Season 2 Episode 1
Flora's back on the livestream with Penrose tiles! In this episode, we'll show you why the Penrose tiling is special, how it's constructed, and why it works. There's also a bit where James and Flora try to remember the digits of pi, with mixed results.
Flora’s title slide included an image of some deformed Penrose tiles from this article, which talks about crystallography, if you’re looking for an application of the maths we talked about on the livestream.
The Wikipedia article on Penrose tiles is pretty good, and includes a clue about why the golden ratio is involved.
For a recap of the composition rules, and to read (much, much) more about the Penrose tiling, see this AMS article. That includes the fact that the ratio of thick tiles to thin tiles in a large patch of Penrose tiling tends to, you guessed it, the golden ratio.
For a time-lapse video of the laying of the Penrose Paving outside the Mathematical Institute, see this video.
Making tiles out of tiles reminded me of rep-tiles! You already know how to put together four equilateral triangles to make a larger equilateral triangle, or how to put nine squares together to make a larger square. Can we do something like that with other shapes? The Wikipedia page on this is worth skimming through just because of the sheer density of different images and animations.
Adding to ten
While we were talking about fitting Penrose tiles together, a question came up along the lines of “how many ways are there to make ten using numbers from 1 to 4”? There are two versions of this problem depending on whether you count 4+2+2=10 to be the same as 2+4+2=10, or whether that counts as two different ways to make 10 (if the order matters). Maybe that looks a little bit odd because, of course, 4+2=2+4. But the order could matter if (for example) we’re thinking about putting Penrose tiles together, where the order matters because it determines which tiles are next to each other (and they have rules about being next to each other). For another example, football managers describe their line-ups in terms of how many players they’ve got at each position, and “4-2-2” is very different from “2-4-2”; one of those has more midfielders. So there are two versions of the problem
- (order doesn’t matter) How many sets of numbers are there such that each member of the set is an integer between 1 and 4 (possibly with repeats*) and such that sum of the elements of the set is 10?
- (order matters) How many finite sequences are there such that each term is an integer between 1 and 4 and such that the sum of the terms equal to ten?
*sets aren’t really supposed to have repeats, so this is actually a “multiset”. It’s not important but I don’t want angry emails from set-theorists.
Hints: for Q1, I’d encourage you to work systematically in terms of how many times each integer appears in the set. For Q2, the answer is huge, probably far too many to work out by hand. I’d encourage you to start with smaller totals (try sequences that add to 5 or 6), and try to spot a relationship between large totals and smaller totals.
Extension: (difficult) The Penrose tiles have rules about which ones can be next to each other. Out of the massive list of possible combinations that add up to 360 degrees (Q2 above), how many correspond to possible corners in a Penrose tiling?
This might look like a random choice of coefficients – try $x=1$, $x=2$, $x=3$ to see what I've done.
Extension: how did I find these coefficients? Can you make a cubic go through points $(1,f(1))$, $(2,f(2))$, $(3,f(3))$ for your own choice of function $f(x)$ ?
If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk.