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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, two quantities are in the silver ratio (or silver mean) if the
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...
, whose value of one plus the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princi ...
is approximately 2.4142135623. Its name is an allusion to the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
; analogously to the way the golden ratio is the limiting ratio of consecutive
Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
s, the silver ratio is the limiting ratio of consecutive
Pell number In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins , , , , an ...
s. The silver ratio is denoted by .
Mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
s 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 number In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are infinitely many square triangular numbers; the first few are: :0, 1, 36, , , , , , , Expl ...
s, Pell numbers,
octagon In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, w ...
s and the like. The relation described above can be expressed algebraically: : \frac = \frac \equiv \delta_S or equivalently, : 2 + \frac = \frac \equiv \delta_S The silver ratio can also be defined by the simple
continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
; 2, 2, 2, ... : 2 + \cfrac =\delta_S The convergents of this continued fraction (, , , , , ...) 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. The silver rectangle is connected to the regular
octagon In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, w ...
. 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:, and the 4 sides of the trapezoids are in a ratio of 1:1:1:. If the edge length of a regular octagon is , then the span of the octagon (the distance between opposite sides) is , and the area of the octagon is ..


Calculation

For comparison, two quantities ''a'', ''b'' with ''a'' > ''b'' > 0 are said to be in the ''
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
'' if, : \frac = \frac = \varphi However, they are in the ''silver ratio'' if, : \frac = \frac = \delta_S. Equivalently, : 2+\frac = \frac = \delta_S Therefore, : 2 + \frac = \delta_S. Multiplying by and rearranging gives :^2 - 2\delta_S - 1 = 0. Using the
quadratic formula In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, ...
, two solutions can be obtained. Because is the ratio of positive quantities, it is necessarily positive, so, :\delta_S = 1 + \sqrt = 2.41421356237\dots


Properties


Number-theoretic properties

The silver ratio is a
Pisot–Vijayaraghavan number In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1, all of whose Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel ...
(PV number), as its conjugate 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 to the nearest integer is . Thus, the sequence of
fractional part The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part ca ...
s of , (taken as elements of the torus) converges. In particular, this sequence is not
equidistributed mod 1 In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequenc ...
.


Powers

The lower powers of the silver ratio are : \delta_S^ = 1 \delta_S - 2 = ;2,2,2,2,2,\dots\approx 0.41421 : \delta_S^0 = 0 \delta_S + 1 = = 1 : \delta_S^1 = 1 \delta_S + 0 = ;2,2,2,2,2,\dots\approx 2.41421 : \delta_S^2 = 2 \delta_S + 1 = ;1,4,1,4,1,\dots\approx 5.82842 : \delta_S^3 = 5 \delta_S + 2 = 4;14,14,14,\dots\approx 14.07107 : \delta_S^4 = 12\delta_S + 5 = 3;1,32,1,32,\dots\approx 33.97056 The powers continue in the pattern : \delta_S^n = K_n\delta_S + K_ where : K_n = 2 K_ + K_ For example, using this property: : \delta_S^5 = 29\delta_S + 12 = 2;82,82,82,\dots\approx 82.01219 Using and as initial conditions, a Binet-like formula results from solving the recurrence relation : K_n = 2 K_ + K_ which becomes : K_n = \frac \left(\delta_S^ - ^\right)


Trigonometric properties

The silver ratio is intimately connected to trigonometric ratios for . :\tan \frac = \sqrt-1= \frac :\cot \frac = \tan \frac = \sqrt+1=\delta_s So the area of a regular octagon with side length is given by :A = 2a^2 \cot \frac = 2\delta_s a^2 \simeq 4.828427 a^2.


See also

*
Metallic mean The metallic means (also ratios or constants) of the successive natural numbers are the continued fractions: n + \cfrac = ;n,n,n,n,\dots= \frac. The golden ratio (1.618...) is the metallic mean between 1 and 2, while the silver ratio (2.41 ...
s *
Ammann–Beenker tiling In geometry, an Ammann–Beenker tiling is a nonperiodic tiling which can be generated either by an aperiodic set of prototiles as done by Robert Ammann in the 1970s, or by the cut-and-project method as done independently by F. P. M. Beenker. ...


References


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. .


External links

*
An Introduction to Continued Fractions: The Silver Means
", '' Fibonacci Numbers and the Golden Section''.

at Tartapelago by Giorgio Pietrocola {{DEFAULTSORT:Silver Ratio Quadratic irrational numbers Mathematical constants Metallic means Ratios