What useless things did you understand recently?

Please re­ply in the com­ments with things you un­der­stood re­cently. The only con­di­tion is that they have to be use­less in your daily life. For ex­am­ple, “I found this idea that defeats pro­cras­ti­na­tion” doesn’t count, be­cause it sounds use­ful and you might be de­luded about its truth. Whereas “I figured out how con­struc­tion cranes are con­structed” qual­ifies, be­cause you aren’t likely to use it and it will stay true to­mor­row.

I’ll start. To­day I un­der­stood how Heyt­ing alge­bras work as a model for in­tu­ition­is­tic logic. The main idea is that you rep­re­sent sen­tences as shapes. So you might have two sen­tences A and B shown as two cir­cles, then “A and B” is their in­ter­sec­tion, “A or B” is their union, etc. But “A im­plies B” doesn’t mean one cir­cle lies in­side the other, as you might think! In­stead it’s a shape too, con­sist­ing of all points that lie out­side A or in­side B (or both). There were some other de­tails about closed and open sets, but these didn’t cause a prob­lem for me, while “A im­plies B” made me stum­ble for some rea­son. I prob­a­bly won’t use Heyt­ing alge­bras for any­thing ever, but it was pretty fun to figure out.

Your turn!

PS: please don’t feel pres­sured to post some­thing su­per ad­vanced. It’s re­ally, hon­estly okay to post ba­sic things, like why a stream of tap wa­ter nar­rows as it falls, or why the sky is blue (though I don’t claim to un­der­stand that one :-))