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GOLDEN CUBOID SEQUENCES

The notion of a golden cuboid as a generalization of a golden rectangle was first introduced by Huntley [1], where it appears as a rectangular parallelepiped with edges 1, (f), and (J). In this paper, we proceed from a somewhat different point of view by first recalling that a golden rectangle may be defined as the unique rectangle with the property that adjunction of a square to the larger side gives a larger rectangle geometrically similar to the first. This definition is generalized to the situation of rectangular parallelepipeds, the results being two new candidates for the title "golden cuboid" (Theorem 1). Theorem 2 establishes a nested sequence of golden cuboids analogous to the well-known sequence of nested golden rectangles. An unexpected application occurs in [2] with the construction of an interpretative model for a disputed passage of Vlato*s Timaeus, lines 31b-32c. In searching for a generalization of the above-mentioned property of golden rectangles, let R be a rectangular parallelepiped with edges a, b, and c, and suppose a < b K e. A larger geometrically similar parallelepiped R' can then always be produced by the adjunction of a single parallelepiped to R, provided hi a = c/b. However, as a generalization of the two-dimensional case, if we in addition insist that a cube appear in the adjunction process, then it is clear that at least two adjunctions must occur. This motivates the following definition.