In mathematics, the plastic ratio is a geometrical proportion close to 53/40.

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

results in the continued reciprocal square root Dividing the defining trinomial

After 29 rotation steps the phases of the inward spiraling conjugate pair – initially close to ⁠

⁠, 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 ⁠

⁠ has closed form expression (which is less than 1/3 the eccentricity of the orbit of Venus).

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.

[8] Requiring proportional continuity, he constructed a geometric series of eight measures (types of size) with common ratio 2 / (3/4 + 1/71/7) ≈ ρ.

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.

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

⁠, the Van der Laan numbers can be computed with the Binet formula [10] Since

[12] The Van der Laan numbers are obtained as integral powers n > 2 of a matrix with real eigenvalue ⁠

⁠ can be interpreted as incidence matrix for a D0L Lindenmayer system on the alphabet ⁠

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.

[13] There are precisely three ways of partitioning a square into three similar rectangles:[14][15] 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.

[16][17] The circumradius of the snub icosidodecadodecahedron for unit edge length is The unique positive node ⁠

⁠ that optimizes cubic Lagrange interpolation on the interval [−1,1] is equal to 0.41779130...

Each term of the series corresponds to the diagonal length of a rectangle with edges in ratio ⁠

The diagram shows the sequences of rectangles with common shrink rate ⁠

⁠ converge at a single point on the diagonal of a rho-squared rectangle with length

⁠, 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 ⁠

[5] French high school student Gérard Cordonnier [fr] discovered the ratio for himself in 1924.

Van der Laan initially referred to it as the fundamental ratio (Dutch: de grondverhouding), using the plastic number (Dutch: het plastische getal) from the 1950s onward.

[23] In 1944 Carl Siegel showed that ρ 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.

Van der Laan designed the 1967 St. Benedictusberg Abbey church to these plastic number proportions.

[25] The plastic number is also sometimes called the silver number, a name given to it by Midhat J. Gazalé[26] and subsequently used by Martin Gardner,[27] 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.

Gardner suggested referring to ρ2 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").