Numb-er. Ouch! Thank you.

I didn't know about that rule. For me, 37A stood out as a bit of a giveaway. I saw there were lots of boxes and just started filling them in, hoping the answer in my head would fit (it did). Like you, I also had to revisit some clues when I was done in order to interpret them. I still don't get 53A.

“Bated breath” not “baited breath”

I like this form over the e^(i pi)+1=0 form because it makes an easier bridge to visualizing how multiplying unit exponentials corresponds to rotation on the complex number plane. On the other hand, getting the 0 in there is pretty elegant, too!

I love those graceful curves, and the ways those challenging shapes interlock in ways not just aesthetically beautiful but anatomically functional. I see these and imagine how gorgeous they would look in motion. Fiddly bits for the win!

Thanks, Kostas!

I liked the sing-ularity. Presumably when there will be more harmonies present then people to sing them?

Of course. My reason for commenting was that you show the points x, f(x), and g(f(x)) explicitly labeled, but the arrows don't visibly align to those points. I can see the tension between generality and a specific example though.

These are wonderful drawings! But I don't get the f and g arrows in the conceptual diagram. I'd expect f to start at x in the data space (the black dot), and the g arrow to end at x-hat (the other, nearby black dot). Currently, f seems to start, and g seems to end, at arbitrary locations on M.

Who is that grizzled man of the woods?