Mathematical Recreations

by Ian Stewart

Tales of a Neglected Number

Last month I described the mathematical sculptures of Alan St. George, who often makes use of the well known "golden number". The catalogue of his Lisbon exhibition mentions a less glamorous relative, referring to a series of articles in which "the architect Richard Padovan revealed the glories of the 'plastic number.' "
The plastic number has little history, which is strange considering its great virtues as a design tool, but its provenance in mathematics is almost as respectable as that of its golden cousin. It doesn't seem to occur so much in nature, but, then, no one's been looking for it.

A Pearly Nautilus and it's logarithmically spiral shell. [Image:"Life on Earth" D.Attenborough p46]

For purposes of comparison, let me start with the golden number: q =1 + 1/q = 1.618034, approximately. The golden number has close connections with the celebrated Fibonacci numbers. This series can be illustrated by a spiralling system of squares [see upper illustration on this page]. The initial square (in gray) has side 1, as does its neighbour to the left. A square of Side 2 is added above the first two, followed in turn by squares of side 3, 5, 8, 13, 21 and so on. These numbers, each of which is the sum of the previous two, form the Fibonacci series. The ratio of consecutive Fibonacci numbers tends to the golden number. For example, 21/13 = 1.615384.

This fact is a consequence of the rule for generating Fibonacci numbers: for large numbers, it leads to the equation q=1 + 1/q . If a quarter circle is added inside each square, the arcs fit together into an elegant spiral. This spiral is a good approximation to the so-called logarithmic spiral often found in nature, such as in the shell of a nautilus mollusc. [Ref: I.Stewart "Nature's Numbers" p88] Successive turns of the spiral grow at a rate approximately equal to the golden number.

That's the golden tale: now for the plastic one. We start with a similar diagram, but composed of equilateral triangles [see lower illustration on this page]. The initial triangle is marked in gray; successive triangles are added in a clockwise direction. The spiral generated is again roughly logarithmic. In order to make the shapes fit, the first three triangles all have side 1. The next two have side 2; then the numbers go 3, 4, 5, 7, 9, 12, 16, 21 and so on. Again there is a simple rule of formation: each number is obtained by skipping the previous one and adding the two before that. Let me call this sequence the Padovan Sequence in honour of Richard Padovan. (It is curious that Padova is the Italian form of Padua, and Fibonacci was from Pisa-roughly 100 miles from Padua.)

SPIRALING SYSTEMS illustrate the Fibonacci numbers (top) and the Padovan sequence (bottom)

In algebraic form the generating rules for the Fibonacci sequence F(n) and the Padovan sequence P(n) are given as follows:
F(n+ 1) = F(n) + F(n - 1) where F(0) = F(1) = 1,
and
P(n + 1) = P(n - 1) + P(n - 2) where P(0) = P(1) = P(2) =1.
The family resemblance is very apparent. The plastic number, which from now on I shall call p and whose approximate value is 1.324718, arises as the limit of the ratio of successive Padovan numbers-just as the golden number arises from the Fibonacci sequence. The formation rule leads to the equation p = 1/p + 1/p², or equivalently p³ - p - 1 = 0; the number p, is the unique real solution of this equation.

The Padovan sequence increases much more slowly than the Fibonacci sequence, because p is smaller than q. There are many interesting patterns in the Padovan sequence. For example, the figure shows that 21 = 16 + 5, because adjacent triangles on the same edge have to fit together. Thus, an alternative rule for deriving more terms for the sequence is P(n + 1) =P(n) + P(n - 4). Some numbers, such as 3, 5 and 21, are both Fibonacci arid Padovan. Are there others? If so, how many, and is that count finite or infinite? Some Padovan numbers, such as 9, 16 and 49, are perfect squares-are there others? The square roots here are 3, 4 and 7-also Padovan numbers. Is this a coincidence or a general rule? These and many other questions deserve further study.

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. Now we get a kind of three-dimensional spiral of boxes [see illustration]. Start with a cube of side 1, placing another adjacent to it. The result is a 1 x 1 x 2 cuboid. On the 1 x 2 face, add another 1 x 1 x 2, getting a 1 x 2 x 2 cuboid. Then on a 2 x 2 face, add a 2 x 2 x 2 cube, to form a 2 x 2 x 3 cuboid overall. To a 2 x 3 face, add a 2 x 2 x 3 to get a 2 x 3 x 4 overall, and so on. Continue the 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 you connect successive square faces of the added cuboids by straight lines, you get a spiral. It even turns out that this spiral lies in a plane. St. George has based a sculpture on this construction, made from rigid rods connected by drilled balls at their corners. (What diagram does the intersection of the system of cuboids with this plane form?)

A sequence with the same rule of formation, but starting with different values, was studied in 1876 by the French mathematician Édouard Lucas. In 1899 his ideas were further developed by R. Perrin, and the sequence is now known as the Perrin sequence A(n). The Perrin numbers differ from the Padovan numbers in that A(0) = 3 ,A(1) =0 and A(2) = 2. Again the ratio of consecutive Perrin numbers tends to become p, but Lucas noticed a more subtle property. Whenever n is a prime number it divides A(n) exactly. For example, 19 is prime, A(19) = 209 and 209/19 = 11.

This theorem provides a curious test for a number to be composite-that is, not prime. For instance, when n = 18, we have A(18) =158 and 158/18 = 8.777, which is not a whole number. Therefore, 18 must be composite. So we can use Perrin numbers to test for nonprimality: any number n that does not divide A(n) is composite.

SPIRALING CUBOIDS also form Padovan numbers

If n divides A (n), must n always be prime? This does not follow from Lucas's theorem-any more than "if it rains, then I get wet" implies if I get wet, then it rains." (I might have fallen into a pond on a perfectly dry day.) Still, it is a fascinating open question. Nobody has ever found a composite n that divides A(n), but nobody has shown that such numbers-known as Perrin pseudoprimes do not exist. In 1991 Steven Arno of the Supercomputing Research Center in Bowie, Md., proved that Perrin pseudoprimes must have at least 15 digits. I would be delighted to hear of any more recent progress.

The conjecture that no Perrin pseudoprimes exist is important, because the remainder on dividing A(n) by n can be calculated very rapidly. If the conjecture is true, this remainder will be 0 if and only if n is prime, thereby providing a speedy primality test. (Indeed, in 1982 William W. Adams and Daniel Shanks of the University of Maryland found a way to calculate this remainder in log n steps.) Thus, the conjecture should have useful applications to secret codes, which nowadays often hinge on properties of large primes.
Just like its glittering golden cousin, the plebeian plastic number generates rich spirals of ideas.

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SCIENTIFIC AMERICAN June 1996 File Info: Created 18/6/2000 Updated 8/8/2003 Page Address: http://members.fortunecity.com/templarser/padovan.html