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Silver means | ||

0: | 0 + √42 | 1 |

1: | 1 + √52 | 1.618033989 |

2: | 2 + √82 | 2.414213562 |

3: | 3 + √132 | 3.302775638 |

4: | 4 + √202 | 4.236067978 |

5: | 5 + √292 | 5.192582404 |

6: | 6 + √402 | 6.162277660 |

7: | 7 + √532 | 7.140054945 |

8: | 8 + √682 | 8.123105626 |

9: | 9 + √852 | 9.109772229 |

⋮ | ||

n: | n + √4 + n^{2}2 |

Binary | 10.0110101000001001111... |

Decimal | 2.4142135623730950488... |

Hexadecimal | 2.6A09E667F3BCC908B2F... |

Continued fraction | |

Algebraic form |

In mathematics, two quantities are in the **silver ratio** if the ratio between the sum of the smaller plus twice the larger of those quantities and the larger one is the same as the ratio between the larger one and the smaller (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 denoted by δ_{S}.

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 covergents, square triangular numbers, Pell numbers, octagons and the like.

The relation described above can be expressed algebraically:

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

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.

The silver ratio is a Pisot–Vijayaraghavan number (PV number), as its conjugate 1 − √2 = −1δ_{S} ≈ −0.41 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 1δ*n**S* ≈ 0.41^{n}. 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.

The lower powers of the silver ratio are

The powers continue in the pattern

where

For example, using this property:

Using K_{0} = 1 and K_{1} = 2 as initial conditions, a Binet-like formula results from solving the recurrence relation

which becomes

The silver ratio is intimately connected to trigonometric ratios for π8 = 22.5°.

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

Silver means | ||

0: | 0 + √42 | 1 |

1: | 1 + √52 | 1.618033989 |

2: | 2 + √82 | 2.414213562 |

3: | 3 + √132 | 3.302775638 |

4: | 4 + √202 | 4.236067978 |

5: | 5 + √292 | 5.192582404 |

6: | 6 + √402 | 6.162277660 |

7: | 7 + √532 | 7.140054945 |

8: | 8 + √682 | 8.123105626 |

9: | 9 + √852 | 9.109772229 |

⋮ | ||

n: | n + √4 + n^{2}2 |

The more general simple continued fraction expressions

are known as the **silver means** or **metallic means**^{[1]} of the successive natural numbers. The golden ratio is the silver mean between 1 and 2, while the silver ratio is the silver mean between 2 and 3. The term "**bronze ratio**", or terms using other names of metals, are occasionally used to name subsequent silver means. The values of the first ten silver means are shown at right.^{[2]} Notice that each silver mean is a root of the simple quadratic equation

- where
*n*is any positive natural number.

These properties are valid only for integers m, for nonintegers the properties are similar but slightly different.

The above property for the powers of the silver ratio is a consequence of a property of the powers of silver means. For the silver mean *S* of *m*, the property can be generalized as

where

Using the initial conditions K_{0} = 1 and K_{1} = *m*, this recurrence relation becomes

The powers of silver means have other interesting properties:

- If
*n*is a positive even integer:

Additionally,

Also,

In general:

The silver mean *S* of *m* also has the property that

meaning that the inverse of a silver mean has the same decimal part as the corresponding silver mean.

where *a* is the integer part of *S* and *b* is the decimal part of *S*, then the following property is true:

Because (for all *m* greater than 0), the integer part of *S _{m}* =

Therefore the silver mean of m is a solution of the equation

It may also be useful to note that the silver mean *S* of −*m* is the inverse of the silver mean *S* of *m*

Another interesting result can be obtained by slightly changing the formula of the silver mean. If we consider a number

then the following properties are true:

- if
*c*is real, - if
*c*is a multiple of*i*.

The silver mean of *m* is also given by the integral

The paper sizes under ISO 216 are rectangles in the proportion 1:√2 (approximately 1:1.4142135 decimal), sometimes called "A4 rectangles". Removing a largest possible square from a sheet of such paper leaves a rectangle with proportions 1:√2−1 which is the same as 1+√2:1, the silver ratio. Removing a largest square from one of *these* sheets leaves one again with aspect ratio 1:√2. A rectangle whose aspect ratio is the silver ratio is sometimes called a **silver rectangle** by analogy with golden rectangles. Confusingly, "silver rectangle" can also refer to the paper sizes specified by ISO 216.

Removing the largest possible square from either kind yields a silver rectangle of the other kind, and then repeating the process once more gives a rectangle of the original shape but smaller by a linear factor of 1+√2.^{[3]}

However, only the Lichtenberg ratio rectangles (rectangles with the shape of ISO 216 paper) have the property that by cutting the rectangle in half across its long side produces two smaller rectangles of the same aspect ratio.

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 inradius of the octagon (the distance between opposite sides) is δ_{S}*t*, and the area of the octagon is 2δ_{S}t^{2}.^{[3]}

**^**Vera de Spinadel (1999)The Family of Metallic Means,*Vismath*1(3) from Mathematical Institute of Serbian Academy of Sciences and Arts**^**Table of silver means from Wolfram MathWorld- ^
^{a}^{b}Kapusta, Janos (2004), "The square, the circle, and the golden proportion: a new class of geometrical constructions",*Forma***19**: 293–313.