Visually Scanning 4-Dimensional Objects with the Aid of Hyperstereograms in Color

Leonardo, Vol. 13,pp. 310-312. Pergamon Press, 1980. Printed in Great Britain. VISUALLV SCANNING 4-DIMENSIONAL OBJECTS WITH THE AID OF HVPERSTEREOGRAMS IN COLOR David W. Brisson* 1. Several years ago, while experimenting with drawings showing stereoscopic pairs of projections of some 4dimensional geometrical objects, hyperobjects, I discovered a new way to look at or scan such hyperobjects . Visualizing certain hyperobjects has been considered by others, too. Recently B. L. Chilton described one method in an article in Leonardo [1]. He presented a set of drawings each showing a view of one regular polyhedron that is a projection (a 3-dimensional shadow in a hyperplane) of a hyperobject. Ordinary stereograms may be employed to visualize objects and scenes in three dimensions, and examples of their use in visual art have been provided by H. A. Layer [2] and by Roger Ferragallo [3]. The stereoscopic pairs of projections that I use in visualizing hyperobjects I call hyperstereograms. Hyperstereograms and ordinary stereograms differ in an important way. In an ordinary stereogram, the corresponding points in the two images, such as in Fig. 1, are displaced equal distances above,or below, a given reference point (for example center point 0) but not equal distances to the right, or to the left of it (parallax) [3]. In binocular vision the unequal horizontal displacements of corresponding points in what is received by the eyes leads to the perception of an object in three dimensions. In such examples of horizontal parallax, two images are 'fused', subjectively in the brain, to produce a single 3-dimensional image. In a hyperstereogram, inequalities occur in both the vertical and horizontal distances of corresponding points from a reference point; there is both horizontal and vertical parallax. This occurs because the rotation of a 4-dimensional form is a rotation around a plane, whereas the rotation of a 3-dimensional form is rotation around a line, that is, the axis of rotation. Perceptual investigations of my own, concerned with retinal rivalry, have convinced me that the fusion that seems to occur in the visual perception of 3dimensional objects is as much a cognitive as it is a physical-optical effect. The optical image in the brain can be observed to be a mosaic of mutually exclusive pieces of the two distinct images on the two retinas. To observe this, it is only necessary to wear differently colored filters over the two eyes and note the mosaic of coloration! What some investigators have done, however, is to project a 4-dimensional shadow in a hyperplane. An ordinary stereoscopic pair was then made of the 3dimensional shadow. But care must be taken with this concept of a shadow. The hyperstereogram in Fig. 2 is •Artist and teacher, Box 85, Rehoboth, MA 02769, U.S.A. (Received 25 Sept. 1979) 310 really not a shadow in Chilton's sense [1]. Chilton's shadow is produced by a parallel, orthogonal projection to a hyperplane (3-dimensional space), and a second projection is then made to a plane. Figure 2 is produced by a non-parallel, orthogonal projection directly to a plane. Although this sounds perhaps like mathematical trivia, perceptually it is not, for the difference is as important as that between a drawing in perspective and an orthogonal projection and is a distinction of the same sort. The direct projection of the 4-dimensional object to a plane offers a closer correspondance to visual experience with 3-dimensional objects and takes more complete advantage of the binocular process. The mathematical basis for this distinction is provided, at least in part, by H. S. M. Coxeter [4]. Although an entire hyperstereogram cannot be fused simultaneously, one is able by my method to produce a fusion locally, and by obtaining successive fusions one can scan the hyperobject. This is analogous to the scanning of a 3-dimensional image that occurs while looking at a 3-dimensional object. The eyes involuntarily and imperceptibly turn inwards and outwards producing a projective correspondance along points describing a circular arc that when extended through the center of the lens of each eye produces a circle called a horopter circle or simply horopter [5]. Thus fusion Fig. 1. A stereoscopic pair...