Khan Amore’s Commentary on The Divine Proportion, RÓŻNOŚCI, Złoty podział

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Khan Amore’s Commentary on
The Divine Proportion
[  = 1.6180339887498948482045868343656381177203…]
(Also known as the Golden Section, the Golden Ratio, the Golden Mean, or the Mean Proportional)
Contents:
§01: Definition of Divine Proportion (or Golden Section, or Golden Ratio, or
Mean Proportional), History of the Divine Proportion, and Symbolic
Representation.
Page
2
§02: Algebraic Derivation of Divine Proportion (or Golden Section, or Mean
Proportional). The derivation is the solution of a simple quadratic equation.
Page
4
§03: How to Geometrically Construct the Divine Proportion (or Golden Section,
or Mean Proportional) using only a Compass and Straight Edge.
Page
14
§04: Analysis of the Construction of the Divine Proportion (or Golden Section, or
Mean Proportional). The construction is based on the Pythagorean
Theorem.
Page
25
§05: The Golden Rectangle (having Height-to-Width Ratio in Divine Proportion)
and How to Construct the Golden Rectangle From a Square.
Page
28
§06: The Divine Proportion’s Connection With the Logarithmic Spiral, and with
the Chambered Nautilus. How to Draw a Spiral Using Golden Rectangles.
Page
31
§07: The Divine Proportion’s Connection with the Fibonacci Numbers and with
Spiraling Natural Structures.
Page
36
§08: The Divine Proportion’s Connection with the Pentagon, the Pentagram, the
Decagon, and with Various Polyhedra.
Page
42
§09: The Divine Proportion’s Representation as a Continued Fraction.
Page
50
§10: Miscellaneous Properties, including Reciprocals and Powers, of the Divine
Proportion.
Page
51
§11: Divine Proportions to be Found in Art, Architecture, and in the Most
Beautiful, Harmoniously-Proportioned Human Bodies (Including a Pictorial
Chart.)
Page
54
The Divine Proportion
1
§1
Definition of Divine Proportion
(or Golden Section, or Mean Proportional, or Golden Ratio:
A line segment is said to be divided in
golden section
(or in
divine proportion
)
if the larger segment is the
mean proportional
between the complete segment
and the smaller segment. In other words, two line segments are in divine
proportion when the ratio of the length of the smaller segment to
the length of the larger segment is equal to the ratio of the length of
the larger segment to the
sum
of the lengths of the smaller and
larger segments, taken together.
The History of the Divine Proportion:
Although
Pythagoras
is believed to have been the discoverer of the divine
proportion, and the ancient Greeks that followed him clearly knew how to
construct it, surprisingly, the ancients seem not to have had a special name for
this fundamental proportionality of Nature. In that definitive masterpiece of
geometry,
The Elements
, Euclid (as well as
Hypatia
) merely call this a division
or section “in the extreme and mean ratio” and this
mean proportional
was
used by them to construct the regular pentagon, the pentagram, the
dodecahedron and its dual, the icosahedron. Indeed, this division in mean
and extreme ratio appears many times in
The Elements
, particularly in Book
XIII, starting with Proposition 1 of that section of this most important book.
The ancient Greeks (particularly the Pythagoreans) regarded the pentagon,
the pentagram, and the Platonic Polyhedra with reverence and awe, and so it
should come as no surprise that they regarded with similar veneration the
“golden mean” proportionality which underlies these shapes, enables their
construction, and makes them what they are.
There is some evidence that the golden ratio was also important to the ancient
Egyptians, for the Rhind Papyrus refers to a “sacred ratio,” and the ratio in the
Great Pyramid at Gizeh of the altitude of a face to half the side of the base is
almost exactly 1.618 (i.e., the golden ratio).
The Divine Proportion
2
It is quite likely that the ancient Greeks used the divine proportion in their
architecture, for one has but to measure the dimensions of their most beautiful
structures (like the Parthenon) to find the golden ratio hiding there, but no
documentary proof remains that this was the result of deliberate calculation
rather than of heightened intuitive esthetic sensibilities. There is no doubt,
however, that this harmonious proportionality was consciously exploited by
Renaissance artists who knew it as the divine proportion. In 1509 Fra Luca
Pacioli published
De Divina Proportione
, illustrated with drawings of the
Platonic solids made by his friend, Leonardo da Vinci. Da Vinci was
probably the first to refer to the mean proportional as the “
sectio aurea
” (i.e.,
the golden section.)
Johannes Kepler (1571-1630), the brilliant discoverer of
Kepler’s Laws
and
of the ellipticity of planetary orbits, also rhapsodized over the Divine
Proportion, declaring that “Geometry has two great treasures: one is the
Theorem of Pythagoras, and the other the division of a line into extreme and
mean ratio; the first we may compare to a measure of gold, the second we
may name a precious jewel.”
Renaissance artists regularly used the golden section in composing paintings
into the most pleasing proportions, just as architects both ancient and modern
have used the golden ratio to plan and analyze the proportions of buildings.
For example, Vitruvius’
De Achitectura
uses the golden ratio to analyze the
elevation of the Milan Cathedral.
The ancient Greeks realized that the most esthetically-pleasing rectangle to the
human eye is one in which the height-to-width ratio is in divine proportion;
that they realized this is evident from the many golden ratios which are to be
found in their sculptures and temples. There have been skeptics, however,
who regarded this connection between esthetics and mathematics as akin to
numerology — an occult pseudo-science. In order to see if there was any
empirical evidence to support any such connection, the Germa n psychologist
Gustav Fechner made a serious and thorough study of the matter. He
made literally thousands of ratio measurements of common rectangular
objects, such as playing cards, windows, writing-paper pads, and book covers,
and found that the average was very close to the golden ratio (1.618…) He
also did an extensive statistical testing of personal preferences and found that
most people prefer a rectangle whose proportions lie between those of a
square and those of a double square. Despite the fact that most of those
tested had never even
heard
of the divine proportion, the plot of the results of
The Divine Proportion
3
these tests shows a sharp spike precisely at a length-to-width ratio of 1.618
(the divine proportion). Of those tested by Fechner, fully 75.6 percent voted
for it (or for a rectangle differing from it by no more than 5%) out of ten
different rectangular shapes having width-to-length ratios ranging from 0.40 to
1.00. These tests have been repeated independently by at least three other
investigators (Witmar, Lalo, and Throrndyke), each in a different decade, and
the results were similar in each case. It seems, there can be little doubt about
it: the divine proportion is quite simply the most pleasing proportionality to the
human eye. [For further corroboration of this connection between esthetics
and mathematics, check out Khan Amore’s own startling findings on the
plurality of divine proportions to be found in his composite of the ideal
woman, to be found at the end of this article.]
A Note on the Symbol Used to Represent the Divine Proportion:
Mathematicians today symbolize the divine proportion either by the lower-
case Greek letter,
tau
(τ), the first letter of
tome
(“to cut”); or (more
commonly) they use the lower-case Greek letter
phi
() (pronounced “
fee
” in
Greek), following the example of the American mathematician, Mark Barr,
who chose this letter because it is the first letter of the name of the greatest
sculptor in history, Phidias, whose masterpieces of sculpture and architecture
(including the Parthenon) seem to have been based upon the divine
proportion.
§2
Algebraic Derivation of the Divine Proportion
(or Golden Section, or Mean Proportional, or Golden Ratio):
The algebraic derivation of the Divine Proportion proceeds in a
straightforward manner from its definition:
The Divine Proportion
4
“A line segment is divided in Divine Proportion or Golden Ratio when the
short part is to the long part as the long part is to the whole segment.” This
can be written mathematically as:
a

b
b)
(Equation
01)
b
(a

Where:
a = the shorter of the two segments which are in Divine Proportion
b = the longer of the two segments which are in Divine Proportion
(a + b) = the length of the whole segment before it is divided into sections
In order to solve for a or b in Equation 01, we must first move the variables
out of the denominators. This can be rather cleverly be accomplished by
multiplying each side of the equation by a fraction that has the same
expression in the numerator as in the denominator (which is permissible
because it is equivalent to multiplying by one), but these “unity-fractions” are
carefully chosen so as to give the expressions on both sides of the equal sign
the same denominator:


 
   

b




a




b




b

(Equation
02)








a

b
b
a

b
b
The Divine Proportion
5
a




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