Although its name is recent, the silver ratio (or silver mean) has been studied since ancient times because of its connections to the square root of 2, almost-isosceles Pythagorean triples, square triangular numbers, Pell numbers, the octagon, and six polyhedra with octahedral symmetry.

It follows that the silver ratio is found as the positive solution of the quadratic equation

The silver ratio can be used as base of a numeral system, here called the sigmary scale.

[b] Every real number x in [0,1] can be represented as a convergent series Sigmary expansions are not unique.

Remarkably, the same holds mutatis mutandis for all quadratic Pisot numbers that satisfy the general equation ⁠

Vera de Spinadel described the properties of these irrationals and introduced the moniker metallic means.

[12] [c] Pell numbers are obtained as integral powers n > 2 of a matrix with positive eigenvalue ⁠

⁠ A rectangle with edges in ratio √2 ∶ 1 can be created from a square piece of paper with an origami folding sequence.

Considered a proportion of great harmony in Japanese aesthetics — Yamato-hi (大和比) — the ratio is retained if the √2 rectangle is folded in half, parallel to the short edges.

[d] If the folding paper is opened out, the creases coincide with diagonal sections of a regular octagon.

⁠ Repeating the construction at successively smaller scales results in four infinite sequences of adjoining right triangles, tracing a whirl of converging silver rectangles.

[16] The logarithmic spiral through the vertices of adjacent triangles has polar slope

The parallelogram between the pair of grey triangles on the sides has perpendicular diagonals in ratio ⁠

Arranging the tiles with the four hypotenuses facing inward results in the diamond-in-a-square shape.

Roman architect Vitruvius recommended the implied ad quadratura ratio as one of three for proportioning a town house atrium.

⁠ The silver mean has connections to the following Archimedean solids with octahedral symmetry; all values are based on edge length = 2.

⁠[19] See also the dual Catalan solids The acute isosceles triangle formed by connecting two adjacent vertices of a regular octagon to its centre point, is here called the silver triangle.

Repeating the process at decreasing scales results in an infinite sequence of silver triangles, which converges at the centre of rotation.

It is assumed without proof that the centre of rotation is the intersection point of sequential median lines that join corresponding legs and base vertices.

The logarithmic spiral through the vertices of all nested triangles has polar slope The long, medium and short diagonals of the regular octagon concur respectively at the apex, the circumcenter and the orthocenter of a silver triangle.

Assume a silver rectangle has been constructed as indicated above, with height 1, length ⁠

⁠ If an horizontal line is drawn through the intersection point of the diagonal and the internal edge of a rabatment square, the parent silver rectangle and the two scaled copies along the diagonal have areas in the ratios

If the diagram is further subdivided by perpendicular lines through U and V, the lengths of the diagonal and its subsections can be expressed as trigonometric functions of argument

Both the lengths of the diagonal sections and the trigonometric values are elements of biquadratic number field

Since the regular octagon is defined by its side length and the angles of the silver triangle, it follows that all measures can be expressed in powers of σ and the diagonal segments of the silver rectangle, as illustrated above, pars pro toto on a single triangle.

⁠ has been dubbed the Cordovan proportion by Spanish architect Rafael de la Hoz Arderius.

According to his observations, it is a notable measure in the architecture and intricate decorations of the mediæval Mosque of Córdoba, Andalusia.

If drawn on a silver rectangle, the spiral has its pole at the foot of altitude of a triangle on the diagonal and passes through vertices of paired squares which are perpendicularly aligned and successively scaled by a factor

The silver ratio appears prominently in the Ammann–Beenker tiling, a non-periodic tiling of the plane with octagonal symmetry, build from a square and silver rhombus with equal side lengths.

Discovered by Robert Ammann in 1977, its algebraic properties were described by Frans Beenker five years later.