The plastic number

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The plastic number

The sequence of side lengths of equilateral triangles in this picture form the Padovan sequence (1,1,1,2,2,3,4,5,7,9…). Just as the Fibonacci sequence is governed by the properties of the golden ratio, the Padovan sequence is governed by the properties of the so-called plastic number.

The Padovan sequence P(n) is sequence is defined by setting P(1)=P(2)=P(3)=1, and then requiring P(n) = P(n–2) + P(n–3) for n > 3. The generating function for the sequence is given by G(x)=(1+x)/(1–x^2–x^3), which means that if this ratio of polynomials is expanded as a power series in x, the coefficient in G(x) of x^n (i.e., x to the nth power) is equal to P(n). 

The denominator in the formula for the generating function, 1–x^2–x^3, can be regarded as an algebraic encoding of the recurrence relation P(n)–P(n–2)–P(n–3)=0. An easy calculation involving polynomials shows that the product (1–x^2–x^3)(1–x+x^2) is given by 1–x–x^5. This means that the generating function G(x) can be rewritten so that the denominator polynomial is given by 1–x–x^5, which in turn means that, if n is large enough, the sequence will satisfy the recurrence relation P(n) = P(n–1) + P(n–5). 

The picture is an illustration of this last relation. Notice that the big triangle with side 16 is bounded on one side by the preceding triangle in the sequence (of side 12) and the triangle five places earlier in the sequence (of side 4). This corresponds to the fact that P(n) = P(n–1) + P(n–5) for n=12 and P(n)=16.

It turns out that the polynomial 1–x^2–x^3 has exactly one real root, and the plastic number is the reciprocal of this root. Another way to say this is that the plastic number is the unique real solution of the equation x^3=x+1. It is not hard to show using abstract algebra that this solution is an irrational number; the same is true for the golden ratio, which is the larger real solution of the equation x^2=x+1. The decimal expansion of the plastic number is therefore non-recurring; the first few digits are 1.324717957…, and over 10,000,000,000 digits have been computed. 

The plastic number is mathematically significant because it is the smallest Pisot number. A Pisot number is a real root of a monic integer polynomial whose other roots are complex numbers of absolute value less than 1. The word monic means that the highest power of x occurring has a coefficient of 1. The connection with the Padovan sequence is that the ratio P(n+1)/P(n), as n becomes large, tends to the plastic number. (In the case of the Fibonacci numbers, the corresponding ratio approaches the golden ratio.)

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