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9 November 2001
Are straight lines curved in the sky? (something to notice toward 23 November)
by Art Winfree
On 23 October I was running outdoors about 4 PM, the half Moon 30 degrees high in the southeastern sky, and the Sun a bit lower in the west. Watch for this in November. The upper right of the Moon is brightly lit, while its lower left is dark.
Moon and Sun diameters are exaggerated here about tenfold

Evidently (one would think) the Sun is somewhere to the upper right to so illuminate that half of the Moon. But look where the Sun really is: at substantially lower altitude! How can this be? Is the Sun not on a straight line symmetrically passing through the bulge of the Moon's brightness? Extricate a shoelace and pull it taut between your outstretched arms. It is straight, and as expected it also looks straight no matter where or how you hold it. Hold it across the Moon's disk, symmetrically bisecting the bright half: the end near your right hand does indeed go through the lowering Sun. This is just as it seemed it must be, but how is this comforting observation compatible with the upper right of the Moon being bright, and the Sun sinking lower toward the western horizon??
This seems very perplexing. A straight line pointing upward and westward seems to bend down to skewer the Sun. Somehow this straight line appears to curve. Let's see if we can figure out how that can be. What else is straight out here? Do any other straight lines curve? The horizon also is straight: hold the string along it, horizontally. Now hold the taut string higher, still horizontal, maybe 30 degrees above the horizon. Still straight, of course, and parallel to the straight horizon... but look how high it is in the middle, and how much lower at the extremities, as though it would intersect the parallel horizon if extrapolated a little further! This string is clearly straight, yet also clearly and acutely bowed upward in the middle.

Whoa! And why don't I see the horizon bowed down, as much as I see the parallel shoestring bowed up? Are these two straight lines somehow not equivalent? Maybe this has something to do with the fact that the horizon has no ends and so resembles a circle in that respect, so it can't be straight even though it looks so?
BTW, the sky presents you this setup twice a month, not just once: at the opposite quarter-moon phase (7-9 November in this case) things look about the same in the early morning, with east and west interchanged.
Here's a related observation. Someday when the air is humid and there are a lot of fluffy clouds around the sunset, notice the rays fanning out from the west as seen in the Arizona state flag. These have to be parallel lines in 3d, as the direction to the Sun is not measurably different from one hole to the next in clouds less than a mile apart. Yet they fan out, all radial from the Sun. Worse than that, if the air is humid or dusty enough, you can follow them overhead and watch them spread apart as they go, and then behold their re-convergence in the dark far distant east!
Maybe this has something to do with optics in the eyeball? The eye is indeed like a ball, and the retina inside it is like a hemisphere, and the world is projected onto it that curved surface through a little hole in the front. Actually it is a fairly big hole, and there is a lens, but never mind: the essential fact is that the world is projected onto a hemisphere. Clearly, straight lines end up curved on the retina. Maybe parallel lines now curve together in this distortion. But how can the physical distortion matter anyhow? We know that images are even further distorted onto the visual cortex of the brain, in fact diversely so on every successive mapping to various parts of the brain. The whole business even starts with splitting the image from each eye as though by a samurai sword stroke to separate it onto to disjoint hemispheres. Yet the image "looks" seamless. Somehow none of this interfered with your doing plane geometry in high school, nor with your present-day concept and evaluation of "straightness". In fact, no one sees with the whole retina anyway. We see mainly with the central fovea, the area that has not just black-white rods but almost all of the eye's color cone cells, and that area is only about 15 degrees wide. I see the bowing of my elevated horizontal shoestring by foveating its middle, then by foveating its left end, then its right end, not by taking it in all at once. I see straightness, sure enough, but I only logically infer curvature by seeing violations of Euclidean geometry in separate foveal fixations. Well, I logically infer it and it does make the string seem curved to me even while it also seems perfectly straight.
This riddle is fun to sort out. I recommend it to your attention, and I will be back in a week's time to make further comments. Maybe you will send me some in the interim.