## Making The End Of The World Quantitative

I haven’t forgot my little problem about working out where the apparent edge of the world was, from my visit to the Sleeping Bear Dunes in northern (lower) Michigan. What I have been is stuck on a way to do all the calculations in a way that’s clear and that avoids confusion. I realized the calculations were reasonably clear to me but were hard to describe because I could put into similar-looking symbols a bunch of things I wanted to describe.

So I’ve resolved that the best thing I can do is take some time to describe the things I mean, and why they’ll get the symbols that they do. The first part of this is drawing a slightly more mathematical representation of the situation of standing on top of the dune and looking out at the water, and seeing the apparent edge of the dune as something very much closer than the water is. This is what’s behind my new picture, a cross-section of the dune and a person looking out at its edge.

## Just How Far Is The End Of The World?

I got to visit the Sleeping Bear Dunes National Lakeshore earlier this month. I thought I knew what dunes were, from the little piles of sand that accumulated on the Jersey Shore sometimes, but, no. These dunes, at the northern end of Michigan’s lower peninsula, look out on Lake Michigan from, at one spot on the Pierce Stocking Scenic Drive, about 450 feet above sea level. That’s a staggering height, that awed me, particularly as we approached it from the drive, and so had a previously quite nice enough drive through lovely forests come to a sudden, almost explosive, panorama of sand towering far above the ocean.

Maybe a quick working definition of a “scenic overlook” is a spot of land from which you can look noticeably down and see birds flying. This is vey scenic: had there been a Saturn V moon rocket, on the launchpad, at sea level, we would have been looking noticeably down at its escape rocket. For that matter, if there had been a Saturn V moon rocket, on top of which was somehow perched a Gemini-Titan rocket, we’d still be … well, we’d have to look up to see the crew in the Gemini capsule, but we would be about eye level with the top of the Titan booster, anyway.

Something that I imagine no picture except a three-dimensional one is going to capture, though, is the sense that one is standing at the edge of the world. From the top of the dune, the end of the sand seems to be very nearby, maybe a couple dozen feet off, and the water below is so clearly distant that it feels impossibly far away. Walking toward that edge makes the edge of the world recede, of course, but it never quite loses that sense of being on the precipice until quite far along.

Some mad souls do follow a trail all the way down to the beachfront of Lake Michigan; I wasn’t among them. The difficulty in walking back up — all on sand, on a pretty significant slope — from just walking a little near the edge and maybe dropping twenty feet or so in altitude convinced me not to carry on. I didn’t know it was a full 450 feet up, but it was obviously far enough.

The geometry of all this, though, has captivated me, and I hope to spend a couple essays here working out such questions as just how the optical illusion of this edge of the world worked, and how its recession works.