Silver ratio

Not to be confused with Silver constant.

For a ratio also known as the silver ratio, see Golden ratio conjugate.

"Silver triangle" redirects here. For other uses, see Kepler triangle.

In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the larger of those two quantities to the smaller quantity is the same as the ratio of the sum of the smaller quantity plus twice the larger quantity to the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δ_S but it can vary from λ to σ.

Silver ratio

Silver rectangle
Representations
Decimal	2.4142135623730950488...
Algebraic form	1 + √2
Continued fraction	$\textstyle 2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{\ddots }}}}}}}}$

Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like.

The relation described above can be expressed algebraically, for a > b:

${\frac {2a+b}{a}}={\frac {a}{b}}\equiv \delta _{S}$

or equivalently,

$2+{\frac {b}{a}}={\frac {a}{b}}\equiv \delta _{S}$

The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ...]:

$2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots }}}}}}=\delta _{S}$

The convergents of this continued fraction (⁠2/1⁠, ⁠5/2⁠, ⁠12/5⁠, ⁠29/12⁠, ⁠70/29⁠, ...) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers.

A regular octagon decomposed into a silver rectangle (gray) and two trapezoids (white)

The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δ_S, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δ_S. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δ_St, and the area of the octagon is 2δ_St².

Calculation

For comparison, two quantities a, b with a > b > 0 are said to be in the golden ratio φ if,

${\frac {a+b}{a}}={\frac {a}{b}}=\varphi$

However, they are in the silver ratio δ_S if,

${\frac {2a+b}{a}}={\frac {a}{b}}=\delta _{S}.$

Equivalently,

$2+{\frac {b}{a}}={\frac {a}{b}}=\delta _{S}$

Therefore,

$2+{\frac {1}{\delta _{S}}}=\delta _{S}.$

Multiplying by δ_S and rearranging gives

${\delta _{S}}^{2}-2\delta _{S}-1=0.$

Using the quadratic formula, two solutions can be obtained. Because δ_S is the ratio of positive quantities, it is necessarily positive, so,

$\delta _{S}=1+{\sqrt {2}}=2.41421356237\dots$

Properties

If one cuts two of the largest squares possible off a silver rectangle one is left with a silver rectangle, to which the process may be repeated...

Silver spirals within the silver rectangle

Other than the silver rectangle, the silver ratio is also related to the regular Octagon, and the following 3D shapes:

1.Truncated cube

2.Deltoidal icositetrahedron

3.Rhombic cuboctahedron

Number-theoretic properties

The silver ratio is a Pisot–Vijayaraghavan number (PV number), as its conjugate 1 − √2 = ⁠−1/δ_S⁠ ≈ −0.41421 has absolute value less than 1. In fact it is the second smallest quadratic PV number after the golden ratio. This means the distance from δ ⁿ
_S to the nearest integer is ⁠1/δ ⁿ
_S⁠ ≈ 0.41421ⁿ. Thus, the sequence of fractional parts of δ ⁿ
_S, n = 1, 2, 3, ... (taken as elements of the torus) converges. In particular, this sequence is not equidistributed mod 1.

Powers

The lower powers of the silver ratio are

$\delta _{S}^{-1}=1\delta _{S}-2=[0;2,2,2,2,2,\dots ]\approx 0.41421$

$\delta _{S}^{0}=0\delta _{S}+1=[1]=1$

$\delta _{S}^{1}=1\delta _{S}+0=[2;2,2,2,2,2,\dots ]\approx 2.41421$

$\delta _{S}^{2}=2\delta _{S}+1=[5;1,4,1,4,1,\dots ]\approx 5.82842$

$\delta _{S}^{3}=5\delta _{S}+2=[14;14,14,14,\dots ]\approx 14.07107$

$\delta _{S}^{4}=12\delta _{S}+5=[33;1,32,1,32,\dots ]\approx 33.97056$

The powers continue in the pattern

$\delta _{S}^{n}=K_{n}\delta _{S}+K_{n-1}$

where

$K_{n}=2K_{n-1}+K_{n-2}$

For example, using this property:

$\delta _{S}^{5}=29\delta _{S}+12=[82;82,82,82,\dots ]\approx 82.01219$

Using K₀ = 1 and K₁ = 2 as initial conditions, a Binet-like formula results from solving the recurrence relation

$K_{n}=2K_{n-1}+K_{n-2}$

which becomes

$K_{n}={\frac {1}{2{\sqrt {2}}}}\left(\delta _{S}^{n+1}-{(2-\delta _{S})}^{n+1}\right)$

Trigonometric properties

See also: Exact trigonometric values § Common angles

The silver ratio is intimately connected to trigonometric ratios for ⁠π/8⁠ = 22.5°.

$\tan {\frac {\pi }{8}}={\sqrt {2}}-1={\frac {1}{\delta _{s}}}$

$\cot {\frac {\pi }{8}}=\tan {\frac {3\pi }{8}}={\sqrt {2}}+1=\delta _{s}$

So the area of a regular octagon with side length a is given by

$A=2a^{2}\cot {\frac {\pi }{8}}=2\delta _{s}a^{2}\simeq 4.828427a^{2}.$

See also

• Metallic means
• Golden ratio
• Supersilver ratio
• Ammann–Beenker tiling

Further reading

• Buitrago, Antonia Redondo (2008). "Polygons, Diagonals, and the Bronze Mean", Nexus Network Journal 9,2: Architecture and Mathematics, p.321-2. Springer Science & Business Media. ISBN 9783764386993.

External links

• Weisstein, Eric W. "Silver Ratio". MathWorld.
• "An Introduction to Continued Fractions: The Silver Means Archived 2018-12-08 at the Wayback Machine", Fibonacci Numbers and the Golden Section.
• "Silver rectangle and its sequence" at Tartapelago by Giorgio Pietrocola