Golden ratio
In
mathematics, two quantities are in the
golden ratio
if their
ratio is the same as the
ratio of their
sum to the larger of the two quantities. The figure on the right
illustrates the geometric relationship. Expressed algebraically, for quantities
a
and
b
with
a
>
b
> 0,
where the Greek letter
phi (
or
) represents the golden ratio.
[1]
It is an
irrational
number with a value of:
[2]
The golden ratio is also called the
golden mean
or
golden section
(Latin:
sectio
aurea
).
[3][4][5]
Other names include
extreme and mean ratio
,
[6]
medial section
,
divine proportion
,
divine section
(Latin:
sectio divina
),
golden proportion
,
golden
cut
,
[7]
and
golden number
.
[8][9][10]
Mathematicians since
Euclid have studied the properties of the golden ratio,
including its appearance in the dimensions of a
regular pentagon and in a golden
rectangle, which may be cut into a square and a smaller rectangle with the same
aspect ratio. The golden ratio has also been used to analyze the proportions of
natural objects as well as man-made systems such as
financial markets, in some
cases based on dubious fits to data.
[11]
The golden ratio appears in some
patterns in
nature, including the
spiral arrangement of leaves
and other plant parts.
Some twentieth-century
artists and
architects, including
Le Corbusier and
Salvador
Dalí, have proportioned their works to approximate the golden ratio—especially in
the form of the
golden rectangle, in which the ratio of the longer side to the shorter is
the golden ratio—believing this proportion to be
aesthetically pleasing.
Calculation
History
Timeline
Applications and observations
Aesthetics
Architecture
Painting
Book design
Design
Music
Nature
Optimization
Line segments in the golden ratio
A golden rectangle with longer side
a
and shorter side
b
, when placed
adjacent to a square with sides of
length
a
, will produce a similar
golden rectangle with longer side
a +
b
and shorter side
a
. This illustrates
the relationship
.
Contents

List of numbers
·
Irrational numbers
ζ
(3)
·
√
2
·
√
3
·
√
5
·
φ
·
e
·
π
Binary
1.1001111000110111011...
Decimal
1.6180339887498948482...
[2]
Hexadecimal
1.9E3779B97F4A7C15F39...
Continued
fraction
Algebraic
form
Infinite
series
Perceptual studies
Mathematics
Irrationality
Minimal polynomial
Golden ratio conjugate
Alternative forms
Geometry
Relationship to Fibonacci sequence
Symmetries
Other properties
Decimal expansion
Pyramids
Mathematical pyramids and triangles
Egyptian pyramids
Disputed observations
See also
References and footnotes
Further reading
External links
Two quantities
a
and
b
are said to be in the
golden ratio
φ
if
One method for finding the value of
φ
is to start with the left fraction.
Through simplifying the fraction and substituting in b/a = 1/
φ
,
Therefore,
Multiplying by
φ
gives
which can be rearranged to
Using the
quadratic formula, two solutions are obtained:
and
Calculation

Because
φ
is the ratio between positive quantities
φ
is necessarily positive:
.

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- Llorando
- Mechanical Engineering, Golden ratio, Golden Triangle