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07 December 2001

A Personal Encounter with Non-Euclidean Space

by Art Winfree

Well, by now you are doubtless personally acquainted with the palpable curvature of straight lines in the sky. (If not yet, please don't yet read further.) The problem is that both of the cartooned observations appear to be correct: On the left, the stick figure looks at the Sun and points parallel to the vector through the Moon, and she thus ends up pointing at the Sun. The shadow of a tree falls at the same angle. But on the right, that upward vector through the Moon also goes right through the lower Sun. 
 

 

Figure 1. Two scenes from the same place at the same time. The white zig-zag tear is supposed to indicate that distances are greater than they look in the right-hand picture.

This observation is probably somehow connected with ancient impressions that the sky is a hemispherical dome sitting over the landscape like a cap, and that the Moon, the Sun, the Planets, and the Fixed Stars are all mounted on concentric crystalline spheres that rotate about some point deep in the Earth that we might call its Center (Thales of Miletus and Pythagoras of Samos 6th century BC, through Aristotle 4th century BC, and ignoring Aristarchus of Samos 3rd century BC, through medieval European scholasticism and Dante). Even in the 21st century I find it hard not to think of the line connecting an airplane to a distant cloud across a background of blue as though it should be a curve on that inverted dome, and I feel the same way about the invisible lines of Right Ascension stretching out from Polaris to other stars across the blackness of that dome at night. Yet there remains the problem that if the sky were a sphere around the observer, then geodesic circles on it would look straight from our perspective at the center. In fact, the lines look both  straight and curved. 

Lines connecting objects in the sky do not curve, as we found by stretching a shoe-string across the Moon to the Sun. If you liked that test, maybe you tried it again with a 12-foot length of rigid pipe from the hardware store: held above the horizon this gives an exceptionally vivid impression that somewhere, somehow, something is curved where I know I am looking only at parallel straight lines. 

Last week I directed the attention of a classroom of university seniors to this matter by asking them about the Moon being illumined in its upper right on 23 November, while the Sun is not higher but rather  lower in the West. Some rationalized that the shadowed half of the Moon must be a shadow cast by the Earth, not indicating the direction of illumination on the Moon. Others that whatever illuminated the Moon may not be the Sun direction, but perhaps somehow a reflection of the Sun. Then I asked them first to write down what they think about the visual appearance of triangle ABC traced between stars: is the sum of its internal angles 180 degrees? Everyone thought so: must be, since the lines are all three visibly straight. Then we considered the case of a star on the southern horizon, one on the western horizon, and one at the zenith. All three angles are 90 degrees. Ooops. Many objections arose about curvature of the sky, about angles seen in tilted perspective, about the horizon really being a circle, and so on. To get away from these tangled verbalizations we crowded out of the classroom into the hall to look at the floor tiles and the baseboards. They converge radially into the long distance. They don't look parallel, though we know they are. At least they are all straight, as lines must be in perspective drawing. Next: turn around and look the other way down the hall. Same vision, but those straight lines converging to a point behind us are still straight but now converging to another point in front. Ooops. Can this happen without something being curved? 

My impression is that we are all so brainwashed by a 10th grade encounter with Euclid that we have trouble even seeing the blatantly different behavior of lines in our visual space. We try to sort out the matter by drawing examples on paper without realizing that once we take that step we are already lost. The analogy to flat paper is in fact the problem. So we make up never-quite-clear verbal excuses and irritably, impatiently, sweep the matter under a rug. 

A thought-experiment: Let's get away from every distraction, at least in imagination. If I think of myself floating in space, forgetting the Earth behind my back and forgetting its horizon, I am just looking at objects like the Moon and the Sun floating in 3d space, and looking at the straight lines my taut shoestring might trace between them. I expect to see straightness and I do. Yet if I look at two parallel straight lines, as though standing between railroad tracks or in a long narrow corridor, I also fully expect to see them converge toward a vanishing point at infinity, and to another in the opposite direction. In between, as they pass by me, they look parallel. How can parallel straight lines converge without something being curved? The lines are not curved, but something must be. Evidently it is the very space in which these lines are embedded. Not Euclidean physical 3-space though. What other possible "space" can be involved? Ah! There is one: the space of some mental map of the visual world inside my head. That must be a curved, non-Euclidean space! 

Is this a strange conclusion to draw? I digress to moralize. The theme of this column is that it is good to notice things and think about them independently so as to become aware of the world in new ways. Maybe one of the university seniors' diverse rationalizations is better, but I think my job as scientist is to try to develop my own way of seeing, initially without biasing my encounter by opinions sought in books. At this stage (weeks after 23 October) it seems to me inadequate to brush this particular matter aside as mere spherical trigonometry. True, what we see has to be consistent with spherical trig, but that does not quite account for the impression of curvature where all lines are straight. It also seems to me that eye anatomy and even brain dynamics have less to do with all this than I first imagined, and perhaps nothing at all. Psychologists doubtless have a way of thinking about it very different from the one presented here, and I am sure it is lots better in every respect. But the confident  expectation that some such better perspective can be had from higher up the mountain trail should not deter us from seeing what we can here and now and taking joy in our own small Discoveries along the way up. We learn more from doing than from reading. So here is my current view of that matter:

Somehow we assemble in our heads a map of the world around us by patching together successive foveal fixations, each about 15 degrees in diameter, and surely no more than 30 degrees. At any one time we actually see, for updating this map, only one such patch, but we remember the others as context around it. We must have a good memory for such things, because our eyes are always flitting about in unconscious saccades; if we really saw what they project on the retina we would be motion-sick in a minute. No, we see the contents of some mental space whose furnishings are cued by fleeting retinal images of small patches. As I imagine it this mental map has two aspects: 

1) First of all, it is 2-dimensional, while the world outside is 3d. This flattening is accomplished by radial projection toward the eye from all directions. There is little more in this assertion than the recognition that light travels in straight lines to the eye. (That may even be the definition of "straight".) It has nothing to do with physiology. 

2) Secondly, this projection is as though onto a spherical bubble around the eye. That 2-dimensional continuum is not in real space, though it could be if I lived inside a big ping-pong ball. Well, I don't, and this space is "in the mind". It has no boundaries: like the ping-pong ball it has the topology and even the geometry of a perfect sphere, consisting only of the observed angles between things as seen in projection. The distances between things in this space are reckoned not in centimeters (as they might be on the big ping-pong ball) but in radians or degrees. Its radius of curvature is given in the same units and might be 1 radian. Its geometric relations are not Euclidean, on account of the uniformly positive curvature of this visual space itself. This is how a straight line ---- the projection of a straight shoestring in Euclidean 3d onto a geodesic great-circle arc in the 2d sphere assembled in my head --- comes to be curved. This is how parallel lines can intersect. Visual 2d space is curved because it inherits the structure of stereographic projection from 3d onto a curved (spherical) 2d surface. 
 

Figure 2. By looking in various directions we acquire impressions of how the world around us fits together. It fits into a continuum without edges: a sphere. The straight lines connecting things as seen from the location of the eye are like arcs on this sphere. Their measure is not centimeters but radians. These straight lines do not follow Euclidean geometry. Note that this is a cartoon only: it makes absolutely no sense to contemplate this ball from such an outside perspective as portrayed here, as though the ball were embedded in physical 3-space.

 This all may be trivially obvious but it remains a marvel to me to have gone so long without noticing that visual space is not Euclidean and it has an intrinsic curvature, in fact a uniform positive curvature of 1 radian! There is an implication, too, that totally escaped me before this little exercise of 23 October. At age 12 or so I figured out how to draw things in 3d by keeping corresponding edges of rectangular prisms parallel or pointing all to one fixed point on the page. Later on I learned that this is a formal subject called "perspective drawing" with clear mathematical rules, and since then I have believed it followed necessarily from projective geometry. But if there really is a distinction to be made between objective 3d space and subjective 2d visual space, and if the latter is indeed strongly curved, then there can be no one right way to draw on flat paper. No way can be right in all respects. It must be that the "rules of perspective" are mere conventions that probably change from century to century with other artistic fashions. Maybe this is wrong, but to me at this moment it seems a wee Discovery, a gratifying reward for the day's "exercise hour" of thinking on things I know nothing whatever about.

It is fun to check this out more quantitatively. If visual space is spherical, and measures "distances" in angles rather than in centimeters, then its relations amount to spherical trigonometry. Not only can parallel straight lines intersect (at infinity in 3-space, we say, but really at a finite point in the 2-space of angles), but also the sum of the angles of a triangle in this space must exceed the familiar  radians or 180 degrees of Euclidean geometry. The excess is proportional to the triangle's area reckoned as a fraction of the whole sphere. Is this observably so or observably false? This question brings us to notice there are two kinds of angles in visual space: 

1) Between one point and another --- e.g., between two bright stars in the night sky --- the separation is an angle, an arc of a great circle on the sphere of visual space, though it looks like a perfectly straight line, and 

2) Between one such straight line joining stars A and B and another joining star B to star C there is another kind of Angle, that I distinguish here with a capital letter. Calling the observer O, there are angles AOB and BOC and COA, each an arc on the sphere, and there are Angles ABC and BCA and CAB between these angles, all six of them measured in radians or degrees. It is an exercise (and not an easy one) in spherical trig to prove the necessary identity, called the Law of Sines by analogy to the one in plane geometry (which you doubtless remember sports analogous ratios but without the sine function on one term): 
 

Figure 3.  Yardsticks or taut shoestrings line up to connect stars A,B,C into a triangle. Its sides cannot be distances, but they are unique angles. There are also Angles between these straight sides, measurable at each vertex with a goniometer. Click image to enlarge.

 sin Angle ACB / sin angle AOB = the same for the two other permutations of corners A,B,C This ratio always exceeds 1, and it can get arbitrarily larger if ABC is a pretty small triangle. 

This and the thing about the sum of the Angles exceeding   might not be fun to derive analytically, but it is at least fun to check by observation. The experience boggles my personal intuition. It requires measuring angles as carefully as you can. One helpful tool is a sextant. You may be able to pick up a nice brass one used from Ebay (www.ebay.com) for about $50, or buy new plastic one: I got a Davis Mark 3 for $38 from Celestaire, Inc. (316-686-9785) or you can go direct to http://www.davisnet.com/marine/
products/marine_product.asp?
pnum=011. 

Next best is a cross-staff, which you can make from a pair of yardsticks, A and B. Strap a pair of two-inch strips of foam-core astride stick A like a moveable sleeve (or use two pieces of wood cut from a third yardstick), wrapping wide heavy tape around the sandwich just tight enough so it can slide up and down the length of A. Make another one. Between them mount yardstick B perpendicularly, well centered as you see in the photo below, then secure the three together by gluing a five-inch long strip of yardstick along their length. (This is invisible on the underside). Use Shoo-goo or epoxy, taking care not to glue the sliding assembly onto A. When well dried, press big push-pins into cross-staff B at equal intervals to left and right. The idea is that you can place A radial to your eye and slide the cross-staff forward and back until two stars, or two mountain peaks, or a distant chimney top and the Moon, are sighted over the tops of a symmetric pair of push pins. You know their half-distance apart or can read it directly from the scale. You know how far the cross-staff is from your eye by reading it direct from scale A at the far edge of the slider, if you hold A with its near end as far from your eye as that edge is from the line of push-pins in the center of the slider. Or after holding the end of A quite close to your eye you know the distance  by reading scale A where the line of pins crosses it. I put a red rubber covering on A's near end in case I poke my eye. Anyhow, the ratio of those two lengths is the tangent of half the angle subtended. 

Sextant or cross-staff provides you angles AOB, BOC, and COA. For Angles ABC, BCA, and CAB the simplest expedient is a goniometer assembled by bolting together two hacksaw blades through the holes at one end (yellow in the adjacent photo). In use, the length of this bolt should be radial to your eye to ensure that the blade angle is seen perpendicularly. Put star or chimney top A in the corner where blades meet near the bolt, and mountain peak B further out along one blade edge, and C similarly on the other blade edge. With the bolt tight enough so you don't lose the angle, lay this yellow wedge on a piece of paper and trace angle CAB with pencil. If you later trace the next two as consecutive wedges you will immediately see the excess beyond 180 degrees. Taking their separate measures beneath a transparent compass rose, you can check the Law of Sines quantitatively. The upshot is that the theorem checks out as exactly as you care to measure. 
 

Figure 4. On the asphalt of my driveway near sunset, a plastic Davis Mark 3 sextant with an extension affixed for other purposes. (The orange shadow is sunlight through one of its color filters.) And a cross-staff made in 15 minutes from two yardsticks. And a goniometer made in 1 minute from hacksaw blades and a 2" bolt. These tools will be useful again in the Adventure of the Rainbow Moon, coming up soon. (Click image to enlarge)

 

Meanwhile if you care to digress from personal experience of Discovery to vicarious experience by looking in "books", the web can be helpful. One site about spherical trigonometry is http://mathworld.rm-f.net/s/s593.htm. And you can find mention of "curvature of visual space" in a www.google.com search. These hits refer to the publications of insightful psychologists since 1948 that seem to mean something quite different: their curvature appears to pertain strictly and exclusively to binocular stereopsis, is reportedly negative and apparently rather slight, is far from uniform across the visual field, and varies substantially from person to person. 

Reverting to the original context of this mystery, I leave you with a teaser: How come one of these clouds casts its shadow on the ground, but the other cannot? 

 

Figure 5. A plane of rays from the Sun, seen edge-on, appears as a yellow line. It would seem that cloud shadows must be in those planes, together with the Sun and whatever dark cloud part. But does that mean some clouds cast no shadow on the ground, and which ones they are depends on the viewer?? A picknicker might be distressed to learn that he is in shadow or not depending on who is looking at the cloud :--) . I don't aim to "answer": this is just for you.

Next week's column will follow-up some loose ends from last week, so plan not to read it if you don't first find time to think through those Adventures in Discovery for yourself. Then, two weeks hence, we take leave of the sky to do an experiment in acoustics inspired by an accident with a Walkman during another afternoon run.