Passages from

"A new visual illusion of direction"

by James Fraser
British Journal of Psychology 2: 307-337, 1908.

Figure III is used by Koeler (1969) to illustrate basic points in gestalt psychology. It is also used by Varenne and McDermott to make a related point for culture theory as it relates to education in America (1998). Further work is being conducted by Michael Bach.
Selected figures:

p. 307-308 (First paragraphes of paper)

In all the hitherto published visual illusions of direction, with the exception of the Chequer-borad or Münsterberg illusion, the illusory lines or bands are definitely continuous, uninterrupted in character of black or white, on a constrasting background. In the illusion of direction here described each illusory band consists of a series of visibly discrete similar parts, all inclined at the same small angle to the line of direction of the series to which they belong. Such visibly discrete similar parts may be conveniently termed 'units of direction.'

Where the illusory band consists of alternating black and white 'units of direction,' it may be conveniently regarded as representing a cord consisting of two strands, black and white, twisted together. This arrangement will, in the descriptions, be referred to as 'the twisted cord.' A twisted cord laid upon a grey or coloured background of intermediate luminosity will apparently deviate from its actual line of direction at an inclination corresponding in trend (but at a smaller angular degree) to the inclination of the units of direction. This illusion is much increased in degree when the twisted cord is laid upon a chequer-work background of squares of white, black, and an intermediately luminous grey or colour in such a way that it bisects diagonally each member of a series of black and white squares, its black and white units bisecting respectively white and black squares (their contrasts) at a small angle with the diagonal line of the square as seen in No. 14, Fig. 15. In such a figure each unit of direction is now, in effect, lengthened by the addition of a triangular area of the same luminosity at each of its ends (No. 14, Fig. 15). These triangular areas are derived or borrowed from the neighbouring squares belonging to the series on which the twisted cord lies, and they lie on opposite sides of the twisted cord. This form of unit of direction may be conveniently (as distinguished from the 'simple unit of direction') termed a 'compound unit of direction,' its middle or rodshaped portion corresponding to the twisted cord and the triangular end portions corresponding to black or white squares of the chequer-work background. This form of the illusion, in which there is a visual fusion of the twisted cord with a chequer-work background, may be termed the 'twisted cord on chequer-work background' illusion.

In some of the figures the twisted cord must be regarded as fixed at certain points, and, starting from these points, to be twisted in an opposite direction.

In Fig. 2 and in the outer and inner pair of curves in Fig. 6 the strands are not twisted but lie uninterrupted, side by side, their marginal dividing line being the diagonal line of a black or white square. This arrangement, which may be named the 'untwisted strands on chequer-work background,' gives a figure which contains a slight illusory element contrasting with the much more definite illusion of the twisted cord.


(pp. 313-315)

V. Construction of the circular figures.

In the construction of the circular figures a circular area is divided by radial lines into a certain number of equal sectors (e.g. 48 in Fig. 5). A series of concentric circles are then described at such intervals that the quadrangular divisions obtained have each three equal sides, the outer circumferential side and the two radial sides, and one side (the inner circumferential) which is slightly shorter. These quadrangular areas are thus approximate squares and on being bisected diagonally in both directions give a new group of approximate squares which is used as the basis of the background of the illusory figure to be produced. The two sets of diagonals of the second group of approximate squares are the original radial and circular lines, and on these lines rest the 'units of direction.' The 'units of direction' all bear the same relation in size to the approximate squares on which they lie, so that in the completed figure the members of each radial series of similar

geometrical parts diminish in size centripetally by geometrical progression, and the members of any circular series are of equal size.

The chequer-work background is composed of a double series of dark and light broad spiral bands running in counter directions, and formed by the visual union of the black and white squares with the grey squares. The units of any one circular series all correspond to the same circle in one of two ways-either the centre of area of a Unit corresponds to the centre of a black or white square; or, the marginal dividing line of two 'units' corresponds at its central part to the diagonal line of a black or white square, that is, to part of the actual circular curve. In Figs. 4 to 7 the curves are arranged in pairs; there being a distance equal to the sum of the radial diagonals of two grey squares between the pairs. The curves in each pair have an identical distribution of the angular inclinations of the units; the units of neighbouring pairs of curves being opposed in angular inclination.

As in Fig. 1 the length of each unit is approximately equal to the sum of the diagonals of three squares, except where a change in the angular inclination takes place. Here either the outer or the inner unit is approximately equal to the sum of five diagonals, as in Fig. 5.

In Fig. 3 the angular inclination of the units is the same throughout.

In Fig. 4 the units in each pair of curves are inclined in an opposite direction to that of the units of a neighbouring pair.

In each curve of Figs. 5 and 9 there are four equidistant squares on which the centre part of the units lie side by side so that their marginal dividing line coincides with the diagonal line of a square, that is, with the circular curve. Starting from the angular points of each of these four squares the units are inclined in opposite directions.

In Fig. 5 the distribution of the angular inclinations of the units is such that the apparent long diameter of each pair of curves is at right angles to the apparent Iona diameter of a neighbouring pair.

In Fig. 8 on each curve there are eight equidistant squares starting from the angles of which the units are inclined in opposite directions.

At the four angular points of the 'squared' circles the central point of the marginal dividing line of the units is very slightly outside the circular line.

In Fig, 9, where there is no arrangement of the curves in pairs, the distribution of the angular inclinations of the units is such that the apparent Iona diameter of each curve forms an angle of 45' with the apparent long diameter of each neighbouring curve.

On ordinary visual examination of this figure, with the exception of the outer curve, it is extremely difficult to isolate individual curves. Neither the actual nor the apparent nature of the curves can be definitely perceived. The general impression is that of an entanglement of curves, or of a single very irregular curve. On steady fixation, however, of some point in the figure, preferably the central point, the curves tend to isolate themselves in a varying sequence, illusory elliptiform curves ' whose long diameters have apparently different angular inclinations, succeeding one another in consciousness, the four angular directions being easily discriminated.

In Fig. 6 the two middle pairs of curves correspond to the curves in Fig. 5. The outer and inner pairs of curves are of the 'untwisted strands' type. There is in them no illusion as to the geometrical nature of the curves as a whole ; but sinuosity or waviness is obtained under the conditions described for Fig. 2.

In Fig. 7, starting from each end of the horizontal diameter of each curve, the angular inclination of the units is reversed.

In Fig. 10, the background of which is similar to that of the other figures of a circular type, an illusion is produced as to the direction of the actually radial 'twisted cords.' There is an opposite angular inclination of the units in neighbouring 'twisted cords,' and a corresponding apparent inclination of each cord.

The 32 radii, instead of appearing to reach the centre of the figure, appear to meet, in pairs, on 16 equidistant points on the circumference of a circle described at some indefinite distance from that cen tre.

An elliptical figure.

In Fig. 11 a concentric series of (geometrically) similar ellipses is treated in the same fashion as the circular curves. The short diameters are to the long diameters as 7 to 8. Starting from the vertical angles of the squares whose horizontal diagonals correspond to the long diameters of the ellipses, the angular inclination of the units is outwards from the elliptical curves. The marginal dividing lines of the central parts of the units which correspond to the ends of the long and short diameters of the ellipses coincide with the elliptical curves. The result of this treatment of the elliptical curves is that from without inwards the curves become apparently more and more circular in character; the inner curves simulating ellipses having long diameters at right angles to the long diameters of the outer curves. If the figure be rotated through half a right angle in the plane of the paper, the apparent gradations are probably better appreciated. Viewing the figure with any angle of rotation the identification of the curve which appears to be the normal circle is impossible.

Last revision: November 7, 1997