Plastic ratio: Difference between revisions

Binary	1.01010011001000001011…
Decimal	1.32471795724474602596…
Hexadecimal	1.5320B74ECA44ADAC1788…
Continued fraction[1]	[1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80 ...] Note that this continued fraction is neither finite nor periodic. (Shown in linear notation )
Algebraic form	${\displaystyle {\sqrt[{3}]{\frac {9+{\sqrt {69}}}{18}}}+{\sqrt[{3}]{\frac {9-{\sqrt {69}}}{18}}}}$

In mathematics, the plastic number ρ (also known as the plastic constant, the plastic ratio, the minimal Pisot number,^[2] the platin number,^[3] Siegel's number or, in French, le nombre radiant) is a mathematical constant which is the unique real solution of the cubic equation

${\displaystyle x^{3}=x+1.}$

It has the exact value^[4]

${\displaystyle \rho ={\sqrt[{3}]{\frac {9+{\sqrt {69}}}{18}}}+{\sqrt[{3}]{\frac {9-{\sqrt {69}}}{18}}}.}$

Its decimal expansion begins with 1.324717957244746025960908854….^[5]

Properties

Recurrences

The powers of the plastic number A(n) = ρⁿ satisfy the third-order linear recurrence relation A(n) = A(n − 2) + A(n − 3) for n > 2. Hence it is the limiting ratio of successive terms of any (non-zero) integer sequence satisfying this recurrence such as the

The plastic number satisfies the nested radical recurrence^[7]

${\displaystyle \rho ={\sqrt[{3}]{1+{\sqrt[{3}]{1+{\sqrt[{3}]{1+\cdots }}}}}}.}$

Number theory

Because the plastic number has the minimal polynomial x³ − x − 1 = 0, it is also a solution of the polynomial equation p(x) = 0 for every polynomial p that is a multiple of x³ − x − 1, but not for any other polynomials with integer coefficients. Since the discriminant of its minimal polynomial is −23, its splitting field over rationals is ℚ(√−23, ρ). This field is also a Hilbert class field of ℚ(√−23).

The plastic number is the smallest

${\displaystyle \left(-{\tfrac {1}{2}}\pm {\tfrac {\sqrt {3}}{2}}i\right){\sqrt[{3}]{{\tfrac {1}{2}}+{\tfrac {1}{6}}{\sqrt {\tfrac {23}{3}}}}}+\left(-{\tfrac {1}{2}}\mp {\tfrac {\sqrt {3}}{2}}i\right){\sqrt[{3}]{{\tfrac {1}{2}}-{\tfrac {1}{6}}{\sqrt {\tfrac {23}{3}}}}}\approx -0.662359\pm 0.56228i,}$

of absolute value ≈ 0.868837 (sequence A191909 in the OEIS). This value is also 1/√ρ because the product of the three roots of the minimal polynomial is 1.

Trigonometry

The plastic number can be written using the

${\displaystyle \rho ={\tfrac {1}{c}}\cosh \left({\tfrac {1}{3}}\cosh ^{-1}(3c)\right),\qquad c=\cos \left({\tfrac {2\pi }{12}}\right)=\sin \left({\tfrac {2\pi }{6}}\right)={\tfrac {\sqrt {3}}{2}}\,.}$

(See Cubic function#Trigonometric (and hyperbolic) method.)

Geometry

There are precisely three ways of partitioning a square into three similar rectangles:^[8][9]

1. The trivial solution given by three congruent rectangles with aspect ratio 3:1.
2. The solution in which two of the three rectangles are congruent and the third one has twice the side length of the other two, where the rectangles have aspect ratio 3:2.
3. The solution in which the three rectangles are mutually non congruent (all of different sizes) and where they have aspect ratio ρ². The ratios of the linear sizes of the three rectangles are: ρ (large:medium); ρ² (medium:small); and ρ³ (large:small). The internal, long edge of the largest rectangle (the square's fault line) divides two of the square's four edges into two segments each that stand to one another in the ratio ρ. The internal, coincident short edge of the medium rectangle and long edge of the small rectangle divides one of the square's other, two edges into two segments that stand to one another in the ratio ρ⁴.

The fact that a rectangle of aspect ratio ρ² can be used for dissections of a square into similar rectangles is equivalent to an algebraic property of the number ρ² related to the Routh–Hurwitz theorem: all of its conjugates have positive real part.^[10][11]

History

Name

Dutch architect and

Martin Gardner has suggested referring to ${\displaystyle \rho ^{2}}$ as "high phi", and Donald Knuth created a special typographic mark for this name, a variant of the Greek letter phi ("φ") with its central circle raised, resembling the Georgian letter pari ("Ⴔ").^[17]

See also

• Snub icosidodecadodecahedron
• Supergolden ratio

Notes