Well, no. You can only do paths

*p* and

*q* separately. If you do them together, the function

*p**

*q* wonâ€™t be well defined. :?

*p**

*q* means: do path

*p* first, then

*q*.

You see,

*t* is a variable that goes continuously from 0 to 1. As it does so, the path

*p* traces out a continuous â€śrouteâ€ť in the topological space

*X* from

*a* to

*b*. Similarly the path

*q* traces out a continuous â€śrouteâ€ť in

*X* from

*b* to

*c*.

Now suppose you want to get from

*a* to

*c*. Since the path

*p* ends at

*b* and the path

*q* starts at

*b*, you can make your journey via

*b* using paths

*p* and

*q*. But thereâ€™s a catch: you still have to make your journey as

*t* moves from 0 to 1 â€“ no faster, no slower. So, what do you do? Well, the trick is to move twice as fast as you would if you were just taking path

*p* or path

*q* alone! :P

So if you go through path

*p* twice as fast, youâ€™ll reach

*b* at

*t* = Â˝ rather than

*t* = 1. This will leave you time to go through path

*q*, moving at the same double speed, and get to

*c* when

*t* hits 1. This is why you have p(2

*t*) rather than p(

*t*) in the formula for the product path â€“ because you want to move at double speed. Same for

*q* â€“ except that for

*q*, you start at

*t* = Â˝ rather than 0, so the formula is q(2(

*t*−Â˝)) = q(2

*t*−1) instead.

Similarly if you have three paths. But note Exercise 1: the product of paths is not associative!