The End Of The World, Qualitatively Explained


So I want to understand the illusion of being at the edge of the world at the Sleeping Bear Dunes in northern Michigan. Since I like doing mathematics I think of this as a mathematics problem; so, I figure, I need to put together some equations. Before I do that, I need to think of what I want the equations to represent, which is the part of the problem where I build a model of the dunes. In the process I should get at least a qualitative idea of the effect; later, I should be able to quantify that.

What’s a dune? Well, it’s a great whomping big pile of sand right next to the water. There’s more to a dune than that, but since all I’m interested in is how the dune looks, I don’t need to think about much more than what the dune’s shape is, and how it compares to the water beside it. If I wanted to understand the ecology of a dune, or fascinating things like how it moves, then I’d have to model it in greater detail, but for now I’m going to try out this incredibly simple model and see what it gets me.

I’ll pretend that the dune is a cylinder. To the extent the word “cylinder” is used outside mathematics that’s usually taken to mean a pipe’s shape. Mathematicians are more lenient; they’ll usually let you call a cylinder any three-dimensional shape that looks the same if, whenever you take a cross-section of it at the same angle, you get the same shape back. In this view, tubes are cylinders, rectangular boxes are cylinders, even (perfectly formed) Bug and Insect shape Macaroni and Cheese are cylinders, since each cross section perpendicular to the straight walls of the macaroni shells are the same shape, at least before cooking.

The dune’s almost certainly not a cylinder, since some parts of it are higher than others, but, is it going to be very different from a cylinder? Well, probably not, if I’m standing in the right spot and only care about the section of dune immediately around me. This is why my sketch has a cross-section of the dune, showing it as some kind of curve sloping from its considerable height (about 450 feet above Lake Michigan), according to the National Park Service) down to Lake Michigan (about 0 feet above Lake Michigan).

The sketch of the person is horribly done, and apparently shirtless, but certainly rather tall if the dune is 450 feet above the water’s edge. No matter; a more accurate drawing would just hard to read at this level, although imagining the more accurate drawing will reveal something important.

The dotted line going from eye level to the edge of the dune to the lake beyond reveals much of the mystery. This is meant to represent what’s directly in front of the person standing on the dune. Looking a little bit down is the sand; looking a little bit up, the water, and the water is obviously much farther away. This seems to explain a lot of the edge-of-the-world illusion: we may not be sure just how far away some bit of sand is, but it’s going to be very obviously a lot closer to the viewer than the water is. And the steady progression of sand down to the water is invisible, so, all we see is the nearby sand and the great beyond.

Imagining a more accurate drawing, incidentally, makes the reasons behind the illusion more obvious at least to me. Grant that the left edge of the picture ought to represent 450 feet; then, any reasonably-sized person is going to be pretty near one-hundredth the height of the left edge above the water. The dotted line we might imagine going from the person’s eye to the edge of the dune to the water beyond, then, last touches the sand incredibly closer to the viewer than even the nearest water beyond is. And that’s going to stand out: as the viewer walks around a little bit, the parallax — the change in where the edge appears to be — is going to be quite prominent compared to the subtle shifts in the water’s position.

There are some other effects at work, surely. One has to be that the imagination pretends that there must be an abrupt drop just past the last bit of visible sand. I’m sure there are people whose imagination doesn’t work that way and they probably can’t quite get what others see in this illusion. They’ll just have to content themselves with more mundane questions like wondering how far away the edge is and how much does it move if the viewer moves.

Another factor is that I imagine it’s hard, at least until you spend a lot of time really studying sand, to say just how far away some bit of sand is. It doesn’t offer much in the way of obvious markers of how big something is, after all, unless there are people or buildings or other size references in the way. With few cues about how big something is, the mind will fill in all sorts of odd and wrong estimates. This was one of the problems afflicting Apollo astronauts: Apollo 14’s Alan Shepard and Edgar Mitchell famously were unable to reach the rim of the Cone Crater, near their Fra Mauro landing spot, in part because the lunar surface gave few hints about how far they had gone in any particular direction.

I’d be delighted to learn of other psychological effects that produce this illusion, which is thrilling and which just can’t be properly bound up in photographs; please feel free to comment about any you know. But with my model in place and the qualitative idea laid down — you can see the sand up to some point, and then the water beyond, and the water is obviously much farther than the sand — I’ll be going over to a quantitative understanding of this thing.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

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