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Difference between revisions of "Golden ratio"

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457013.png" /> are the [[Fibonacci numbers|Fibonacci numbers]].
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457013.png" /> are the [[Fibonacci numbers|Fibonacci numbers]].
  
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<center><asy>
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{{:Golden ratio/Fig1}}
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pair A=(-1,0);
 
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pair B=(0,0);
 
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pair E=(0,0.5);
 
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pair C=A+(0.5*(sqrt(5)-1),0);
 
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pair D=(-1/sqrt(5), 0.5*(1-1/sqrt(5)));
 
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draw( A--B--E--cycle,currentpen+1.5 );
 
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dot(A,currentpen+3.5); dot(B,currentpen+3.5); dot(E,currentpen+3.5); dot(C,currentpen+3.5); dot(D,currentpen+3.5);
 
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draw( shift(E)*scale(0.5)*unitcircle,currentpen+1 );
 
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draw( shift(A)*scale(0.5*(sqrt(5)-1))*unitcircle,currentpen+1 );
 
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// draw( shift(B)*scale(0.5)*unitcircle, dashed+red );
 
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clip(A+(-0.15,-0.15)--B+(0.15,-0.15)--E+(0.15,0.15)--A+(-0.15,0.15)--cycle);
 
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label("$A$",A,S); label("$B$",B,S); label("$C$",C,S);
 
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label("$E$",E,N); label("$D$",D,N);
 
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label( "\small Golden Ratio construction", (-0.5,0.8) );
 
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shipout(scale(100)*currentpicture);
 
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</asy></center>
 
  
 
The golden ratio of a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457014.png" /> may be found by a geometric construction (see Fig.). Draw the perpendicular to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457015.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457016.png" />; on the perpendicular, mark off the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457017.png" />; then join <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457019.png" />, mark off <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457020.png" /> and finally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457021.png" />. Then
 
The golden ratio of a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457014.png" /> may be found by a geometric construction (see Fig.). Draw the perpendicular to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457015.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457016.png" />; on the perpendicular, mark off the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457017.png" />; then join <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457019.png" />, mark off <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457020.png" /> and finally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044570/g04457021.png" />. Then

Revision as of 11:51, 13 December 2014

golden section, sectio aurea, sectio divina, divina proportio, golden mean, harmonic division, division in extreme and mean ratio

The division of a line segment into two parts the greater of which, , is the mean proportional between the whole segment and the smaller part , i.e.

(*)

To find , one has to solve a quadratic equation,

the positive solution of which is

Condition (*) may also be written as

or

i.e. representing as a continued fraction, the convergents of which are

where are the Fibonacci numbers.

The golden ratio of a segment may be found by a geometric construction (see Fig.). Draw the perpendicular to at ; on the perpendicular, mark off the segment ; then join and , mark off and finally . Then

The golden ratio was already known in Antiquity. Its first appearance in the extant classical literature is in Euclid's Elements, Book II, 11 (3th century B.C.).

Principles based on the golden ratio or similar proportional ratios provided the basis for the compositional design of many works of art (mainly architectural works of Antiquity and the Renaissance).

Comments

The golden ratio occurs in many places in nature. It is assumed to have been known and used by the Pythagoreans (500 B.C.).

The term "divina proportio" (sectio divina, divine section) has been introduced by Fra Luca Pacioli in his book: De Divina Proportione (1509). Later on he collaborated with Leonardo da Vinci on this topic. The term "golden section" (sectio aurea) is from a later date (probably 18th century).

How to Cite This Entry:
Golden ratio. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Golden_ratio&oldid=35614
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