There’s usually more than one way to get from A to B.
The way this is treated in algebraic topology
Poincaré considered [in Analysis Situs] the question of what essentially different surfaces are possible. We’re not interested … to distinguish … surfaces [that] … happen to be scaled differently, but we are interest in distinguishing if they have differences that are independent of scaling factors.
To attack the question, Poincaré studied closed curves on the surface. A closed curve is something that is parametrized by the interval [0,1] and such that it starts and ends in the same point. So, mathematically, it’s a function ƒ: [0,1] → 𝐗, with f(0)=f(1).
The normal way to study these entities is to also require all the curves we’re looking at to start at one specific point x₀, so we require ƒ(0)=ƒ(1)=x₀. Should we have two such functions, ƒ and g, we can compose them by taking h(t)=ƒ(2t) during [0,½] and h(t)=g(2t−1) during [½,1].
If you’re thinking about the Ministry of Silly Walks, then
you want to not only parameterise hands and elbows and necks with paths (not a huge deal; we’re simply repeating the same conceptual process as Poincareacute;
but you also want to notice that paths can take a few steps backwards or spin around in a circle and wait for a while, or take a small detour
whilst remaining essentially the same or at least the same in some way.
