Daniel DOBRE
MATHEMATICAL PROPERTIES OF THE GOLDEN RATIO –
A FASCINATING NUMBER
Abstract: The Golden Ratio appears in many situations: in geometry, in lists with special numbers drawn
up by mathematicians, in art history lectures, in rose petals arrangement or spiral shells of mollusks. This
paper presents definitions and algebraic properties of Golden Section and their applications in geometry,
in the calculation pentagon and decagon sides and area, and volume of Platonic solids. Golden Ratio
occurrences are analyzed in geometric shapes as Golden Triangle, Golden Rectangle and Golden
Gnomon. Different expressions involving radicals that continue indefinitely and continued fractions with
Golden Section value are given and discussed. It is also revealed the connection between the Fibonacci
series and the Golden Ratio. Are presented two patterns that have symmetry of order five and that have
incorporated in them the Golden Section.
Key words: proportion, pentagram, pentagon, Golden Triangle, Golden Gnomon, Golden Ratio.
1. INTRODUCTION
“Golden Number” represents a geometrical
proportion known since antiquity. This number has
received in the nineteenth century the honorary name
“Golden Ratio” or “Golden Section”.
In the mathematical literature, the common symbol
for the Golden Ratio is the Greek letter tau (τ) which
means “the cut” or “the section”. At the beginning of the
twentieth century, the mathematician Mark Barr gave the
ratio the name of phi (ϕ), the first Greek letter in the
name of Phidias, the great Greek sculptor. Barr decided
to honor the sculptor because a number of art historians
maintained that Phidias had made frequent use of the
Golden Ratio in his sculptures [1]. Golden Section is the
core of an infinite number of phenomena, it can be found
at the intersection of Euclidean geometry with fractal
geometry or in a variety of human made objects and
works of art.
The paper describes the fascinating properties
assigned of number phi, or 1.6180339887…, widely
known as “the Golden Ratio”, “the Golden Section” or
“the Golden Mean”. This mathematical relationship was
discovered by Euclid more than two thousand years ago
because of its principal role in the construction of the
pentagram. This number appears in the most astonishing
variety of places, from mollusk shells, sunflower florets,
and rose petals to the shape of the galaxy.
This paper develops the relationship between the
Golden Ratio and triangles (equilateral triangle, pentagon
triangle, isosceles triangle, and whirling triangle),
rectangles, pentagrams and Platonic solids.
The geometrical plane figure most directly related to
the Golden Ratio is the regular pentagon, which has a
fivefold symmetry. Also, the five regular polyhedra are
intimately connected with the Golden Section.
2. ALGEBRAIC PROPERTIES OF THE GOLDEN
RATIO
The first definition of the Golden Section was given
around 300 B.C. by Euclid of Alexandria. The proportion
derived from a simple division of a line with intermediate
point C (fig. 1) such as:
AC
AB
=
ϕ =
(1)
CB
AC
with
AB>AC; AC>CB
(2)
was called by Euclid “extreme and mean ratio”.
Fig. 1 Dividing a line segment according to the Golden Ratio
In other words, if the ratio of the length of AC to that
of CB is the same as the ratio of AB to AC, then the line
has been cut in a Golden Ratio.
In Euclid’s Elements (a fundamental work on
geometry and number theory) the Golden Ratio appears
in several places. The first definition of the Golden Ratio
(“extreme and mean ratio”), in relation to areas, is given
in Book II. A second definition, in relation to proportion,
appears in Book VI. Then, the Golden Ratio is used
especially in the construction of the pentagon (in Book
IV) and in the construction of the icosahedron and
dodecahedron (in Book XIII) [1].
The precise value of the Golden Ratio (the ratio of
AC to CB in fig. 1) is never–ending and never–repeating
number:
ϕ = 1.6180339887...
(3)
Being an irrational number, the Golden Ratio cannot
be expressed as a fraction (as a rational number). In other
words, we cannot find some common measure that is
contained in the two lengths AC and CB.
a) The Golden Ratio possesses unique algebraic
properties. In fig. 1 we may take the length of the shorter
segment, CB, to be 1 unit and the length of the longer
one, AC, to be x units:
CB = 1; AC = x.
JOURNAL OF INDUSTRIAL DESIGN AND ENGINEERING GRAPHICS
(4)
11
Mathematical Properties of the Golden Ratio – a Fascinating Number
The point C was chosen so that:
AC
AB
AC + CB
=
=
CB
AC
AC
(5)
x 2 = 1 + 1 + 1 + 1 + 1 + ...
By substitution we get:
x
x +1
=
1
x
(6)
Multiplying both sides by x, we obtain x 2 = x + 1
or the simple quadratic equation:
(7)
x2 − x −1 = 0
The two solutions of the equation for the Golden
Ratio are:
1+ 5
(8)
x1 =
=ϕ
2
x2 =
x = 1 + 1 + 1 + 1 + ...
Let us square both sides of the last equation:
1−
5
2
=−
1
ϕ
(9)
x2 = 1 + x
1
ϕ
= 0 . 6180339887 ...
(11)
both having precisely the same digits after the decimal
point. The Golden Ratio has the unique properties that
we produce its square by simply adding the number 1
and its reciprocal by subtracting the number 1. The
negative solution of the Eq. (7), x2, is equal precisely to
the negative of 1/ϕ.
We defined ϕ to be:
ϕ2 = ϕ +1
(12)
If we divide both sides of Eq. (12) by ϕ, we obtain:
ϕ = 1+
1
ϕ
(13)
Here is another definition of ϕ – that number which is
1 more than its reciprocal.
b) The value of the following expression that involves
square roots that go on forever is equal to ϕ:
1 + 1 + 1 + 1 + ...
(14)
We denote the value of the expression by x. We
therefore have:
12
VOLUME 8 │ ISSUE 2 │ NOVEMBER 2013
(17)
that is precisely the equation that defines the Golden
Ratio.
c) The value of the following expression that involves
continued fractions is equal to ϕ:
1
1+
1
1+
1+
1
1+K
(18)
Based on previous procedure, we denote the value by
x. Thus:
5
= ϕ = 1 . 6180339887 ...
2
gives the value of the Golden Ratio, that is irrational,
being half the sum of 1 and the square root of 5.
The Golden Ratio has some unique properties:
(10)
(16)
We therefore obtain the quadratic equation:
The positive solution 1 +
ϕ 2 = 2 . 6180339887 ...
(15)
1
x = 1+
1
1+
1+
1
1+K
(19)
But the denominator of the second term on the righthand side is in fact identical to x itself. We therefore
have the equation:
x = 1+
1
x
(20)
Multiplying both sides by x, we obtain x 2 = x + 1
which is again the equation defining the Golden Ratio.
We find that this continued fraction is also equal to ϕ.
d) Also, we may use other definitions of the Golden
Section:
ϕ = 1 + 2 sin (π / 10 ) = 1 + 2 sin 18 °
(21)
ϕ = 2 cos (π / 5 ) = 2 cos 36 °
(22)
1
1
csc (π / 10 ) = csc 18 °
2
2
(23)
ϕ = 2 sin (3π / 10 ) = 2 sin 54 °
(24)
ϕ =
e) The Fibonacci sequence is intimately connected
with the Golden Section. The Fibonacci series:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …
has the general property that each term of the sequence
(from the third) is equal to the sum of the two preceding
terms:
Mathematical Properties of the Golden Ratio – a Fascinating Number
Fn + 2 = Fn + 1 + Fn
(25)
where Fn is the n-th term of the sequence.
If we calculate the ratio of two successive numbers of
the Fibonacci sequence, the ratio of the two numbers
oscillate around Golden Section (being alternatively
higher or lower than it), but tends to the Golden Section
(fig. 2).
lim
n→∞
Fn+1
=ϕ
Fn
(26)
So the ratio of successive Fibonacci numbers Fn+1/Fn
converges on ϕ when n increases.
Fig. 3 Golden Triangle (left) and Golden Gnomon (right)
Figure 4 demonstrates a unique property of Golden
Triangles and Golden Gnomons – they can be dissected
into smaller triangles that are also Golden Triangles and
Golden Gnomons. If we continue in this fashion we get a
set of Whirling Triangles.
2.25
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
0
2
4
6
8
10
12
Fig. 4 Dividing a Golden Triangle
Fig. 2 Ratio of two successive numbers
in the Fibonacci sequence
In the Fibonacci series, the numbers increase rapidly
as a power of the Golden Section.
The value of any Fibonacci number, Fn, is entirely
based on the Golden Section (Binet's formula):
Fn =
=
1
5
1
5
 1 + 5  n  1 − 5  n 
 −
 

 2  =
  2 

 

 n  1 n 
1  n
1 
ϕ − (− 1)n n 
ϕ −  −   =

ϕ 
5 

 ϕ  
(27)
Figure 4 shows how constructing a sequence of
smaller Golden Triangle yields a spiral.
Figure 5 shows a way to draw a Golden Rectangle
with the Golden Ratio:
• Draw a square (of size 1);
• Place a dot half way along one side;
• Draw a line from that point to an opposite corner (it
will be 5 / 2 in length);
• Turn this line so that it runs along the square's side.
Then you can extend the square to be a Golden Rectangle
with the Golden Ratio.
For relatively large values of n, the second term in
brackets becomes very small, so that Fn can be
n
determined as the integer number nearest to ϕ
5.
3. THE APPEARANCE OF THE GOLDEN
SECTION IN GEOMETRICAL SHAPES
In a regular pentagon, the ratio of the diagonal to the
side is equal to ϕ, this fact providing a simple means of
constructing the regular pentagon.
The triangle in the middle of figure 3, with a ratio of
side to base of ϕ, is known as a Golden Triangle. This
isosceles triangle has the two base angles of 72 degrees.
The two triangles on the sides, with a ratio of side to base
of 1/ϕ, are called Golden Gnomons. The 36°–72°–72°
triangle occurs in both the pentagram and the decagon.
Fig. 5 Drawing a Golden Rectangle
The self–generating capability of the Golden
Rectangle is frequently cited [1]. The lengths of the sides
of the rectangle are in a Golden Ratio to each other. If we
cut off a square from this rectangle we will obtain a
smaller rectangle that is also a Golden Rectangle. The
dimensions of this rectangle are smaller than those of the
first rectangle by factor of ϕ. We can now cut a square
from the second Golden Rectangle and we will obtain
again a Golden Rectangle, the dimensions of which are
smaller by another factor of ϕ. Continuing this process
ad infinitum, we will produce smaller and smaller
Golden Rectangle. Figure 6 shows how constructing a
sequence of smaller Golden Rectangle yields a spiral.
JOURNAL OF INDUSTRIAL DESIGN AND ENGINEERING GRAPHICS 13
Mathematical Properties of the Golden Ratio – a Fascinating Number
The Golden Rectangle is the only rectangle with the
property that cutting a square from it produces a similar
rectangle. If we draw two diagonals of any pair of
rectangles in the series, as in figure 6, they will all
intersect at the same point. The series of continuously
diminishing rectangles converges to that never–reachable
point.
The ratio of the length of diagonal AC to the length of
side CD of first pentagon is ϕ, the ratio of the lengths of
CD to DR is ϕ, the ratio of the lengths of DR to RQ is ϕ,
and so on.
If we inscribe a regular decagon in a circle, the ratio
of the radius of the circle to the side of the decagon
forms the Golden Ratio (fig. 9):
r
l10
=ϕ
(28)
Fig. 6 Constructing a sequence of smaller Golden Rectangle
On a five–pointed star pattern (fig. 7), each of the five
isosceles triangles that make the corners of a pentagram
has the property that the ratio of the length of its longer
side to the shorter one (the base) is equal to the Golden
Ratio, ϕ.
Fig. 9 Golden Ratio in a regular decagon
Figure 10 shows how we can use any circle to
construct on it a hexagon and an equilateral triangle.
Joining a pair of three points, it reveals a line and its
Golden Section point as follows:
• On any circle, construct a regular hexagon ABCDEF;
• Choose every other point to make an equilateral
triangle ACE;
• On two of the sides of that triangle (AE and AC),
mark their mid–points M and N by joining the centre
of the circle to two of the unused points of the
hexagon (F and B);
• The line MN is then extended to meet the circle at
point P. N is the Golden Section point of the line MP.
Fig. 7 Pentagram and regular pentagon
The pentagram is related to the regular pentagon. If it
connects all the vertices of the pentagon by diagonals, we
obtain a pentagram (fig. 8). The diagonals also form a
smaller pentagon at the center, and the diagonals of this
pentagon form a pentagram and a yet smaller pentagon.
This progression can be continued ad infinitum, creating
smaller and smaller pentagons and pentagrams. All of
these figures have the property that every segment is
smaller than its predecessor by a factor that is precisely
equal to the Golden Ratio, ϕ.
Fig. 8 Relationship between pentagram and regular pentagon
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VOLUME 8 │ ISSUE 2 │ NOVEMBER 2013
Fig. 10 Golden Ratio in a regular hexagon
The diagram shown in figure 11, containing only
equilateral triangles, has many Golden Sections.
Fig. 11 Golden Ratio in equilateral triangles
Mathematical Properties of the Golden Ratio – a Fascinating Number
Figure 12 shows the sketch of a quadrilateral pyramid
(tetrahedron) in which a is half the side of the base BC, s
is the height of the triangular face VBC and h is the
pyramid’s height VO.
five polyhedra are also known as the platonic or
Pythagorean solids, and are intimately connected with
the Golden Section, ϕ.
The ancients considered the tetrahedron to represent
the element of fire; the octahedron, air; the icosahedron,
water; and the cube, earth. The dodecahedron symbolized
the harmony of the entire cosmos [3].
In particular, the surface area and volume of a
dodecahedron and an icosahedron with an edge length of
one unit are defined in Table 1.
Table 1
Solid
dodecahedron
Fig. 12 Quadrilateral pyramid
If the square of the pyramid’s height is equal to s× a
(the area of the triangular face), then the ratio s/a is
precisely equal to the Golden Ratio. We have that:
2
h = s×a
(29)
Using the Pythagorean theorem in the right angle
triangle VOM, we have:
s2 = h2 + a2
(30)
We can substitute for h2 from the Eq. (29) to obtain:
s2 = s × a + a2
(31)
Dividing both sides by a2, we get:
(s/a)2 = (s/a) + 1
(32)
In other words, if we denote s/a by x, we have the
quadratic equation:
x2 = x + 1
(33)
which is precisely the equation defining the Golden
Ratio. The Great Pyramid of Cheops, built before 2000
B.C., has been measured and the ratio s/a = 1.62 which is
very close to the Golden Section (differing from it by
less than 0.1 percent). However cannot say that the
ancient Egyptians knew the significance of the Golden
Ratio, ϕ.
icosahedron
Surface area
Volume
15 ϕ
3−ϕ
5ϕ 3
6 − 2ϕ
5 3
5ϕ5/6
The symmetry of the Platonic solids leads to other
interesting properties. For example, the cube and the
octahedron have the same number of edges (twelve), but
their number of faces and vertices are interchanged (the
cube has six faces and eight vertices and the octahedron
eight faces and six vertices).
The same is true for the dodecahedron and
icosahedron. Both have thirty edges, and the
dodecahedron has twelve faces and twenty vertices,
while it is inversely for the icosahedron. Because these
two solids are dual to each other they have the same
symmetry group. The order of the group of direct
symmetries (all rotations) is S = 60. The elements are: 4
rotations (by multiples of 2π/5) about centers of 6 pairs
of opposite faces; 1 rotation (by π) about centers of 15
pairs of opposite edges; 2 rotations (by ± 2π/3) about 10
pairs of opposite vertices. Together with the identity,
these rotations totalize 60 elements [2]. These similarities
in the symmetries of the Platonic solids allow for
mappings of one solid into its dual (reciprocal) solid.
We define the dual of a regular polyhedron to be
another regular polyhedron, which is formed by
connecting the centers of the faces of the original
polyhedron [1]. For example, if we connect the centers of
all the faces of a cube, we obtain an octahedron, while if
we connect the centers of the faces of an octahedron, we
obtain a cube (fig.13).
4. THE GOLDEN SECTION AND THE PLATONIC
SOLIDS
Platon and the Golden Section are linked through the
Platonic solids. A regular polyhedron is a threedimensional shape whose edges are all of equal length,
whose faces are all identical and equilateral, and whose
vertices all touch the surface of a circumscribing sphere.
There are only five regular polyhedra: the tetrahedron (4
triangular faces, 4 vertices, 6 edges); the
cube/hexahedron (6 square faces, 8 vertices, 12 edges);
the octahedron (8 triangular faces, 6 vertices, 12 edges);
the dodecahedron (12 pentagonal faces, 20 vertices, 30
edges); and the icosahedron (20 triangular faces, 12
vertices, 30 edges). In each case, the number of faces
plus corners equals the number of edges plus 2. These
Fig. 13 Dual of a regular polyhedron
The same procedure can be applied to map an
icosahedron into a dodecahedron and vice versa, and the
ratio of the edge lengths of the two solids (one embedded
JOURNAL OF INDUSTRIAL DESIGN AND ENGINEERING GRAPHICS 15
Mathematical Properties of the Golden Ratio – a Fascinating Number
in the other) can be expressed of the Golden Ratio, as
ϕ2 / 5
The tetrahedron is self–reciprocating (joining the four
centers of the tetrahedron’s faces makes another
tetrahedron). A pyramid (a tetrahedron), an octahedron,
an icosahedron and a dodecahedron are all beautiful
because of the symmetries and equalities in their
relations.
The twelve vertices of any icosahedron can be
divided into three groups of four, with the vertices of
each group lying at the corners of a Golden Rectangle.
The rectangles are perpendicular to each other, and their
one common point is the center of the icosahedron (fig.
14). Similarly, the centers of the twelve pentagonal faces
of the dodecahedron can be divided into three groups of
four and each of those groups forms a Golden Rectangle.
Finally, two isosceles triangles (the 36°-72°-72°
triangle and 36°-36°-108° triangle) are the basic building
shapes of Penrose tilings [5]. Penrose tiles can be pairs of
several different shapes, though the two most interesting
are those presented in figure 15a, known as “darts” and
“kites” [1], [4]. The two shapes were derived from a
pentagon as is shown on the fig. 15c. The dart is
produced by adding two of the central triangles together
and the kite by the addition of the two side triangles. The
dart and kite shapes, when added together, form the
rhombi that are known as Penrose rhombi.
The darts and kites can be obtained from a rhombus
with degree measures of 72° and 108° by dividing the
long diagonal into two segments in the Golden Ratio ϕ =
1.618… then joining the dividing point to the obtuse
corners as shown in fig. 15c. Both rhombuses are
composed of two Golden Triangles. The ratio of the
areas of the kite and dart is the Golden Ratio as well.
5. CONCLUSION
Fig. 14 Perpendicular Golden Rectangles in an icosahedron
a
The Golden Ratio is a very special number. The
Golden Ratio can be found not only in natural
phenomena but also in a variety of human–made objects
and works of art.
This number appears in numerous situations: in
geometrical constructions, in lists of “favourite numbers”
processed by mathematicians, in the works of many
artists, architects and designers, in the animal kingdom,
and even in famous musical compositions [1].
The Golden Ratio combines two definitions of a
proportion: to express the comparative relation between
parts of objects with respect to size or quantity or to
describe a harmonious relationship between different
parts (to one another and to the whole).
REFERENCES
[1] LIVIO, M.: Secţiunea de aur – Povestea lui phi, cel
mai uimitor număr, Ed. Humanitas, Bucureşti, 2005.
[2] DOBRE, D., SIMION, I.: Symmetry - Mathematical
concepts and applications in technology and
engineering. Journal of Industrial Design and
Engineering Graphics, no. 5, 2009, pp. 21-26, ISSN
1843-3766.
[3] WEYL, H.: Simetria, Ed. Ştiinţifică, Bucureşti, 1966.
[4] LOCKERBIE, J.: Notes for a Study of the Design and
Planning of Housing for Qataris, London, 2010.
[5] PENROSE, R.: The role of aesthetics in pure and
applied mathematical research, Institute of
Mathematics and its Applications Bulletin 10, no. 7/8,
pp. 266-271, 1974.
b
c
d
Fig. 15. Penrose patterns [5]
a – Dart and kite; b, c – Penrose rhombi;
d –Penrose patterns
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VOLUME 8 │ ISSUE 2 │ NOVEMBER 2013
Author:
Eng. Daniel DOBRE, Ph.D., Associate professor,
Department of Engineering Graphics and Industrial
Design, University Politehnica of Bucharest, Romania.
E-mail: [email protected]