Wednesday 14 October 2020

The math of butane

 . . . That would be now, 11-18th October 2020. There's a heckuvalot of events round the country, mostly aimed at schools; because if you haven't "got" math by the time you leave school, people will tend to give up on you - and you may even have given up on yourself as a numerate person. 

Mathematics is not about numbers, or that is only a tiny part of the circus. Maths is more about patterns and connexions and relationships. I have found it really helpful, teaching remedial math these last 8 years, to recognise that the ancient Greeks did all their maths scratching patterns in the sand of the Agora. Algebra came much later and was able to embrace levels of abstraction / unreality which were beyond geometry. By coincidence, I was browsing through Alex’s Adventures in Numberland by Alex Bellos [prev] and found him citing The On-Line Encyclopedia of Integer Sequences. I've grazed past this planet before.

Sequences are the epitome of patterns and so feature quite strongly on intelligence tests. "what are the next two numbers in this series: 1 4 9 16 25, _, _ ?" You'd think that a database of counting-number sequences would start simple and get more challenging but not so. Frankly Scarlett, I've no idea what the first 3 even mean

  • A000001: Number of groups of order n  
    • 0, 1, 1, 1, 2, 1, 2, 1, 5,
  • A000002: Kolakoski sequence: a(n) is length of n-th run 
    • 1, 2, 2, 1, 1, 2, 1, 2, 2
  • A000003: the class number of the quadratic order of discriminant D = -4n  
    • 1, 1, 1, 1, 2, 2, 1, 2, 2,
  • A000004: The Zero Sequence
    • 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
  • A000005: the number of divisors of n.
    • 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2
There are now more than 300,000 different integer sequences. Some more familiar than others: Fibonacci, squares, cubes, odd-numbers, primes . . . some children who have been taught maths by bores can be quite excited by discovering unexpected relationships: between odd numbers and squares for a simple example: 1 (+3) 4 (+5) 9 (+7) 16 . . . which I riffed on a month ago in my bonnets off to Grace Cunningham.

You could do worse than using using OEIS as a sort of divination to see how you should approach the day. Can't be less informative that reading "your" horoscope in the Daily Express. You could pick a random OEIS series every day for 100 years without needing to repeat the cycle. My "favorite" number is 59 and A000059 turns out to be graspable: 
  • A000059: Numbers n such that (2n)^4 + 1 is prime  
    • 1, 2, 3, 8, 10, 12, 14, 17, 23, 24, 27 . . .
When I was 17, I switched rather abruptly from The Arts Block to Science at school. The first lesson in Organic Chemistry was about alkanes: methane, ethane, propane, butane some of which you'll recognise if you're cooking' on gas. Here are them boys are with their structures:
That's a series of sorts: each member of the alkane family has 1 extra Carbon and 2 extra Hydrogens; so far so boring. Mr Yates the chemistry teacher pointed out that butane was interesting because there were two structural forms of the molecule: n-butane, shown above, with all the Cs in a row . . . but also iso-butane where the backbone is branched as shown [R]. "Before the next lesson you may like to think about how many different structures are possible for the higher alkanes", said Mr Yates with a brief rotation of his left hand. I was on! and spent the next couple of days doodling out the permutations of structures for pentane through octane. As the carbon chain gets longer the number of structures grows exponentially and these are in OEIS:
  • A000602: number of n-carbon alkanes C(n)H(2n+2)
    • 1 methane, 1 ethane, 1 propane, 2 butanes, 3 pentanes, 5 hexanes, 9, 18, 35, 75, 159, 355, 802, 1858
Like I said, math / science is largely about recognising connexions. That's why it is a desperate shame that Chemistry is taught in a different room to Mathematics and quite probably by a different person. This is not the best way to light school kids up to reflect about how their world [and mine, and yours] is arranged.

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