The plastic number is a little-known number that has much in common with the golden ratio in that it is closely linked to architecture and to aesthetics. The concept of the plastic number was first described by the Dutchman Hans van der Laan (1904–1991) in 1928, shortly after he had abandoned his architectural studies and become a novice monk, and has subsequently been explored by the English architect Richard Padovan (1935–). It is derived from a cubic equation, rather than a quadratic in the case of the golden ratio, and is intimately linked to two ratios, approximately 3:4 and 1:7, which van der Laan considered fundamental in the relationship between human perception and shape and form. These ratios, he believed, express the lower and upper limits of our normal ability to perceive differences of size among three-dimensional objects. The lower limit is that at which things differ just enough to be of distinct types of size. The upper limit is that beyond which they differ too much to relate to each other; they then belong to different orders of size. According to van der Laan, these limits are precisely definable. The mutual proportion of three-dimensional things first becomes perceptible when the largest dimension of one thing equals the sum of the two smaller dimensions of the other. This initial proportion determines in turn the limit beyond which things cease to have any perceptible mutual relation.

 

In mathematical terms, the plastic number is the unique real number solution to the equation x ³ – x – 1 = 0 and has the approximate value 1.324718.... Just as the golden ratio is approximated better and better by successive terms of the Fibonacci sequence, F(n + 1) = F (n) + F (n – 1), where F (0) = F (1) = 1, so the plastic number arises as the limit of the ratio of successive numbers in the sequence P (n + 1) = P (n – 1) + P (n – 2) where P (0) = P (1) = P (2) = 1. This sequence has been called the Padovan sequence, and its members the Padovan numbers. The Padovan sequence increases much more slowly than does the Fibonacci sequence. Some numbers, such as 3, 5, and 21, are common to both sequences; however, it is not known if there are finitely many or infinitely many such pairs. Some Padovan numbers, such as 9, 16, and 49, are perfect squares; the square roots of these – 3, 4, and 7 – are also Padovan numbers, but it is not known if this is a coincidence or a general rule. Another way to generate the Padovan numbers is to mimic the use of squares for Fibonacci numbers, but with cuboid structures – boxes with rectangular faces. A kind of three-dimensional spiral of boxes emerges. Start with a cube of side 1 and place another adjacent to it: the result is a 1 × 1 × 2 cuboid. On the 1 × 2 face, add another 1 × 1 × 2 box to produce a 1 × 2 × 2 cuboid. Then on a 2 × 2 face, add a 2 × 2 × 2 cube to form a 2 × 2 × 3 cuboid overall. To a 2 × 3 face, add a 2 × 2 × 3 to get a 2 × 3 × 4 box overall, and so on. Continue this process, always adding cuboids in the sequence east, south, down, west, north and up. At each stage the new cuboid formed will have as its sides three consecutive Padovan numbers. Moreover, if successive square faces of the added cuboids are connected by straight lines, the result is a spiral that lies in a plane. The Padovan sequence is very similar to the Perrin sequence.

 

Reference

1. Stewart, Ian. "Tales of a Neglected Number." Scientific American, 274: 102–103 (June 1996).