Plastic number

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

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

This article is about the irrational number. For numbers stamped on plastics for recycling identification, see Resin identification code.

List of numbers Irrational numbers ζ(3) √2 √3 √5 φ e π
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	${\displaystyle {\sqrt[{3}]{\frac {9+{\sqrt {69}}}{18}}}+{\sqrt[{3}]{\frac {9-{\sqrt {69}}}{18}}}}$

Triangles with sides in ratio of ${\displaystyle \rho }$ form a closed spiral

Squares with sides in ratio of ${\displaystyle \rho }$ form a closed spiral

It has the exact value[3]

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

Its decimal expansion begins with 1.324717957244746025960908854….[4]

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 Cordonnier numbers (better known as the Padovan sequence), the Perrin numbers and the Van der Laan numbers, and bears relationships to these sequences akin to the relationships of the golden ratio to the second-order Fibonacci and Lucas numbers, akin to the relationships between the silver ratio and the Pell numbers.[5]

The plastic number satisfies the nested radical recurrence[6]

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

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 Pisot–Vijayaraghavan number. Its algebraic conjugates are

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

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 hyperbolic cosine (cosh) and its inverse:

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

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

Geometry

Three partitions of a square into similar rectangles

There are precisely three ways of partitioning a square into three similar rectangles:[7][8]

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.[9][10]

History

The 1967 St. Benedictusberg Abbey church by Hans van der Laan has plastic-number proportions.

Further information: Mathematics and art

Name

Dutch architect and Benedictine monk Dom Hans van der Laan gave the name plastic number (Dutch: het plastische getal) to this number in 1928. In 1924, four years prior to van der Laan's christening of the number's name, French engineer Gérard Cordonnier had already discovered the number and referred to it as the radiant number (French: le nombre radiant). Unlike the names of the golden ratio and silver ratio, the word plastic was not intended by van der Laan to refer to a specific substance, but rather in its adjectival sense, meaning something that can be given a three-dimensional shape.[11] This, according to Richard Padovan, is because the characteristic ratios of the number, 3/4 and 1/7, relate to the limits of human perception in relating one physical size to another. Van der Laan designed the 1967 St. Benedictusberg Abbey church to these plastic number proportions.[12]

The plastic number is also sometimes called the silver number, a name given to it by Midhat J. Gazalé[13] and subsequently used by Martin Gardner,[14] but that name is more commonly used for the silver ratio 1 + √2, one of the ratios from the family of metallic means first described by Vera W. de Spinadel in 1998.[15]

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").[16]

See also

• Snub icosidodecadodecahedron
• Supergolden ratio

Notes

1. Sequence OEIS: A072117 in the OEIS
2. Choulet, Richard (January–February 2010). "Alors argent ou pas ? Euh … je serais assez platine" (PDF). Pour chercher et approfondir. Le Bulletin Vert. Association des Professeurs de Mathématiques de l'Enseignement Public (APMEP) Paris (486): 89–96. ISSN 0240-5709. OCLC 477016293. Archived from the original (PDF) on 2017-11-14. Retrieved 2017-11-14.
3. Weisstein, Eric W. "Plastic Constant". MathWorld.
4. Sequence OEIS: A060006 in the OEIS.

5. Shannon, Anderson & Horadam (2006).

6. Piezas, Tito III; van Lamoen, Floor; and Weisstein, Eric W. "Plastic Constant". MathWorld.CS1 maint: multiple names: authors list (link)
7. Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275, No. 5, November 1996, p. 118
8. de Spinadel, Vera W.; Antonia, Redondo Buitrago (2009), "Towards van der Laan's plastic number in the plane" (PDF), Journal for Geometry and Graphics, 13 (2): 163–175.
9. Freiling, C.; Rinne, D. (1994), "Tiling a square with similar rectangles", Mathematical Research Letters, 1 (5): 547–558, doi:10.4310/MRL.1994.v1.n5.a3, MR 1295549
10. Laczkovich, M.; Szekeres, G. (1995), "Tilings of the square with similar rectangles", Discrete and Computational Geometry, 13 (3–4): 569–572, doi:10.1007/BF02574063, MR 1318796
11. Padovan (2002); Shannon, Anderson & Horadam (2006).
12. Padovan (2002).
13. Gazalé, Midhat J. (April 19, 1999). "Chapter VII: The Silver Number". Gnomon: From Pharaohs to Fractals. Princeton, N.J.: Princeton University Press. pp. 135–150. ISBN 9780691005140. OCLC 40298400.
14. Martin Gardner, A Gardner's Workout (2001), Chapter 16, pp. 121–128.
15. de Spinadel, Vera W. (1998). Williams, Kim (ed.). "The Metallic Means and Design". Nexus II: Architecture and Mathematics. Fucecchio (Florence): Edizioni dell'Erba: 141–157.
16. "Six challenging dissection tasks" (PDF). Quantum. 4 (5): 26–27. May–June 1994.

References

• Aarts, J.; Fokkink, R.; Kruijtzer, G. (2001), "Morphic numbers" (PDF), Nieuw Arch. Wiskd., 5, 2 (1): 56–58.
• Gazalé, Midhat J. (1999), Gnomon, Princeton University Press.
• Padovan, Richard (2002), "Dom Hans Van Der Laan And The Plastic Number", Nexus IV: Architecture and Mathematics, Kim Williams Books, pp. 181–193.
• Shannon, A. G.; Anderson, P. G.; Horadam, A. F. (2006), "Properties of Cordonnier, Perrin and Van der Laan numbers", International Journal of Mathematical Education in Science and Technology, 37 (7): 825–831, doi:10.1080/00207390600712554.

External links

• Tales of a Neglected Number by Ian Stewart
• Plastic rectangle and Padovan sequence at Tartapelago by Giorgio Pietrocola
• Harris, Ed. "The Plastic Ratio" (video). youtube. Brady Haran. Retrieved 15 March 2019.