Monday, May 22, 2006

Do Tilted Coins Look Elliptical? (Part One)

Put a coin on your desk and look at it from an angle. Is there some sense in which it looks elliptical? Look out your window at a row of receding streetlights. Is there some sense in which the farther ones look smaller?

Many philosophers have said such things -- from Malebranche in the 17th century through contemporary philosophers of perception Michael Tye (e.g., here) and Alva Noe. But is this right?

One reason to have some doubts is this: It's just not clear what the geometry of such a view is supposed to be.

Tye and others have suggested that it involves something like projection onto a plane perpendicular to the line of sight: If you drew a straight line from your eye to the object, then interposed between yourself and that object a plane perpendicular to the line, what kind of shape would you have to put on that plane to perfectly occlude the object? An ellipse, in the case of the coin. A smallish figure in the case of a distant streetlight, a larger figure in the case of a nearer streetlights.

So far, so good. But the problem with doing the geometry that way is that lines projected onto the plane from objects off to the side will intersect the plane obliquely, with the consequence that they will appear much larger in the plane than their straight-ahead counterparts -- weirdly larger, if projective size is supposed to correspond to experienced size. My friend Glenn Vogel drew me up a figure that very nicely illustrates this point:

See how the sphere to the right makes a much bigger shadow in the plane?

One could avoid the problem of the projective size of objects off to the side by making the projective surface, between you and the objects, a sphere rather than a plane. Imagine bending the plane in the figure back to the right, wrapping it around until it was an even sphere encircling the point where the lines converge. Then the projective shadows would be the same size. Projecting onto a sphere rather than a plane would also respect the idea that apparent size varies with visual angle subtended.

But now we've lost our ellipse! The ellipse is a planar figure. Projecting onto a sphere generates not an ellipse but rather a concave ellipsoid.

Should we say, then, that strictly speaking the coin looks concave? That would be strange! In fact, that seems just plainly, observably false -- even with a larger circular object, like a plate, held close to the face so that its concavity given a spherical projection would be considerable.

So the puzzle remains: Is there some way to make sense of the geometry of a view according to which circular objects, viewed obliquely, look elliptical and distant things look smaller than their nearer counterparts? Despite the frequency with which philosophers say such things, no one has adequately explained how this is supposed to work.

(Steven Lehar has perhaps come closest, attempting to get very clear about the geometry; but I suspect that his view will end up making the coin look concave.)

In Part Two, I'll say a bit about what cultural influences and metaphors might be driving all this. You can also look at my essay here.

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