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Wikipedia / Plastic ratio

In mathematics, the plastic ratio is a geometrical proportion, given by the unique real solution of the equation Its decimal expansion begins with . It is the smallest Pisot number.

The adjective plastic does not refer to the artificial material, but to the formative and sculptural qualities of this ratio, as in plastic arts.

Definition

Three quantities This ratio is commonly denoted (rho).

It follows that the plastic ratio is the unique real solution of the cubic equation

Solving with Cardano's formula or, using the hyperbolic cosine,

is the superstable fixed point which is the update step of Newton's method applied to

by and the conjugate elements of

Properties

The plastic ratio and golden ratio are the only morphic numbers: real numbers for which there exist natural numbers m and n such that and Morphic numbers can serve as basis for a system of measure.

Properties of (m=3 and n=4) are related to those of (m=2 and n=1). For example, The plastic ratio satisfies the continued radical

The plastic ratio can be expressed in terms of itself as the infinite geometric series

For every integer from this an infinite number of further relations can be found.

The algebraic solution of a reduced quintic equation can be written in terms of square roots, cube roots and the Bring radical

Continued fraction

The plastic ratio is the smallest Pisot number. By definition of these numbers, the absolute value of the algebraic conjugates is smaller than 1, thus powers of generate almost integers . After 29 rotation steps the phasess of the inward spiraling conjugate pair initially close to nearly align with the imaginary axis.

The infinite radicals with Perron number The other real root is the negative reciprocal of the second Pisot number .

lie on a circle with radius lie on a circle with radius . In both cases, the complex conjugate zeros have approximate argument .

The minimal polynomial has discriminant The Hilbert class field of imaginary quadratic field can be formed by adjoining a generator for the ring of integers of , one has the special value of Dedekind eta quotient

Expressed in terms of the Weber-Ramanujan class invariant Gn

Properties of the related Klein j-invariant The difference is .

The elliptic integral singular value for (which is less than 1/3 the eccentricity of the orbit of Venus).

Van der Laan sequence

In his quest for perceptible clarity, the Dutch Benedictine monk and architect Dom Hans van der Laan (1904-1991) asked for the minimum difference between two sizes, so that we will clearly perceive them as distinct. Also, what is the maximum ratio of two sizes, so that we can still relate them and perceive nearness. According to his observations, the answers are and , spanning a single order of size. Requiring proportional continuity, he constructed a geometric series of eight measures (types of size) with common ratio Put in rational form, this architectonic system of measure is constructed from a subset of the numbers that bear his name.

The Van der Laan numbers have a close connection to the Perrin and Padovan sequences. In combinatorics, the number of compositions of n into parts 2 and 3 is counted by the nth Van der Laan number.

The Van der Laan sequence is defined by the third-order recurrence relation

The first few terms are 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86,...

The first 14 indices n for which is prime are n = 5, 6, 7, 9, 10, 16, 21, 32, 39, 86, 130, 471, 668, 1264 . The last number has 154 decimal digits.

which is proved by mathematical induction on This relation also holds for The order of the coefficients corresponds to the bottom row of matrix below.

The generating function of the Van der Laan sequence is given by

The sequence is related to sums of binomial coefficients by

The characteristic equation If the three solutions are real root and conjugate pair and , the Van der Laan numbers are given by the Binet formula with real and conjugates and

, the number is the nearest integer to , with

The first few terms are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, 51, 68, 90, 119,... .

This Perrin sequence has the Fermat property The converse does not hold, but the small number of pseudoprimes makes the sequence special. The only 7 composite numbers below to pass the test are n = 521, 904631, 16532714, 24658561, 27422714, 27664033, 46672291.

The Van der Laan numbers are obtained as integral powers of a matrix with real eigenvalue

The trace of gives the Perrin numbers.

Alternatively, can be interpreted as incidence matrix for a D0L Lindenmayer system on the alphabet with corresponding substitution rule and initiator . The series of words produced by iterating the substitution have the property that the number of and

Associated to this string rewriting process is a set composed of three overlapping self-similar tiles called the Rauzy fractal, that visualizes the combinatorial information contained in a multiple-generation letter sequence.

Geometry

Partitioning the square

There are precisely three ways of partitioning a square into three similar rectangles:

  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 lengths of the other two, where the rectangles have aspect ratio 3:2.

  3. The solution in which the three rectangles are 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.

Cubic Lagrange interpolation

The unique positive node that optimizes cubic Lagrange interpolation on the interval is equal to The square of with discriminant which is verified by insertion into

the Lebesgue function at critical point

and can be expressed as infinite geometric series Each term of the series corresponds to the diagonal length of a rectangle with edges in ratio with odd. The diagram shows the sequences of rectangles with common shrink rate

Plastic pentagon

A spiral of equilateral triangles with edges in ratio tiles a plastic pentagon with four angles of 120 and one of 60 degrees. The initial triangle is positioned at the left-hand side of a parallelogram with base to side ratio and left base angle 60 degrees, so that two edges of the triangle are collinear with sides of the parallelogram. Scaling the parallelogram in ratio accompanied with a clockwise rotation by 60 degrees, the horizontal base is mapped onto the third edge of the triangle. The centre of rotation is on the short (falling) diagonal, dividing it in ratio , the expansion rate for a half-turn. Iteration of the process traces an infinite, closed sequence of equilateral triangles with pentagonal boundary.

The logarithmic spiral For parallelogram base

In the vector image, the construction is repeated on each side of a triangle. John Rutherford Boyd discovered a related figure, build on the sides of the triangle.

Plastic spiral

A plastic spiral is a logarithmic spiral that gets wider by a factor of for every quarter turn. It is described by the polar equation with initial radius If drawn on a rectangle with sides in ratio , the spiral has its pole at the foot of altitude of a triangle on the diagonal and passes through vertices of rectangles with aspect ratio which are perpendicularly aligned and successively scaled by a factor

In 1838 Henry Moseley noticed that whorls of a shell of the chambered nautilus are in geometrical progression: "It will be found that the distance of any two of its whorls measured upon a radius vector is one-third that of the next two whorls measured upon the same radius vector ... The curve is therefore a logarithmic spiral." for a quarter turn. Considering the plastic ratio a three-dimensional equivalent of the ubiquitous golden ratio, it appears to be a natural candidate for measuring the shell.

Polyhedra

The plastic ratio is important in the study of the snub icosidodecadodecahedron.

History and names

was first studied by Axel Thue in 1912 and by G. H. Hardy in 1919. French high school student discovered the ratio for himself in 1924. In his correspondence with Hans van der Laan a few years later, he called it the radiant number (). Van der Laan initially referred to it as the fundamental ratio (), using the plastic number () from the 1950s onward. In 1944 Carl Siegel showed that is the smallest possible Pisot–Vijayaraghavan number and suggested naming it in honour of Thue.

Unlike the names of the golden and silver ratios, 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. This, according to Richard Padovan, is because the characteristic ratios of the number, and , 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.

The plastic number is also sometimes called the silver number, a name given to it by Midhat J. Gazalé and subsequently used by Martin Gardner, but that name is more commonly used for the silver ratio , one of the ratios from the family of metallic means first described by Vera W. de Spinadel. Gardner suggested referring to 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").

See also

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Notes

References

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