How I was confused and wrong, and I love it

Mathematical lore has it that, at one point, the mathematician Julia Robinson was asked to keep a timesheet detailing how she was spending her work days. Apparently, one week her time log looked like this: 

• Monday: tried to prove theorem.
• Tuesday: tried to prove theorem.
• Wednesday: tried to prove theorem.
• Thursday: tried to prove theorem. 
• Friday: Theorem false.

So much this. I think the one of the stumbling blocks for people who fear math is that they feel -- for good reason -- that not knowing the answer, or (worse) thinking you know the answer but are mistaken, . . . that those are bad places to be.  But as a person who spends my life thinking about math, I've learned to be comfortable in those very spots.  It's when I am happily exploring the playground of my ignorance that I get to learn the coolest new things.   

So it is that, after my meet-up in Seattle stared me on a new collaboration, that I tried to figure out a geometric construction for something that those algebraists indicated would be possible -- something about turning one conic (like an ellipse or hyperbola) into 5 new conics that had a nifty coincidence. 

The picture I was hoping to construct
(and that eventually, I did!)

My time sheet looked a bit different.

• Thursday:  Rode the train all day, pondering the stuff the algebraists had told me.  They speak another language.
• Friday:  Tried unsuccessfully to construct the conics. None of my techniques worked.
• Saturday:  Figured out a construction method, very involved, but doable, and successfully constructed the conics.
• Sunday:  After zoom church, spent the entire day documenting the construction in excruciating detail.
• Monday: Realized my construction was WAY overly complicated, and there's a much (MUCH) easier way to do it.  Decided to scrap all that past work and start over.

I'd spent a full day trying to carefully document the
construction techniques I'd been using. 
Can you see the influences of the comic-book writer I'd met with?

One of the things this description doesn't explain is how insights tend to come, not while I'm beating my head against the wall, but in times of relative rest.  The breakthrough for my PhD thesis came -- after months of intense work -- while I was washing a kitchen floor.  This week, both of my insights came not while I was cranking through the math itself, but as I was drifting off to sleep.  Or rather, they came as I was trying to drift off to sleep, but then I was so math-excited that it was hard to relax, because I just wanted to jump up and do math again.

My brain at night. 
Or, an overly complicated geometric construction.

Much of the early parts of my trip have been taken up with meetings with people, and this particular Minnesota layover is much less scheduled.  That's turned out to be excellent planning on my part, because having these days completely to myself has been such a gift, getting to live in my own head like this. 

And fortunately for me, I like those intense periods, even when (and sometimes because) I'm doing things in a fog of ignorance that eventually, I hope, clears. The insights come in a flash during the quiet times, but they need those periods of intense work beforehand, or they don't come at all.