On the evening of November 16th, a group of Radley’s young mathematicians met to conclude the sequence of this term’s Maths Society lectures. Four members of the Sixth Form took turns to draw us into an area of mathematics that was of particular interest to them.
James Mocatta initiated the proceedings with Graham’s Number: the largest number to have been used in a mathematical proof. This number is so monstrous that it requires an entirely new method of notation, arrow notation, just to write the number down. Even if every atom in the universe were turned to ink, it would still not be possible to write Graham’s Number in conventional form. Despite its size, the number does have its uses, being required to solve a problem of multiple dimensionality.
Next up was Henry Rees, accompanied by an entirely new question: what is the smallest space required to rotate a rod 360˚? However, this was no ordinary rod. Being a mathematician’s rod, it had zero width and therefore no area. The apparently obvious answer of spinning it in a circle was quickly dismissed in light of a triangular space which, in turn, was left behind in the wake of an alternative shape created by cutting a triangle into n strips and overlaying them.
Ever wondered how you would paint an infinite surface with a finite volume of paint? Ed Day has. He turned to volumes of revolution to tackle the Painter’s Paradox. The volume of the solid generated by rotating 𝑦=1𝑥 360˚ about the 𝑥-axis is known as Gabriel’s Horn. Since the value of 𝑦 tends to zero as 𝑥 tends to infinity, the solid can be said to have a finite volume. All you need to do is fill it with paint.
Closing the evening was Jamie McCulloch, sharing with us his delight of factorials. Once we had covered that 𝑥!=0 (essentially, because it has to), the talk quickly turned to gamma functions. These functions (Euler’s very own) can be exploited to yield answers to the likes of 12! and −2! as well as helping to elegantly solve some pretty tricky integrals.
This evening emphasised how much more there is to explore beyond the classroom. If your head did not hurt you had not been paying attention.
Report by Felix Barbour, F Social, 6.2