What
is Sacred Geometry?
To
me, sacred means something that is symbolic of a trancendent truth,
and worthy of study, contemplation, and veneration. When I refer to
Sacred Geometry, I am talking about geometry that is derived from
or directly related to the structure of nature. Our universe is structured
in a highly complex yet sublimely ordered manner. This is a truth
that is readily felt by sensitive people, and has also been demonstrated
by science and mathematics. Structural forms seen at the microscopic
level are repeated at other scales, and the laws of fractional symmetry
appear to apply throughout. So, geometry that refers to the structural
unity of nature is a powerful metaphor for the mystery of life, and
thus sacred. One of the best examples of Sacred Geometry are forms
based on the Golden Ratio.
The
Golden Ratio
Knowledge
of the Golden Section, ratio, or proportion has been known for a very
long time. The Egyptians knew about it and the Greeks learned about
it from them. It is called phi, Φ , in honor of Phideas, the architect
of the Parthenon, and is approximated by the irrational fraction 0.618034...
Shamans, priests, and artists throughout the world and across history
have understood and applied Φ to ritual, architecture, art, and the
crafting of musical instruments and everyday objects.
Φ
shows up throughout nature. Recall the famous drawing by Da Vinci
showing man within the circle and the Golden Ratios in the human body,
and more recently, Le Corbusier's The Modular. For example, the finger
bones are in Φ ratio to each other, and the position of features
on the human face follow Φ. The major 6th harmony interval
in music is in Φ ratio to the octave.
The
Golden Rectangle
The
Golden Rectangle (GR), the organizing form in my current work, is
a rectangle with a short to long side ratio of 1: 1.618, or 1: (1
+ Φ). An interesting property of GR's is that if you cut out
a square starting from one of the short sides of the GR, you will
be left with another GR. You can continue to cut out short side squares
for each successively smaller GR and another smaller GR will remain.
And the dimensions of each successively smaller rectangle will be
in Φ ratio to the previous larger size. A series of my collage
explore this Φ -ratio "coiling" property of GR's,
seen here in Figure 1:
The
Logarithmic Spiral and the Golden Rectangle
Figure
2 shows a logarithmic spiral superimposed on a coiled GR. This study
shows the Φ -ratio sectioning of the GR with short side squares
and the diagonals of the original seed GR (GR0) - outside boundary
lines) and the diagonal of the first Φ -sectioned GR (GR1).
Note that the two diagonals intersect at a point called the "Eye
of God," the origin of the logarithmic spiral.
Logarithmic
spirals are natural forms (remember the chambered nautilus?) and many
natural forms will fit neatly with a GR. Examples include bird eggs,
human heads, and spruce trees. So, it would seem that the distinction
between a simple "geometric" form like a GR and an organic
form like a logarithmic spiral is superficial. Both forms imply and
can be derived from the other!
Symmetry
of Golden Rectangles and the Square-Root of 5
The
symmetry of GR's means that the coiling can proceed in 4 different
ways, explored in Figure 3. Here we can see that there are a lot of
implied diagonal lines (gray) and shapes within a GR - and that there
are four Eyes of God that are connected by another GR (shaded darker).
The sides of the inner Eyes of God GR have a special ratio to the
sides of GR0 - it has length and width that are in square-root of
5 (1: 2.236...) ratio to the original seed GR0 sides. The square-root
of 5, also an irrational fraction, appears often, both in the calculation
of Φ (see below) and when GR forms are combined.
The
Square-Root of 5 Rectangle
Figure
4 shows overlapping GR's which I use in a series of collage and mandala
square collages in my recent work. Overlapping GR's that share a middle
square are in the aspect ratio of 1: (square root of 5).
Fractals
and the Golden Rectangle
A
final note about the GR involves fractals. Fractals are geometric
forms that look the same no matter what the size scale. They are composed
of repeating units that combine to make larger and similar units at
larger scales. This property is called self-similarity, and it is
also a huge property of nature and natural forms. Fractals also have
a property called fractional symmetry, which means that the self-similar
units are in non-integer fractional proportion to each other. The
square is the self-similar shape that is repeated in the GR, and Φ
is a non-integer proportionality ratio, so GR's qualify as basic fractals.
All of the abstract images in my collages also contain fractal form
elements, for example, coastlines and rock fractures have a fractal
structure.
Deriving
and Calculating Φ
There
are 2 general ways to derive Φ. One approach uses the Fibonacci
numbers, and the other is geometrical and algebraic. Fibonacci numbers
are easier to comprehend, so let's start there.
Fibonacci
Numbers and the Golden Ratio
Fibonacci was the pen name of Leonardo of Pisa, a 13th century mathematician
whose book, Liber Abaci, introduced western civilization to Arabic
numerals (replacing Roman numerals), and a special sequence of numbers
named after him. Fibonacci raised rabbits and observed their population
numbers over successive generations. They increased in a peculiar
"additive" way, and from this he surmised the more abstract
number sequence. Starting with 0 and 1 as the first two numbers in
the sequence (or 1 and 1), each successive number is determined by
adding the previous two numbers. Starting with 0 and 1, the series
goes like this:
0,
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377...
It
turns out that this sequence of integers is much more than an arithmetic
game. These numbers turn up all the time in nature, and are observed
in the dimensions and branching of all plants, animals, as well as
crystals. This happens because when things grow, they often grow on
top of a previous structure, so that the new growth is "added
to" the existing structure (like new offspring are added to the
existing population). Many plants exhibit Fibonacci numbers in branching,
and in spiral structures like the arrangement of rows of bracts on
pinecones, petals on an artichoke, and scales on a pineapple.This
would just be a logical curiosity about growing things until we start
calculating the ratio between adjacent Fibonacci numbers. Table 1
lists the Fibonacci numbers in the left column, the ratio calculations
in the middle column, and the results in the right column. It turns
out that after the fifth Fibonacci number, the ratio begins to get
very close to the algebraic solution for Φ = 0.6180339. If
you figure that two decimal places are as much as even the most careful
craftsman can measure and cut, then the 6th Fibonacci ratio, 8 ÷
13, will do as an approximation of 0.62 for Φ. After the 16th
Fibonacci number, the ratio approximates Φ to 6 decimal places.
Higher Fibonacci number ratios yield changes in only in the 7th decimal
place and beyond. Table 1 - Ratios between adjacent numbers in the
Fibonacci series start to approximate the Golden Ratio, Φ
|
Fibonacci number
|
Ratio Calculation
|
Result
|
|
Fibonacci number
|
Ratio Calculation
|
Result
|
|
1
|
|
|
|
55
|
34 ÷ 55
|
0.618181
|
|
1
|
1 ÷ 1
|
1.000000
|
|
89
|
55 ÷ 89
|
0.617977
|
|
2
|
1 ÷ 2
|
0.500000
|
|
144
|
89 ÷ 144
|
0.618055
|
|
3
|
2 ÷ 3
|
0.666666
|
|
233
|
144 ÷ 233
|
0.618025
|
|
5
|
3 ÷ 5
|
0.600000
|
|
377
|
233 ÷ 377
|
0.618037
|
|
8
|
5 ÷ 8
|
0.625000
|
|
610
|
377 ÷ 610
|
0.618032
|
|
13
|
8 ÷ 13
|
0.615384
|
|
987
|
610 ÷ 987
|
0.618034
|
|
21
|
13 ÷ 21
|
0.619047
|
|
1,597
|
987 ÷ 1,597
|
0.618033
|
|
34
|
21 ÷ 34
|
0.617647
|
|
2,584
|
1,597 ÷ 2,584
|
0.618034
|
So,
Fibonacci numbers are related to Φ and can be used to derive
the Golden Ratio. I use frame dimensions of 21" x 13" for
coiled GR collages (Figure 1) which are adjacent Fibonacci numbers.
Saves a lot of trouble trying to get precise odd fractions of an inch
cut by the frame cutter!
Geometric
Method: Using a Compass and Straightedge to Draw a Golden Rectangle
The geometric approach to drawing a GR is fairly simple using a straightedge
and compass. Refer to Figure 5. Basically, draw a straight base line
across the bottom of a sheet. Draw a fairly large square near the
middle of the line segment. Make sure the sides are perpendicular
and equal. Find the midpoint of the bottom side of the square and
mark it. Take the compass and stick the sharp point at the midpoint
of the bottom of the square. With the compass point at the marked
midpoint, open the compass until the pencil point touches the upper
right hand corner of the square, radius r in figure 5. Now draw a
big half-circle that intersects the baseline. Draw a line extending
the square and a perpendicular line that meets the point where the
big circle intersected the baseline. You have created a Golden Rectangle!
Algebraic
Method:
For the algebraic method (which can be skipped if you are not mathematically
inclined), consider the geometric definition of Φ on a line
segment, AB, seen in Figure 6. We want to find the Golden Section,
the point on AB, call it C, such that the ratio of the big chunk AC
to the whole segment AB is the same as the ratio of the small chunk,
CB, to the big chunk, AC. For C to be the Golden Section point, (AC
divided by AB) must equal (CB divided by AC).
To simplify matters (or to confuse the less mathematically gifted),
we will do some substitutions and let the whole segment AB = 1, and
we will let AC be phi, Φ. Now, it should be clear that the
whole segment AB is the sum of the 2 parts, AC and CB, right? So we
can set up an equation that says the same thing:
and
subtracting AC from both sides gives us
and
substituting 1 for AB and Φ for AC, we get
Figure
7 shows how the substitutions have changed the problem. Here we show
the line segments with the new substituted terms.
With
me so far? OK...now we are going to set up an algebraic equation to
solve for Φ. Remember our definition: for C to be the Golden
Section of segment AB, the ratio of the big chunk AC to the whole
segment AB, has got to be the same as the ratio of the little chunk
CB to the big chunk AC. So now we can set this up as an equation:
OK... So now let's substitute 1 for AB, Φ for AC, and (1 -
Φ) for CB, and we get:
We are almost there! Now, we need to "multiply the extremes and
means" to simplify this equation and solve for Φ. Looking
at the equation like a box, the extremes are the upper left and lower
right terms, the means are the lower left and upper right terms. Multiplying
these gives you:
Subtracting (1 - Φ) from both sides of the equation gives us
the characteristic equation for solving for Φ:
The following two values will solve this equation:
We ignore the first solution because it is negative, and the length
of a line segment can't be a negative number. So the value of Φ
calculates to 0.6180339..., an irrational non-repeating digit ratio.
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Bibliography
These
books have been helpful in my study of the Golden Proportion and form
in nature and art. I recommend Garland's Fascinating
Fibonaccis to anyone who would like a general overview. It's easy
to understand and gives a lot of good visual examples. Teachers may
want to consider Garland as an introduction and Runion's The
Golden Section for a more math-oriented approach with problems
in each chapter. Garland is suitable for middle school kids while
Runion is high school algebra level.
Natural
Form and Sacred Geometry
Briggs,
John, 1992,
Fractals: The Patterns of Chaos
Cook, Theodore Andrea, 1979,
The Curves of Life
Doczi,
György, 1994, The
Power of Limits: Proportional Harmonies in Nature, Art, and Architecture,
Ghyka,
Matila, 1946, The
Geometry of Art and Life
Godwin,
Joscelyn, 1987,
Harmonies of Heaven and Earth: Mysticism in Music from Antiquity to
the Avant-Garde
Hambidge,
Jay, 1919,
The Elements of Dynamic Symmetry
Lawlor,
Robert, 1982, Sacred
Geometry: Philosophy and Practice
Pennick,
Nigel, 1982, Sacred
Geometry
Pennick,
Nigel, 1995, The
Oracle of Geomancy
Thompson,
Sir D'Arcy W., 1946, On
Growth and Form
Mandlebrot,
Benoit, 1977, The
Fractal Geometry of Nature
Runion,
Garth E., 1990, The
Golden Section
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Math-Oriented
Books
Garland,
Trudy H., 1987, Fascinating
Fibonaccis: Mystery and Magic in Numbers
Gazalé,
Midhat, 1999, Gnomon:
From Pharaohs to Fractals
Herz-Fischler,
Roger, 1987, A
Mathematical History of the Golden Number
Huntley,
H.E., 1970, The
Divine Proportion: A Study in Mathematical Beauty
Lauwerier,
Hans, 1991, Fractals: Endlessly Repeating Geometrical Figures
Mandlebrot,
Benoit, 1977, The
Fractal Geometry of Nature,
Runion,
Garth E., 1990, The
Golden Section
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Philosophy
These
books are not specifically about the Golden Ratio, but provide integrative
links to the ideas of harmony, physical law, and self-similarity.
They have been a huge inspiration and source of insight to me.
Bohm,
David, 1980, Wholeness
and the Implicate Order,
Campbell,
Joseph, all of his work.
Eco,
Umberto, and Hugh Bredin (translator), 1988, The
Aesthetics of Thomas Aquinas
Firth,
Florence (ed), 1996, The
Golden Verses of Pythagoras and other Pythagorean Fragments
Jung,
Carl, 1959,
Mandala Symbolism
Kandinsky,
Wassily, 1977,
Concerning the Spiritual in Art
Schroedinger,
Erwin, 1969, What
is Life - Mind and Matter
Talbot,
Michael, 1991, The
Holographic Universe
Wilbur,
Ken, 1992, A
Brief History of Everything
Sheldrake,
Rupert, 1981, A
New Science of Life: The Hypothesis of Morphic Resonance
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Sacred
Geometry
