Great Words In Science Number Two: Winding (number)
A lot of people’s view of topology stops short at “a cup of coffee is a doughnut”, and that’s a perfect place to stop, because it seems like nonsense/obvious to pretty much everyone. The people that start asking about why, if we have some magic substance that you can transform between one thing and another, you can’t stick holes in it, are often treated as not getting it. They are the people to watch.
This is most of science – the twin questions “How is this like that?” and “How are this and that different?” Super extra hard science may involve three things.
An visual way of looking at it is that every space is viewable as a table(a plane) with some or none spikes sticking out of it. A loop of yarn (a path) can be threaded around on the table, and you can move it around as you want, but if you can’t lift it off the table, you can’t change a loop that goes around a spike once to one that goes around it twice.
This is topology’s refinement/reversal of the questions – is thing1 the same as thing2 in the same way as thing3 is the same as thing4 Vs if these things are the same, how many ways can we write the rules (“You can’t lift the thread from the table” = “You can’t push holes in the mathematical playdough”)?
The winding number is just what it says, the number of times that a loop winds around a spike, the irreducible part. The city I walk around in is a big table with hundreds of spikes. Through habit, there must be buildings around which my life has a winding number in the thousands.