Download The Golden Section

The Golden Mean
1
5

 1.618033989... , called the Golden Mean, or the Golden
2 2
Section, has many interesting and useful properties, some of which have been known
since the time of the ancient Greeks.
The number  
1. A regular five-pointed star, such as those appearing on the U. S. flag, may be
considered to be composed of a central pentagon and five isosceles triangles of equal size
conjoined so that the base (shorter side) of each triangle coincides with a side of the
pentagon. The ratio of the length of each of the two equal sides of one of the isosceles
triangles to its base is  .
2. If we divide a line segment into two pieces so that the ratio of the length of the whole
segment to the length of the longer piece is the same as the ratio of the length of the
longer piece to the length of the shorter piece, then this ratio is  as shown in the picture
below.
x 1 x

, or
1
x
1
5
  , and
equivalently x 2  x  1  0 , a quadratic equation whose solutions are x  
2 2
1
5
1
x 
  . Two interesting properties of the number  are apparent from this
2 2

equation: a) the square of  may be found by adding 1 to  , and b) the reciprocal of 
may be found by subtracting 1 from  .
This may be expressed mathematically in the form of the equation
3. If we construct a regular (equal-sided) pentagon, then construct an isosceles triangle
by extending two line segments from one vertex of the pentagon to the two opposing
vertices, we obtain what is called a Golden Triangle. It is called this because the ratio of
the length of either of the two equal sides to the base is  :
aba a
1
. If we let b = 1, then this equation is equivalent to the

 
ab
b 2 cos72
quadratic equation in the preceding paragraph.
4.  is the simplest continued square root, and also the simplest continued fraction. I.e.,
1
  1  1  1  1  .... , and   1 
1
1
. Either of these two identities may
1
1  ....
be verified by using the algebraic formulae from paragraph 2 (try it).
1
5. If we construct a rectangle for which the ratio of the longer side to the shorter side is
 , as shown in the figure below, and then partition the rectangle into a square and
another rectangle, the new rectangle will also have the property that the ratio of the
1
longer side to the shorter side is  , since
 .
 1
6. The Golden Section is also involved in the characteristics of two of the Platonic solids
(convex regular polyhedrons). There are exactly five Platonic solids: 1) the regular
tetradhedron, 2) the hexahedron (cube), 3) the regular octahedron, 4) the regular
dodecahedron, and 5) the regular icosahedron. 1“The five Platonic solids are
distinguished by the following properties: they are the only existing solids in which all
the faces (of a given solid) are identical and equilateral, and each of the solids can be
circumscribed by a sphere, (with all of its vertices lying on the sphere).” As a
consequence of these characteristics, it is possible to construct a fair die based on each
Platonic solid; for the regular tetrahedron, the die would have four faces, labeled 1, 2, 3,
4; for the cube, the die would be the ordinary die used at gaming tables; for the
octahedron, the die would have eight sides, labeled 1, 2, 3, 4, 5, 6, 7, 8; for the
dodecahedron, the die would have 12 sides, labeled 1, 2, 3, 4, …, 11, 12; and for the
icosahedron, the die would have 20 sides, labeled 1, 2, 3, 4, …, 19, 20.
Pictures of the Platonic solids are given below.
1) The tetrahedron:
2) The hexahedron:
3) The octahedron:
4) The dodecahedron:
5) The icosahedron:
If a dodecahedron has an edge length of 1 unit, then its total surface area is 15 3   ,
and its volume is
5 5
5 3
. An icosahedron with unit edge length has volume
.
6
6  2
7. The Fibonacci sequence is a sequence of natural numbers in which each number,
starting at the third place in the sequence, is the sum of the two preceding numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …. This sequence occurs in
nature in some surprising places.
“Suppose we have two glass plates made of slightly different types of glass (different
light refraction properties, or ‘indices of refraction’) mounted face to face. If we shine
light through the plates, the light rays can (in principle) reflect internally at four reflective
surfaces before emerging. More specifically, they can either pass through without
reflecting at all, or they can undergo one internal reflection, or two internal reflections,
three internal reflections, and so on, potentially an infinite number of internal reflections
before re-emerging. All of these are paths allowed by the laws of optics. Now count the
number of beams that emerge from this two-plate system. There is only one emerging
beam in the case of no reflections at all. There are two emerging beams when all the
possibilities for the rays to undergo precisely one internal reflection are considered,
because there are two paths the ray can follow. There are three emerging beams for all
the possibilities of two internal reflections, five beams for three internal reflections, eight
paths if the ray is reflected four times, thirteen paths for five reflections, and so on. The
number of emerging beams – 1, 2, 3, 5, 8, 13, … – form a Fibonacci sequence.”
1
If we form the sequence of ratios of pairs of successive Fibonacci numbers, we obtain
1/1
1.000000000
2/1
2.000000000
3/2
1.500000000
5/3
1.666666666
8/5
1.600000000
13/8
1.625000000
21/13
1.615384615
34/21
1.619047619
55/34
1.617647059
89/55
1.618181818
144/89
1.617977528
233/144
1.618055556
377/233
1.618025751
610/377
1.618037135
987/610
1.618032787
Etc.
The limit of the (oscillating) sequence of ratios of successive Fibonacci numbers is  .
1
The Golden Ratio, by Mario Livio, 2002, Broadway Books.
8. Consider the sequence of powers of a real number x: 1, x, x2, x3, x4, …. There are
exactly two such numbers that satisfy the condition that the sequence has the Fibonacci
property, i.e., that each number in the sequence, starting at the third, is the sum of the two
preceding numbers. The two numbers that satisfy this condition are ϕ and -1/ϕ.
9. The Fibonacci sequence (and thus the Golden Mean) also occurs in many other natural
systems:
The distribution of seeds in a sunflower is spiral. The seeds of the sunflower spiral
outwards in both clockwise and counterclockwise directions from the center of the
flower. The number of clockwise and counterclockwise spirals are two consecutive
numbers in the Fibonacci sequence.
Pine cones are one of the well-known examples of the Fibonacci sequence. All cones
grow in spirals, starting from the base where the stalk is, and going round and round the
sides until they reach the top.
Another notable example is the human body. In the human body, the ratio of the length
of forearm to the length of the hand is equal to 1.618, that is, Golden Ratio. Other wellknown examples for the human body are:
1. The ratio between the length and width of face
2. Ratio of the distance between the lips and where the eyebrows meet to the length
of nose
3. Ratio of the length of mouth to the width of nose
4. Ratio of the distance between the shoulder line and the top of the head to the head
length
5. Ratio of the distance between the navel and knee to the distance between the knee
and the end of the foot
6. Ratio of the distance between the finger tip and the elbow to the distance between
the wrist and the elbow