Funny Discrete Pattern

Random number: 19

19th April 2017

Yes, it's been a century since I last wrote something. No; that's not because I haven't been doing math, rather I've been doing so much homeworkey math in actual math classes. Good life.

Anyhow, look at this! This pattern can be represented as a matrix of infinite size, whose entries \(a_{ij}\) are described by the following rule:

Let \(\alpha\) be the entry \(a_{(i-1)j}\) and let \(\beta\) be the entry \(a_{i(j-1)}\). Then let \(\gamma_\alpha\) be the number of consecutive terms \(a_{(i-1)j}, a_{(i-2)j},...,a_{(i-\gamma_\alpha)j}\) such that they are all equal to \(\alpha\). Define \(\gamma_\beta\) in the natural dual manner. Then set \(a_{ij} = (\gamma_\alpha + \gamma_\beta) \text{ mod } 2\). This defines a matrix of zeroes and ones. Also, set \(a_{00} = 1\) (this is important!)

Or; as a special friend put it, : "Parity of runs top and left XORed"

Funny pattern! Many natural questions to ask; about repetition, etc. Also been thinking about how eigenvalues/eigenvectors generalise to infinite matrices; but that might be out of my depth. We can consider a family of patterns by changing the modulus: see here. PSYCHE! Actually, when you try modulus by anything 3 or greater, you get a completely regular pattern! Two is the lucky; chaotic nugget! What a wonderful world we live in. And also; maybe think about the density of 0's versus the density of 1's. There's heaps of ideas.