 > Note (d) Previous note-----Next note
Notes for: Two Dimensions and Beyond | Substitution Systems and Fractals

*Penrose tilings

The nested pattern shown below was studied by Roger Penrose in 1974 (see page 943).

The arrangement of triangles at step t can be obtained from a substitution system according to

With[{φ = GoldenRatio}, Nest[# /. a[p_, q_, r_] :>
With[{s = (p+φ q) (2 - φ)}, {a[r, s, q], b[r, s, p]}] /.
b[p_, q_, r_] :> With[{s = (p+φ r) (2 - φ)}, {a[p, q, s], b[r, s, q]}] &, a[{1/2, Sin[2 Pi/5]φ}, {1, 0}, {0, 0}], t]]

This pattern can be viewed as generalizations of the pattern generated by the 1D Fibonacci substitution system (c) on page 83. As discussed on page 903, this 1D sequence can be obtained by looking at how a line with GoldenRatio slope cuts through a 2D lattice of squares. Penrose tilings can be obtained by looking at how a 2D plane with slopes based on GoldenRatio cuts through a lattice of hypercubes in 5D. The tilings turn out to have approximate 5-fold symmetry. (See also page 943.)

In general, projections onto any regular lattice in any number of dimensions from hyperplanes with any quadratic irrational slopes will yield nested patterns that can be generated by subdividing some shape or another according to a substitution system. Despite some confusion in the literature, however, this procedure can reproduce only a tiny fraction of all possible nested patterns.


Page image


Pages related to this note:


All notes on this page:

* Sierpinski pattern
* [2D substitution systems with] non-white backgrounds
* Higher-dimensional generalizations [of substitution systems]
* [Substitution systems based on] other shapes
* Penrose tilings
* Dragon curve
* All notes for this section
* Downloadable programs for this page
* Downloadable images
* Search Forum for this page
* Post a comment
* NKS | Online FAQs
From Stephen Wolfram: A New Kind of Science [citation] Previous note-----Next note