Great lecture. Explains how when complex numbers were defined (a + bi) we had to work in a space that was not the 3D space we live in, these imaginary numbers were often intermediates in real world 3D calculations. So then we tried 4D and 5D. Turns out only 1, 2, 4 and 8D work and latter is the math of quantum mechanics.
The spooky part is one diagram he shows that's a digraph of the algebra of 4D space. It looks for all the world like the all seeing eye. The 8 dimensional one looks like a Spirograph picture.
I'd considered that there may be more dimensions than 3 - actually ours is 4, not 3, time is the 4th a picture of a tulip only looks that way at that exact instant in time. And the math works so why couldn't there be more dimensions. What I hadn't thought was there may not be 5, 6, 7, 8, 9 etc, there may only be 1, 2, 4, 8, 16...
The other thing I thought of is if we're looking for experimental proof of 8th dimensional strings, then we need to look in the 7th dimention for effects because that's where we'll "see" it frozen in it's dimension of time. That is, I suspect for any dimension n, you can see a "slice" of it in dimension n-1, and if subtracting one from the dimention gives is that thing frozen in one sense of its "time" then this gives rise to the idea (which seems obvious now) that there are not only other dimentions of space, but that there are other dimesions of time as well. It's hard enough to picture a 4 dimentional shape in 3D, trying to wrap ones head around greater dimensions of time is a bit of a mindfuck, but, it would define a new set of natural laws as abstract to us now as octonion math and quantim strings would have been to Newton. Ok, bad example, he would have probably have got it if it were explained to him but you get the idea. Bazinga.