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How We See Straight Lines

In looking at a straight line, the eye can detect a lateral break
thatforms an image only .00001 centimeter wide on the retina. A new
hypothesis holds that this ability is due to rapid scanning motions

by John R. Platt

On first thought it would seem quite impossible for human beings to see whether a line is straight or not. Our visual mechanism is apparently unsuited to the task.
Consider what happens when we look at a straight line: Its image falls on the

curved surface of the retina at the back of the eye. Here the light pattern stinlU­ lates the tiny receptor elements-rods and cones-that lie underneath it, and they fire off a volley of electrical signals to vaguely defined regions on both sides of the cortex of the brain. Surely the

cerebral pattern is not "straight." It is often argued, however, that the brain somehow knows the location of the rods and cones whose stimulation gave rise to the sensation. If these particular rods and cones lie along an appropriate curve, the object they record is a straight line.

CELLS IN RETINA of the human eye form a mosaic pattern of
light-sensitive receptors. This photomicrograph shows the cells en-

larged about 3,000 diameters. It appears in The Vertebrate Visual System, by S. L. Polyak, published by University of Chicago Press.

121

© 1960 SCIENTIFIC AMERICAN, INC

Let us examine this reasoning a little more closely. The retina is a layer of tis­ sue about one inch square, containing something like 10 million receptor cells arranged in a closely packed mosaic. How can the brain know where each cell is? Hermann von Helmholtz, the great pioneer in the theory of vision, thought the knowledge might be provided by spe­ cific "local signs"-possibly chemical in nature-from cells at different positions. His idea has recently been confirmed in some elegant experiments performed by Jerome Y. Lettvin and his colleagues at the Massachusetts Institute of Technolo­ gy. When they cut a frog's optic nerve and allowed it to regenerate, they found that the neuron from each point of the retina grew back to its proper point in the brain.
But this specificity can hardly be in­ definitely fine. Seen under a microscope, the mosaic of retinal cells looks random, and, as living tissue, it has been subject to all the accidents and irregularities of biological growth. Surely the cells must be subject to some microscopic un­ certainty of location. Thus a line that ap­ pears straight to one man should appear full of little wiggles to his twin brother. The amplitude of the wiggles would in­ dicate the limits of accuracy of the gen­ etic or local sign-specification.
Yet the fact is that we can tell when a line is straight, and none of us ever sees any such wiggles. Our actual precision in certain visual observations is fantastic. Our vernier acuity, or ability to detect a lateral break in a straight line, is about two seconds of arc. This corresponds to a distance of a little more than a hundred thousandth of a centimeter on the retina, about a 30th of the diameter of a cone cell! Even in mechanical construction this precision is almost impossible; a hundred thousandth of a centimeter can­ not be measured in the finest machine shops except by optical methods. In a biological system such as the eye the 10cation of every tissue cell to such an ac­ curacy, 30 times finer than the size of the cell, is quite unbelievable.

VISUAL PATHWAYS of the human central nervous system (solid colored lines) carry
stimuli from the eye (center) to the visual areas of the brain. This horizontal cross·section view shows how the image of a straight line (AG) falls on the curved surface of the retina and is projected onto convoluted areas of the visual cortex (heavy colored lines at bottom).

I puzzled over this paradox for a long time, until I finally began to wonder if we were not looking at the problem in the wrong way in emphasizing the precise location of the individual cells. We were unconsciously assuming that the brain can somehow examine its as­ sociated retina, as if through an external microscope, and locate each of the rods or cones in space.
Thinking about the microscope fallacy, as it might be termed, led me to wonder whether there could not be some high-

122

© 1960 SCIENTIFIC AMERICAN, INC

THE RA W MA TERIALS OF PROGRESS in the rubber industry

How a 3M Chemical helps prevent disastrous dumping of jet fuel

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The jet aircraft roars out over the ocean. Inside its powerful engine, hot fuels of high solvency swirl around the seal of the fuel dumping device. Should the seal weaken, the fuel will dump into the sea, and the jet aircraft's flight will end disastrously.

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The fuel dumping device might never be operated during the life of the air­ craft. So the seal must perform con­ tinuously, and at normal temperatures of 400°F. Even hotter surges can be expected occasionally. To put even further demands on the seal, the JP-4 and JP-5 fuels used swell and degrade most elastomers excessively. Thus, ordinary rubber just can't work as the sealing agent-too short a life expectancy.
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Readily Processed
With all these outstanding properties, you might imagine that FLUOREL 2141 Elastomer would be difficult to process. Actually, it is readily proc­ essed on standard rubber compound­ ing and fabricating equipment. You can mold it, extrude it, and bond it to most metals. And its ideal Mooney Scorch Rating permits reprocessing with greater safety ... gives you the extra margin required by such exact­ ing military specifications as MIL­ R-25897B.
Despite its many exciting applications in the aircraft and missiles field, FLUOREL 2141 Elastomer has even more down-to-earth uses. These in­ clude gaskets and "0" rings for chemical processing, fuel and hydraulic hose, valve diaphragms and linings, and other applications where rubber is presently being used with limited success because of heat or corrosion.
One of Many 3M Products
This new elastomer is but one of a

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PARALLEL SCANNING along a straight line (top row) or a curved line (bottom row) excites a single pattern of receptors on the surface of the retina (large circles). Small black circles indicate
124

excited receptors. If eye scans a broken line (middle row), three different patterns of receptors are stimulated. The brain interprets this shift from one pattern to another as a lack of straightnes•.

© 1960 SCIENTIFIC AMERICAN, INC

precIsIon physical method that would enable a system consisting of 10 million elements to make acute discriminations without knowing exactly where its in­ dividual sensory elements were located. I finally found one, a method that I call functional geometry. As its name implies, the method generates spatial relations in the course of the normal functioning of the visual system rather than through the static, point-by-point location of im­ ages.
The essence of this functioning is mo­ tion, or scanning. Several years ago ex­ perimenters discovered that, in order to keep a static pattern steadily in view over even a short period of time, a per­ son must continuously shift his eyes in tiny scanning motions. If he does not, the image fades away. I suggest that the same scanning can provide the sense of straightness.
The basic idea is as follows. Imagine the image of a scene-any scene-pro­ jected on the retina. The arrangement of light and dark areas stimulates a particu­ lar set of rod and cone cells, which then transmit a specific array of signals to the brain. If the eye scans the scene, mov­ ing so as to shift the image slightly, a somewhat different set of receptors is stimulated, and the signal array changes accordingly. But suppose the scene is a straight line, and the scanning is parallel to the line. Then the motion does not change the set of stimulated receptors, and the signal array remains constant. This constancy, or "self-congruence," after displacement is what the brain rec­ ognizes as straightness.
Evidently an ability to detect the sameness of an array is about the weak­ est demand one could make of a com­ munications network, however it may operate. Moreover, the perception can be made without knowledge of where the individual receptor-cells are located. All that is necessary is an external object that is congruent to itself under a dis­ placement such as the eye can carry out. A straight line fulfills the condition. So do parallel lines. A crooked line, or a set of nonparallel lines, does not.
One of the important features of the method is that the images of these lines on the retina or on the cortex can be as crooked as you please without de­ stroying the self-congruence; all that is required is that the image fall on the same locus after displacement, and it makes no difference what that locus is. The discrimination is therefore for straightness or parallelism in the external field. Clearly the brain does not know­ and, if it uses functional geometry, does not need to know-how the images on its

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125

© 1960 SCIENTIFIC AMERICAN, INC

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TRANSVERSE SCANNING is another method of determining the straightness of a line. By comparing the patterns produced on the surface of the retina each time the eye sweeps across the line, the hrain can form a very accurate judgment of its relative straightness.

cortex would look when observed from the outside. The fact that we see self­ congruence in the external field is what makes straightness (and the other self­ congruent pattern properties I shall mention) matters for public discussion. We do not see objects, but relationships; and the relationships are public. This is a point of considerable importance in linguistics and in theories of knowledge.
Another important feature of the self­ congruent method of perceiving pat­ terns is that it is not affected by damage or loss of receptor cells, or by blind spots. An array of signals can be the same after displacement as before, regardless of what cells have high or low sensitivity. This means we do not have to assume uniform sensitivity in all the cells of the eye. And it is consistent with the fact that we do indeed perceive patterns as passing straight across our blind spots.
In addition to scanning back and forth along a line, our eyes may move trans­ versely across it. By making a series of such perpendicular passes at various points along a line, and by comparing the times at which signals come from different receptors, we can also form a judgment of straightness. Here we are limited only by the time taken in obser­
I- vation. The longer the time, the closer
the check on possible discrepancies be­ tween time-sequences at various points along the line. It is probably this mech­ anism by which we make visual judg­ ments of the highest acuity.
Either type of scanning of straight lines, or of parallels, is carried out by combinations of two rotations of the eye­ ball: around a horizontal axis (looking up or down) and around a vertical axis (looking left or right). However, our eyes are capable of still another motion, though it is sharply limited: rotation around a longitudinal axis pointing
along the line of sight [see bottom of il­ lustration on page 128]. (To observe
this rotation, closely examine a marking on the iris of your eye in the mirror as you tip your head from side to side.) By adding a component of this twisting mo­ tion we can scan along a curve, and, if it has constant curvature, keep the image over the same set of receptors on the retina. Thus arcs of circles exhibit the same sort of self-congruence as straight lines, and concentric arcs the same sort of self-congruence as parallels.
As a matter of fact, it is not easy to judge whether a gentle arc is curved or straight. Uniformity of curvature (in­ cluding the zero curvature of a straight line) is more readily perceived than is straightness or curvedness. If the cur­ vature is sharp enough, however, we

126

© 1960 SCIENTIFIC AMERICAN, INC

Day by day the Polaris missile gets nearer to its first sub­ marine test firing ... nearer the day when, operational, it will become one of our nation's most formidable deterrents to aggression. For with submarines serving as mobile missile launching pads, any target on earth is within deadly striking range if retaliation becomes necessary.
The Polaris-launching submarines are splendidly fitted out not only to aim and fire and accurately guide the missile, but also to defend themselves. Advanced Sperry submarine equipment contributes to both these functions. For precise navigation there is SINS (Ship's Inertial Navigation System), automatic steering and stabilization, depth detectors, gyro-

compasses, diving and maneuvering controls, instrumenta­ tion, and computers . . . and the NA VDAC computer which correlates all navigation data. For anti-submarine warfare the subs have Sperry torpedo fire control systems, sonar sub detection equipment, the attack periscope itself. At two spe­ cial laboratories both aspects of the Polaris program are being refined and integrated: one of which simulates sub­ marine navigation, the other the environments of the sea.
Sperry's role in the Polaris program is typical of the Company today, achieving through specialized divisions an integrated capability that is contributing to every major arena of our environment. General offices: Great Neck,N. Y.

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© 1960 SCIENTIFIC AMERICAN, INC

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EYE ROTATES aboul three axes as it scans lines and geometric shapes. It can turn rather freely aboul Ihe x·x' and the y-y' axes, but can rotate only slightly about the z·z' axis.

probably become aware of it through muscular cues arising out of the twisting motion.
Both the method of self-congruence and the approach to greater accuracy through repeated trials are central con­ cepts in high-precision optical work. Every amateur who has made his own telescope mirror knows that spherical and plane mirrors and precision screws can be brought virtually to perfection by being polished with a matching tool until they are self-congruent under lat­ eral or rotational displacement. Within a finite time the error can be made less than any preassigned value.
In biology the principle of self-con­ gruence generates perfectly helical el­ bow-joints and spherical hip-joints, and eyeballs in spherical sockets. The spheres are self-centering; they know nothing about the point centers and fixed radii of Euclidean geometry. This sug­ gests a new approach to the study of geometry. It might be more natural to start not with points, distances, lines and coordinates, but with self-congru­ ences, which are biologically more prim­ itive.
D o we actually apply functional geometry to every judgment of straight­ ness (and other patterns)? The experi­ enced adult eye may not 'need to scan every new line afresh to determine its approximate straightness. Possibly cer­ tain receptors on the retina have been as­ sociated so often in past straight-line per­ ceptions that when these elements are excited again and give off the same chorus of signals, we are satisfied of the straightness of the new object without further scanning. It is self-congruence to an old straight line, with a long time­ delay. In a sense, the pattern has been learned.
If this is the mechanism of pattern perception, then we should expect to find that an infant or a visually naive adult (for example, a person who has had congenital cataracts removed) would require long scanning and study to determine the straightness of a line. Such, in fact, appears to be the case. The finding is consistent not only with the theory of perception developed here, but also with the doctrine of D. O. Hebb and his school at McGill University. They hold that perceptual organization of even such apparently primitive re­ lationships as straightness or triangulari­ ty is acquired-learned-only through visual experience.
Arthropods (such as insects and spi­ ders) can learn almost nothing, and

128

© 1960 SCIENTIFIC AMERICAN, INC

birds can learn only certain things. It follows that much, if not all, of their pattern-perceiving system must be pre­ located and preconnected, determined by genetic information alone. Pattern­ perceiving that involves learning, per­ haps using methods such as functional geometry, is a way of escaping this genet­ ic limitation. Such an escape is obvious­ ly needed for a really big brain with more inputs. This suggests that pattern­ learning may be the faculty that grew most rapidly in the sudden evolutionary expansion of our own brain and cortical capacity in the last few hundred thou­ sand years.
Perhaps the most noteworthy feature of functional geometry with a mosaic system is that it necessarily picks out cer­ tain patterns as fundamental or primi­ tive. A mathematician of curved spaces might say that an S-curve in one curved

space is a straight line in another, and that these are equally good descriptions of the line. But a functional mosaic will accept as straight only those Euclidean .lines that satisfy self-congruence under displacement. Thus straightness is a primitive and unique category of per­ ception for all mosaic systems. So is parallelism, concentricity and so on. It is interesting to note that the various re­
lationships belong to the "synthetic a priori" categories of Immanuel Kant­
unique categories that impose them­ selves on all minds regardless of pm·ticu­ lar experiences and yet cannot be learned without experiences and com­ parisons.
I suggest that there is only a small number of unique symmetry categories for a visual mosaic receptor, and that they are determined by the three possi­
ble rotations of the eyeball [see table

below]. 'When the rotations are con­
tinuous, we get straightness, parallel­ ism and the like. vVhen the eye moves in discrete jumps, it perceives relationships such as equidistance, congruence and the equality of angles. On this view ev­ ery visual pattern-relationship that can be perceived is some combination of the primitive elements.
It would be interesting to try to con­ struct artificial mosaic receptors, com­ plete with scanning motions, that might be able to make discriminations similar to those our eyes make. Evidently any such system would have to be able to learn. The receiving network would somehow have to grow or to establish new connections guided by experience. If we could design such a system, it might teach us far more than we now know about how the human eye and brain organize external information.

PATTERN OBSERVED STRAIGHT LINE

CURVATURE OF ARC

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AXIS OF EYE MOVEMENT

PATTERN OBSERVED

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PRIMiTIVE PATTERNS nre self.congruent under the types of rotation the eye cnn perform. Patterns at left are perceived

by continuous rotations; relationships at right, by discrete rota­ tions. Axes of rotation are listed in second and fourth columns.

129

© 1960 SCIENTIFIC AMERICAN, INC

... parallel production, development

expedited production has been applied successfully in these major areas
at Melpar

Printed circuitry and materials
Acoustic, audio and ultrasonic systems
Electronic training devices Countermeasures systems Ordnance electronics Aids to navigation Flight simulators Radar systems Telemetry systems Antennas

130 © 1960 SCIENTIFIC AMERICAN, INC

FROM MINIATURE RADAR beacons to complex mission simulators, Melpar has successfully employed its own techniques of paralleling pro­ duction and equipments development.
This concept of paralleling production and development is successful at Melpar because of a wholly-integrated production division and broad experience producing a wide range of electronic and electromechanical equipments and systems_ This method of development and manufacturing control permits system monitoring from primary design to completed production, shipment, installation, and field service_
MELPAR ADDS to its "quick reaction" capability with extensive produc­ tion facilities, permitting specialized operations such as dip-brazing, printed circuitry, automatic dry screen etching, and electroplating proc­ esses for base and precious metals-all contributing to the efficiency and dependability of Melpar's production division.

mean "quick reaction"

Direction finders Fire control systems Microwave components Communications equipment An alog and digital computers Automated assembly units Data handling equipment Satellite electronics Reconnaissance equipment Ground support equipment Detection an d identification
systems

For details on provocative job openings in advanced scientific, engineering area.•, write to: Department 1-10, Profes­
sional Employment Supervisor, 3632 Arlington
Blvd., Falls Church, Virginia-in historic Fairfax County, 10 miles from Washington, D. C.

An Arsenal of Technology <20>r!"!.Y !

M . ELPAR

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Please send me a copy of your descriptive brochure, which outlines your full capabilities and resources for original concep­ tion, design, and production of complete weapons systems.

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13 1 © 1960 SCIENTIFIC AMERICAN, INC