September 1969
Three Kinds of Equivocal Information in Line Drawings
J. J. Gibson, Cornell University
The World Wide Web distribution of James Gibson’s “Purple Perils” is for scholarly use with the understanding that Gibson did not intend them for publication. References to these essays must cite them explicitly as unpublished manuscripts. Copies may be circulated if this statement is included on each copy.
I have argued that the so-called illusions of reversible perspective and reversible figure-ground are misnamed and misconceived (Senses Considered as Perceptual Systems, p. 246 ff). Far from teaching a lesson about perception in general, they are merely cases of equivocal (ambiguous or conflicting) information in line-drawings. They are anomalies of pictorial perception. They arise because a line, or in some cases a contour, can convey information for two incompatible “forms of layout,” that is, two surface-arrangements that could not coexist in the same place. The contradiction is in the picture and in the light from the picture, not in us. The question we must ask is, What can a line-segment represent? (Kennedy, Thesis proposal).
We shall not consider cases of “multivocal” information from a display (e.g. an ink blot) but only cases of equivocal information. Besides the reversible “perspective” of the Schroeder staircase and the reversible “figure-ground” of Rubin’s displays, there are the “impossible objects” such as the two-pronged or three-pronged tuning fork and, more generally, the drawings of Escher (cf. pp. 247, 258 in Senses Considered).
Given certain definitions (below) from a geometry-with-points-of-observation, there are three ways in which a line segment can be an equivocal representation: 1. A line can represent either a convex corner or a concave corner. This rule applies to the reversible book and the reversible staircase. In the latter drawing each of the internal lines represents the apex of a dihedral angle. The illusion does not have to do with “perspective” but rather with convexity – concavity.
2. A line can represent either an occluding edge that covers (hides) one way or that covers the other way . This rule applies to all the drawings that illustrate reversible figure-and-ground. We might suspect that the information for occlusion in a frozen picture constitutes all that is important in the figure-ground phenomenon.
3. A line can represent either an occluding edge or a convex corner . This rule explains why two of the six lines in the “impossible tuning-fork” function as they do, that is, specify a different layout at one end of the line than at the other end. Two more of these lines provide examples of rule no. 2 above. The two outermost lines of this drawing, however, are univocal; they each represent an occluding edge over the entire length of the line. The reader is invited to check this explanation of the incompatibility of one end of this virtual object with the other end, that is, the “impossibility” of the object.
Definitions . In solid geometry-with-points-of-observation (the kind of geometry required for ecological optics) the following definitions can be given for “corner” and for “occluding edge.”
A corner is the apex of a dihedral angle whose two plane surfaces both face the point of observation. It may be convex or concave.
To face the point of observation a surface must be projected in the optic array at that point. Unprojected surfaces exist for a fixed point of observation but all surfaces are projected to some point of observation (and all the “faces” of a stationary object).
An occluding edge is the apex of a dihedral angle one of whose plane surfaces does not face the point of observation. (It may be obtuse, acute, or wholly acute, as with the “cut edge” of a sheet.) This is one kind. Another kind of occluding edge, for a curved surface, is the margin between the projected and the unprojected parts of the surface (the locus of the tangents of sight-lines to the surface).
The counterbalancing of contrary stimulus-information in a drawing or a choreographic picture (as distinguished from a photographic picture) is an old endeavor of artists and psychologists. They did not, however, formulate what they were trying to do in that way. The hundreds of demonstrations and experiments on reversible or ambiguous or indefinite phenomena in pictures have been taken as showing that perception does not depend simply on “the stimulus.” They are supposed to be routes to the understanding of perception as a special process. New interpretations of these old experiments are now possible. (Cf. also Memorandum of November 1968 entitled The Perception of Surface layouts: A Classification of Types, and papers by John Kennedy.)
Restrictive note . The information considered in this memorandum is information about a limited class of environmental facts, namely, facts of layout. The coloration of surfaces (pigmentation or reflectance) and the illumination of surfaces (lighting and shadowing) have been neglected in favor of the geometrical properties of surfaces. Line drawings, as distinguished from paintings, tend to deal with a sort of colorless shadowless world that is abstracted from the real environment. The perception of “space,” accordingly, has been considered separately from the perception of colors and of shadows. Can this separation, in the last analysis, be maintained?