Silver cut

The silver section in a regular octagon

The silver section (based on the term golden section ) is the division ratio of a route or other size, in which the ratio of the sum of the doubled larger and the smaller part to the larger part is equal to the ratio of the larger to the smaller part.

Definition and characteristics

Silver cut, proportions of the track parts:
${\displaystyle \delta _{S}={\frac {a}{b}}={\frac {2a+b}{a}}}$

With a larger and a smaller part as well as a silver cut applies: ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \delta _{S}}$

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

The silver cut therefore satisfies the equation ${\displaystyle \delta _{S}}$

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

or transformed results in the quadratic equation

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

Because of this it follows ${\displaystyle \delta _{S}>1}$

${\displaystyle \delta _{S}=1+{\sqrt {2}}\approx 2{,}414\,213\,562\,373\,095\,048\,801\,688\,724\,210}$.

The golden and silver ratio can be determined by the function

${\displaystyle f(x)={\frac {x}{2}}+{\sqrt {\left({\frac {x}{2}}\right)^{2}+1}}}$

where the golden ratio is a functional value for , the silver ratio is for . ${\displaystyle x=1}$ ${\displaystyle x=2}$

The silver section can also be expressed by trigonometric functions and is associated with the angle : ${\displaystyle \pi /8=22{,}5^{\circ }}$

${\displaystyle \delta _{S}=\cot {\frac {1}{8}}\pi }$

${\displaystyle \delta _{S}=\tan {\frac {3}{8}}\pi }$

In addition, like the golden ratio, the silver ratio has a particularly simple representation as a chain fraction :

${\displaystyle \delta _{S}=[2;2,2,2,\dotsc ]=2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\dotsb }}}}}}}$

construction

initial situation

Silver section in half a regular octagon according to the Pythagorean theorem :
${\displaystyle {\overline {AB}}\;{\widehat {=}}\;{\sqrt {2}}=a+b}$
${\displaystyle {\overline {IJ}}\;{\widehat {=}}\;1+{\sqrt {2}}-1+1=1+{\sqrt {2}}=\delta _{S}}$

Based on the regular octagon with one side length (as in the above sketch The silver section in a regular octagon ), the following description of the adjacent construction should clarify the division of a line in the ratio of the silver section. ${\displaystyle b_{0}=1}$

• The geometric derivation of the numerical value of with a compass and ruler is created in the course of the construction. It turns out to be a practicable alternative to arithmetic derivation. ${\displaystyle 1+{\sqrt {2}}}$ ${\displaystyle \delta _{S}}$

First, after the establishment of a perpendicular on a straight line , in each case from the point to the point a quarter circle with a radius drawn, so that the yield points and . If you now halve the right angle , you get the angle , the point of intersection and thus the first side of the half octagon as a line . There follows a circular arc having the radius around the point and parallel to the track from ; both intersect and form the second side of the half octagon. In order to get the center of the half octagon, one constructs the two vertical lines and the two sides of the octagon. Then the center axis is drawn through the center point parallel to the line and around the semicircle with the radius . The result is the point of intersection and thus the third side of the half octagon as a line . The connections of the points and with the center result in the point of intersection and the central angle of the octagon side. To complete the half octagon, you need two more perpendiculars to the route , each from the points and down to the central axis. This finally results in the two intersections and . ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle r=1}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle CAD}$ ${\displaystyle 45^{\circ }}$ ${\displaystyle E}$ ${\displaystyle {\overline {AE}},}$ ${\displaystyle {\overline {AE}}}$ ${\displaystyle E}$ ${\displaystyle {\overline {AC}}}$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle {\overline {EF}}}$ ${\displaystyle M}$ ${\displaystyle Ms_{1}}$ ${\displaystyle Ms_{2}}$ ${\displaystyle M}$ ${\displaystyle {\overline {AC}}}$ ${\displaystyle M}$ ${\displaystyle {\overline {AM}}}$ ${\displaystyle H}$ ${\displaystyle {\overline {FH}}}$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle M}$ ${\displaystyle B}$ ${\displaystyle 45^{\circ }}$ ${\displaystyle {\overline {AH}}}$ ${\displaystyle A}$ ${\displaystyle H}$ ${\displaystyle I}$ ${\displaystyle J}$

The side of the triangle intersects the line , the length of which corresponds, at the point and divides it there in the ratio of the silver section. ${\displaystyle {\overline {EM}}}$ ${\displaystyle MFE}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle {\sqrt {2}}}$ ${\displaystyle C}$

The result shows that each of the two distances and has the length  . ${\displaystyle {\overline {AH}}}$ ${\displaystyle {\overline {IJ}}}$ ${\displaystyle 1+{\sqrt {2}}-1+1=1+{\sqrt {2}}}$

If you put in the general formula for  results ${\displaystyle \delta _{S}={\frac {a}{b}}={\frac {a_{0}}{b_{0}}}}$

${\displaystyle {\overline {IJ}}={\frac {a_{0}}{b_{0}}}={\frac {1+{\sqrt {2}}}{1}}=1+{\sqrt {2}},}$ it follows

${\displaystyle {\overline {IJ}}=\delta _{S}=1+{\sqrt {2}}\approx 2{,}414\,213\,562.}$

• In words, the length of the route  corresponds to the numerical value ${\displaystyle {\overline {IJ}}}$ ${\displaystyle \approx 2{,}414\,213\,562.}$

Inner division

Silver cut, internal division
${\displaystyle {\frac {a}{b}}=1+{\sqrt {2}}=\delta _{S}}$

For the inner division of the line in the ratio of the silver section, the following construction elements can be derived from the drawing of the half regular octagon: ${\displaystyle {\overline {AB}}}$

• Green triangle ${\displaystyle ACE}$
• Midpoint of the route ${\displaystyle G}$ ${\displaystyle {\overline {AB}}}$
• Circular arc around , creates dividing point ${\displaystyle CED}$ ${\displaystyle A}$ ${\displaystyle C}$

At the beginning, the distance is halved, the center point results . Then you draw the semicircle with the radius around the point . A vertical line follows the line through the point , resulting in the intersection with the semicircle. The final circular arc around the point with the radius divides into the distance in the ratio of the silver section as a larger and as a smaller part. ${\displaystyle {\overline {AB}}}$ ${\displaystyle G}$ ${\displaystyle {\overline {GA}}}$ ${\displaystyle G}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle G}$ ${\displaystyle E}$ ${\displaystyle A}$ ${\displaystyle {\overline {AE}}}$ ${\displaystyle C}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle a}$ ${\displaystyle b}$

External division

Silver cut, outer division
${\displaystyle {\frac {a}{b}}=1+{\sqrt {2}}=\delta _{S}}$

Similar to the golden section, the silver section can also be constructed with an external division by extending the specified distance.

For the outer division of the line in the ratio of the silver section, the following construction elements can be derived from the drawing of the half regular octagon: ${\displaystyle {\overline {AB}}}$

• Green triangle ${\displaystyle ACE}$
• Arc around ${\displaystyle CED}$ ${\displaystyle A}$
• Arc around , creates the segment ${\displaystyle ABF}$ ${\displaystyle E}$ ${\displaystyle {\overline {AB}}}$

It begins with the construction of a right angle (a vertical line ) on the given route at the point . Then a quarter circle is drawn around the point from the point up to the vertical, resulting in the intersection point . Now halve the right angle , resulting in the angle and the point of intersection . It continues with the extension of the route from the point by about half the distance . The final arc of a circle around the point of radius extended the predetermined distance in the length of the track . Thus, the route is divided in the ratio of the silver section with a larger and a smaller part. ${\displaystyle {\overline {AC}}}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle CAD}$ ${\displaystyle 45^{\circ }}$ ${\displaystyle E}$ ${\displaystyle {\overline {AC}}}$ ${\displaystyle C}$ ${\displaystyle {\overline {AC}}}$ ${\displaystyle E}$ ${\displaystyle {\overline {AE}}}$ ${\displaystyle {\overline {AC}}}$ ${\displaystyle B}$ ${\displaystyle {\overline {CB}}}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle a}$ ${\displaystyle b}$

Silver rectangle

Silver rectangle,
based on the silver cut with internal division
${\displaystyle {\frac {a}{b}}={\frac {a_{1}}{b_{1}}}=1+{\sqrt {2}}=\delta _{S}}$

A rectangle with the side lengths and is called a silver rectangle if the quotient of the side lengths is just the silver section: ${\displaystyle a}$ ${\displaystyle b<a}$

${\displaystyle {\frac {a}{b}}={\frac {1}{{\sqrt {2}}-1}}=1+{\sqrt {2}}=\delta _{S}}$

The silver rectangle can be constructed with a compass and ruler .

In order to find the two side lengths and , you first have to divide any line in the ratio of the silver section using one of the two methods described above (inner division or outer division). The side thus determined is now folded up into the vertical and then the silver rectangle is completed. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle b}$

The adjacent construction shows that the midpoint of the line now divides the side length in the ratio of the silver section. This creates the side lengths and another silver rectangle, which of course can be continued as far as you want with the construction of new "silver" pairs . ${\displaystyle G}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle a}$ ${\displaystyle a_{1}}$ ${\displaystyle b_{1}}$ ${\displaystyle (a_{2},b_{2}),\ (a_{3},b_{3}),\dotsc }$

In contrast to the golden ratio and the golden rectangle, there are only a few examples from everyday life where you can observe this quotient. For example, there are cars whose length and width correspond to the silver cut. A simple way to create a silver rectangle yourself is with the help of a DIN A4 sheet. This has an aspect ratio of . Such a silver rectangle can be constructed by bending and cutting. ${\displaystyle 1:{\sqrt {2}}}$

literature

• Donald B. Coleman: The Silver Ratio: A Vehicle for Generalization . In: The Mathematics Teacher , Vol. 82, No. 1 (January 1989), pp. 54-59 ( JSTOR 27966097 ).

Web links

• An Introduction to Continued Fractions. The Silver Means .
• Continued Fractions and the Fibonacci Numbers .
• Eric W. Weisstein : Silver Ratio . In: MathWorld (English).
• The silver cut . Folding of a silver square.
• Numberphile: The Silver Ratio on YouTube , May 11, 2018, accessed January 19, 2019.

Individual evidence

1. ↑ See also Hans Walser: Colloquium on mathematics, computer science and teaching. In: 4.3 Diagonal intersection angle in the silver rectangle, 5. The regular octagon. November 20, 2014, accessed June 20, 2017 .