### The Maths of Star Trek: The Original Series (Part II)

This is the second in a series of posts about the maths of Star Trek. Part I covered the probability of survival while wearing a red shirt.

### The Maths of Star Trek: The Original Series (Part I)

As you may well know, Star Trek was a science fiction TV show in the late 1960s. It featured futuristic technology and science fiction ideas such as warp drives, transporters, strange new worlds, time travel, and green alien space babes. And the possibility of all these things has, in the past, been discussed by experts, and nerds, in great detail. Especially that last one about green space babes.

But dammit, I’m a mathematician, not a physicist. So, instead of talking about the science of Star Trek yet again, what about the maths of Star Trek? After all, Star Trek is science fiction, but there is no such thing as maths fiction – so any mathematics featured on the show is sure to be on firmer ground. Right? Or as Spock himself says in ‘The Conscience of the King’;

SPOCK: Even in this corner of the galaxy, Captain, two plus two equals four.

Should we even expect much maths to feature on a simple space adventure show? In fact, many interesting mathematical ideas were raised during the show’s short run of 79 episodes, including; the probability we are alone in universe; a paradox that upset 20th century mathematicians as well as 23rd century androids; the mathematics of alien and Earth biology; and the most important question of all – when on a dangerous away mission, does the colour of your shirt really affect your chances of survival?

### The perfect formula for mathsiness

It’s mid-January, which means it’s time for the tabloids to trot out their annual “this is the most miserable day of the whole year!” story — before they spend the rest of the year blaming immigration, youth and political correctness for problems they’ve spent the last year stoking up.

Ahem.

### AS maths results and batting averages

Phil Harvey gave a talk on this subject at last November’s MathsJam conference. We liked it so much, we asked him if we could put it on the site. Phil’s kindly written his talk up as an essay for us.

I am 64¼ years old and I’ve been a maths teacher all my working life. In that time things have changed. Long gone are the days when gowned masters would sweep in, silence any murmur with half a raised eyebrow, and delight compliant uniformed schoolchildren with chalk-covered boards of mathematical exposition.

No, you’re right. That never happened outside the covers of Goodbye Mr Chips, even in my day.

The reality then. Schoolchildren have morphed into learners. Exam results rule. Quality (in the sense that Orwell might have used the word) is managed by quality managers. And so our working lives are driven by the pursuit of Ofsted targets, success rates, achievement rates, benchmarks, observation grades, results. And every joyless lesson has its own lesson plan, with aims, objectives, learning outcomes and action points. But above all, those damned results – and every year, year after year, they had to IMPROVE.

Well I was no good at any of this stuff – and consequently I always got on very badly with my managers. Until one year…

### Deck the halls with τ of holly, formula-la-laaa!

Christmas is a time for giving, celebrating, family and magic. But did you know it’s also a time for equations? Department store Debenhams has decided to honour this recent Christmas tradition by tasking at least two members of Sheffield University’s undergraduate maths society to come up with formulae for ‘a perfectly decorated Christmas tree‘, picked up by The Sun, The Metro and others.

Photo by Aleksandar Cocek, used under a Creative Commons licence.

Previous festive howlers include ‘the formula for the perfect family Christmas‘ (sponsored by The Children’s Society to promote a book) and a prior stab at ‘the equation for the ideal Christmas tree‘ (sponsored by B&Q), which are just nonsensical strings of abbreviations. However, unlike those examples of naff-ematics, the Sheffield tree-decorating equations make enough sense for me to take a critical, overly-serious look at them on their own merits, and show how you might begin to come up with something more rational.

The formula for the surface area of a sphere, $A=4\pi r^{2}$, is the derivative of the formula for the volume of a sphere: $V=\frac{4}{3}\pi r^{3}$.
This result does not hold for a cube with side length $a$ if the surface area and volume are written in terms of $a$. However, if the surface area and volume are written in terms of half the side length, $r=\frac{a}{2}$, you get the surface area $A=24 r^{2}$, which is the derivative of the volume, $V=8 r^{3}$.