let's take a hypercube; you're going to build a hypercube. >> i'm going to build a hypercube out of a yardstick. >> out of a yardstick. so you're some four-dimensional being who has this luxury. you've built the hypercube and now we're kind of used to looking at this picture. >> there it is. so i'm going to guess, okay, that i'm going to need three raised to the fourth power of your hypercubes to fill up my hypercubes. so three to the fourth is 81. so that's my guess. >> and let's watch. eighty-one of my cubes fill up your hypercube. so this is -- >> it's all true. i couldn't have seen it, but it's all true. >> and this notion that we're using an analogy to understand dimensions is going to be very powerful for our next understanding because we've just gone through some nice friendly dimensional counts. let's build something a little bit more interesting. let's build something called a koch snowflake. so what i'm going to first have you do is take your yardstick and build an equilateral triangle. all the sides are the same. >> that i can do. >> that we can do. and now i'm going to hav