Tetraconic Perspective for a Complete Sphere of Vision

Leonardo, Vol. 9, pp. 289-291. Pergamon Press 1976. Printed in Great Britain TETRACONIC PERSPECTIVE FOR A COMPLETE SPHERE OF VISION Kenneth R. Adams* 1. Tetraconic perspective I have made a study of perspective since 1957 and two reports giving some of its results have been published in Leonardo [l, 21. In this note, I discuss what I believe is a new kind of perspective, tetraconic perspective, that provides simultaneously a view of all that can be seen in all directions from a point in space (say the spatial centre of a room). Fig. 1 shows four views (an infinite number is possible) of a cubical room in tetraconic perspective with no apparent doors or windows as ‘seen’ by a hypothetical monocular viewer whos eye is located at its spatial centre. The room, for example, measures 4 x 4 x 4m and the floor has a uniform checkerboard pattern, each square measuring 50 x 50 cm. The four walls are marked with equally spaced (50 cm) vertical lines. The ceiling is marked with dashed lines forming a grid of squares, each measuring 1 x 1m. A dashed horizontal line is drawn around the room, 2m above the floor, at the eye level of the hypothetical viewer; it represents a projection of the horizon on the walls. Anthony Green, who has often represented a complete room * Artist living at 19 Dartmouth Park Rd., London NW5 lSU, England. (Received 20 Mar. 1975.) in a single painting, used in ‘My Mum’s Dream’ (Fig. 2) a system of perspective that is approximate to the systems shown in Fig. 1. The range of what can be seen by moving one’s eyes with the head fixed can be indicated roughly by moving across the drawings in Fig. 1 a sheet of paper having as a frame a circular hole of a diameter equal to the height of the drawing. If the circular frame sufficiently avoids enclosing what I call the hub points HI, H1, H, and H,, then what is represented within the circle is recognized easily. The views in Fig. 1 are based on terraconic projection, which I shall explain here. The cubical room is assumed to be filled by an imaginary sphere that touches the central point of each of the six sides (Fig. 3 (a)). Four points (hub points) are equally spaced from each other on the surface of the sphere. When the four points are connected by straight lines, a regular tetrahedron is formed. A regular tetrahedron is a geometrical body having four identical faces, each an equilateral triangle. (Alternatively stated, a regular tetrahedron is a pyramid all of whose faces (including the base) are identical equilateral triangles.) Fig. 1. Four tetraconic projections of a cubical room. 289 290 Ketitieth R.Adatns Fig. 2. Anthony Green. 'My Mum's Dream', oil on board, 220 x 240 crn, 1974. (Gallery Dieter Brusberg, Hanover, Fed. Rep. Ger.) (Photo: John Webb, London.) 1-12 ..-. . . . . 6 Y Fig. 3 (a) and (b). The constriictiori of a tetraconic projection (cf. Fig. 3 (c) and (d)). Four points, 0,,0,, 0, and 0,, are also equally spaced on the surface of the sphere; they are at the centres of the spherical triangles located by the hub points. 0, is diametrically opposite HI, 0, is opposite H,, and so on. The sphere with all these points fixed on it can be rotated in any way about the central viewpoint. For constructing Figs. 1(a) and (b), H, coincides with the central point of the floor and 0, coincides with the central point of the ceiling; for Fig. 1 (a), the sphere is rotated about the axis H,O,, so that the face of the tetrahedron defined by 0, is directed towards the wall that is to form the upper centre of the picture; for Fig. l(b), the edge of the tetrahedron H,H, is directed at the same wall. For Figs. l(c) and (d), H, coincideswith the central point of the wall and 0, coincides with the central point of the opposite wall; for Fig. I(c), the face of the tetrahedron defined by 0, is directed towards the floor; for Fig. l(d), the...