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Golden ratio

The number is called the golden ratio. It is also known as the golden mean, the golden section and divine proportion or ratio (this name appears for the first time in the 1500's and was used till the 19th century. Martin Ohm (1792-1872), a younger brother of physicist Georg Ohm, is believed to be the first to use the term golden ratio  ).

To compute the golden ration with higher precision go to .

Thus the Golden ratio is an algebraic number of order 2. Its minimal polynomial is (see below). The second root of this polynomial is  

Note that the reciprocal called golden ratio conjugate (or also silver ratio)  has minimal polynomial  . The first known approximate decimal expansion of the (inverse) golden ratio as "about 0.6180340" was given in 1597 by M.Maestlin (1550-1631) in a letter to J.Kepler (1571-1630).

The ancient Greeks  believed that the proportion is the most pleasing, and therefore all of their sculpture ¹, architecture elements, etc. made use of this proportion. A rectangle whose sides had this proportion was called the Golden Rectangle.

Plato in his Timaeus, considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos.

In Definition 3 of Book VI of Euclid’s Elements we can read: A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.

In terms of an equation we get

(1)

If we set the length to be and to represent the length of , then

(2)

This also gives the equation , or , that is    for .

Euclid in Proposition 30 of Book VI of his Elements referred to dividing a line at the point 0.6180399...  as dividing a line in the extreme and mean ratio. A construction how to cut a line segment in this manner appears earlier in Proposition 11 of Book II in an equivalent form stated in terms of rectangles: To cut a given straight line so that the rectangle contained by the whole and one of the segments equals the square on the remaining segment. The construction is then used in Book IV in order to construct regular pentagons and 15-sided polygons (Propositions 10 through 12 and 16). Another substantial part of Elements related to pentagon is in Book XIII. The important information for determination of the diagonal of pentagon is contained in Proposition 8 of Book XIII: If in an equilateral and equiangular pentagon straight lines subtend two angles are taken in order, then they cut one another in extreme and mean ratio, and their greater segments equal the side of the pentagon.

The blue triangle left has its sides in the golden ratio with its base, and the red triangle in the middle has its base in the golden ratio with one of the sides. The last fact can used to construct a regular pentagon

To divide a given line segment into extreme and mean ratio follow the following procedure:

• bisect , the midpoint of is
• erect a perpendicular at
• let be the intersection of the circle and the perpendicular and the circle with center at and radius
• let be the intersection of the line and the circle with center at and radius
• the intersection be the intersection of the circle with center at and radius with
• is divided internally into extreme and mean ratio at

The numerical value of the golden ratio to decimal places can be obtained by command:

If

then , that is

Along similar lines we can prove the continued fraction expansion

Ramanujan gave the following continued fraction expansion involving the golden ratio

Notes

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