Sacred Geometry
Ngilcach
The term "sacred geometry" is used by archaeologists, anthropologists, and
geometricians to encompass the religious, philosophical, and spiritual beliefs
that have sprung up around geometry in various cultures during the course of
human history. It is a catch-all term covering Pythagorean geometry and
neo-Platonic geometry, as well as the perceived relationships between organic
curves and logarithmic curves.
Here are a few examples of how the "sacred" has entered into geometry in
different eras and cultures:
1) The ancient Greeks assigned various attributes to the Platonic solids and to
certain geometrically-derived ratios, investing them with "meaning." For
example, the cube symbolized kingship and earthly foundations, while the Golden
Section (or Mean) was seen as a dynamic principle embodying philosophy and
wisdom. Thus a building dedicated to a god-king might bear traces of cubic
geometry, while one
dedicated to a heavenly god might have been constructed using Golden Section
proportions.
2) When Hindus (ancient and modern) plan to erect any edifice for religious
purposes, from a small wayside shrine to an elaborate temple, they first
perform a simple geometric construction on the ground, establishing due East
and West and constructing a square therefrom. (It's a simple, elegant piece of
work, at about the level of high school geometry). Upon this diagram they lay
out the entire building. The making of this geometric construction is
accompanied by prayers and other religious observances.
3) The Christian religion uses the cross as its major religious emblem, and in
geometric terms this was elaborated during the Medieval period to the form of
an unfolded cube (reminiscent of example #1 above, where the cube was equated
with kingship). Many Gothic cathedrals were built using proportions derived
from the geometry inherent in the cube and double-cube; this tradition
continues in modern Christian churches to the present time.
4) The ancient Egyptians discovered that regular polygons can be increased
while still maintaining the ratio of their sides by the addition of a strictly
constructed area (which was later named the "gnomon" by the Greeks); the
Egyptians assigned the concept of the ratio-retaining expansion of a
rectangular area to the god Osiris, who was, therefore, often shown in ancient
Egyptian frescoes seated on a square throne
(square = kingship again) in which the original square and its L-shaped gnomon
are clearly delineated, but the geometrical construction used to create the
gnomon is not shown. It is, in fact, the absence of the attendant arcs and
extension lines used in the creation of geometric forms that has led art
historians and iconographers such a merry chase through history. It often
takes the eye of a geomterician to spot the tell-tale signs of construction.
5) One of the best-known pieces of detective work in this regard was the
discovery by Jay Hambidge, an art historian at Yale University during the
1920s, that the spirals on the Ionic column capitals of ancient Greek temples
were laid out by the so-called "whirling rectangle" method for creation of a
logarithmic spiral. He realized this by examining numerous Ionic capitals in
art museums until he located some in
which the holes made by the placement of compass points had not been
obliterated over time. (One of these capitals was an unfinished, broken piece,
dug up from a rubbish heap near a temple -- it had apparently been damaged
during manufacture and was discarded; its burial preserved it from the
elements, and the marks of the geometric layout were remarkably clear upon it.)
No "sacred meaning" for the log spiral form of the Ionic column capital has
been determined from Greek writings, but the use of other log spirals in Greek
temple architecture (for instance in floor-block proportions and their
placement in relation to overall floor area) indicates that Greek architects,
unlike the Romans who came after them,
deliberately constructed their temples according to "whirling rectangle"
geometric ratios.
There are endless examples. Not everyone who catalogues and writes about
sacred geometry considers geometry itself to be inherently spiritual; for some
of us, sacred geometry is an adjunct to the study of archaeology, architecture,
art history, comparative religion, anthropology, archaeoastronomy, or geometry
itself.
Sadly, many books on the subject of sacred geometry are chock full of
extraneous blather about UFOs and perpetual motion and Atlantean Science
(whatever that is). Even worse, unexpected encounters with such drivel tends
to discourage research into the subject by steely-eyed geometricians -- who
are, of course, those most qualified to undertake the work.
Over the years I have compiled a theory that encompasses three interlocked
reasons for the unfortunate state of affairs by which the once-honoured field
of sacred geometry is now often perceived as something akin to pyramidology or
spirit-channeling. They are:
1) There was an actual loss of general geometric knowledge during the Dark Ages
- old Egyptian and Greek geometry was no longer passed along as it had been;
instead it became the secret of such trade guilds as made use of it. Thus
geometry became "mysterious."
2) Although interest in geometry revived during the Renaissance, the adoption
of the Arabic numbering system had already led many Europeans onto a different
way of thinking when it came to numbers. Specifically, because the irrational
numbers that are so common in geometrical proofs are difficult to handle
arithmetically, they became the domain of academic mathematicians. Thus sacred
geometry --
which allows one to rattle off irrational number formulae like ".618... : 1 ::
1 : 1.618..." before having one's morning coffee -- seemed rather hard to
master. And, the joke of it all is that "1.618..." is but a rude approximation,
anyway -- just something for folks with rulers to measure after the
geometricians have put
down their compasses.
3) During the 19th century the sizes of construction materials became quite
standardized for the first time. A common brick was 2 x 4 x 8 inches; lumber
came in 12 foot lengths that were 1 or 2 or 4 or 8 or 12 inches wide and a
similar choice of numbers deep. Construction therefore took on a more
arithmetic aspect than it had when geometric ratio was the prime mover behind
design. In a wooden frame house of typical Victorian style, a vernacular
builder could lay out the work using simple arithmetic. An "architect-designed"
house of an earlier period might have included a spiral staircase -- a test of
geometric knowledge -- but by the late 19th century tables of angles printed on
steel framing squares obviated the need for carpenters to study even the small
amount of geometry used in figuring out the area of a roof gable.
I believe that the combination of these three factors led many 19th century
scholars (especially those who were culturally bound up in colonialist feelings
of superiority toward conquered races) to decide that it was inconceivable that
ancient cultures could have known enough "math" to have used "irrational
numbers" to construct architectural monuments. One thing these writers failed
to consider was that a
culture that relied on compass-and-straight-edge geometry rather than
arithmetic to design structures would not give a fig that some of the lengths
turned out to be irrational numbers. The numbers (or rather, the lengths they
represented) would simply appear during the course of construction and that
would be that.
Jay Hambidge pioneered the technique of searching for certain typically
"sacred" geometric ratios among the arithmetic measurements of ancient
artifacts. Like all sacred geometry detectives, he had to work backwards -- he
took arithmetic measurements of Greek vases and temples and derived from them
their geometric construction. This is not as simple as it sounds, because many
times an arithmetical construction will duplicate the results of a geometric
construction -- in fact, in order to derive even the faintest proof that
geometric construction underlay Greek architecture, he had to perform
calculations on dozens of items of differing size, establishing beyond doubt
that it was RATIO, not measurement, that
determined the relative lengths of crucial dimensions.
The hardest battle Hambidge fought was convincing academics of his day that the
ancient Greeks were "sophisticated" enough to have used geometry to lay out
their temples. For someone to double check his figures meant the work of years,
and few wanted to devote the time to it. (Only recently, with the
lightning-quick calculations offered to us by computers, has it become possible
to "deconstruct" an ancient temple into its geometric basis with anything
approaching efficient speed or to determine the intended astro-calendrical
orientation of a temple constructed thousands of years in the past.)
Hambidge did fine work -- and one would think that he and his students would
have proved the case for the legitimacy of sacred geometry beyond reasonable
doubt -- but he wrote at a peculiar time in history. It was then, during the
inter-War period -- when Hambidge's 1925 book "The Parthenon and Other Greek
Temples: Their Dynamic Symmetry" was in print, when King Tut's tomb had been
discovered, when
Theosophy and Rosicruciansism were at the height of their popularity -- that
the "mystery" of sacred geometry became bound up with the writings of people
whose interests lay far afield from the use of the compass and the
straight-edge.
It's hard to say where the art deco theories of Hambidge's students Edward B.
Edwards (author of "Dynamarhythmic Design") and Walter Dorwin Teague (author of
"Design This Day" and designer of Texaco Gas Stations) leave off and the
metaphysical theories of fellow Hambidge student - and Theosophist -
Claude Bragdon (author of "The Beautiful Necessity" and designer of railroad
terminals) begin. Bragdon seems so ... norma l...as he writes about the Golden
Mean and creates a new design style based on projective geometry, that it comes
as quite a shock to find him edging into the theory that Man is a Cube,
crucified in Time. Still, to his credit, Bragdon, despite dedicating one of his
volumes to "The Delphic Sisterhood," was a practicing architect.
Beyond Bragdon, however, a line is crossed - and one finds onself confounded by
writers of that period such as Manley P. Hall, who lumps sacred geometry
together with belief in Lemuria, spirit channeling, Enochian magic, and
Rosicrucianism. And he wasn't the worst, by far. The 1960s hippie interest in
the occult and the 1990s New Age interest in spirituality have both given
library shelf space to authors intent on inventing or perpetuating imputed
connections between sacred geometry, metaphysics, fringe archaeology, magic,
and eccentric religions.
In the landscape of Sacred Geometry, there are many books that present mystical
and even other-worldly claims, but they have been clearly labeled as such. To
ignore spiritually oriented or even scientifically lunatic texts would be to
ignore a vast part of the writing on the subject of sacred geometry and sacred
architecture. So drink deep, but do not swallow every draught uncritically, for
to do so would be to drink from a polluted pool.
_______________________
Prepared by Saillewen Darkwater, a teaching elder in Thistle Down Sword. She
lives in Colorado Springs, Colorado.
Forwarded with permission from the author.
________________
heart in hand ...
nGilcach
PSmith9626
I liked your sacred geo post,especially the idea that standardized
building materials removed geometry from construction.I have seen a similar
thing happen in electronics in my lifetime.
However ,in regard to the greeks.
1) the log spiral is called the archimedian spiral because archimedes
discovered the construction and also the arclength( by a form of calculus,which
he called the method).
2) the pythagoreans discovered the irrational
numbers and irrational numbers are much discussed in Euclid.
These facts were well known to any nineteenth century schoolkid.Greek
study was much stressed then.
It was also known that the problems of angle trisection,circle
squaring, and cube duplication-the three famous geometry problems of
antiquity-were inspired by temple
tasks and priestly requirements.
I didnt know about the kingship stuff and the cube but the
duplication of the cube was required to double a certain temple. Thanks for
that insight.
There is a nice article on Shinto (nature spirits) temple geometry in
a recent issue of scientific american.
in service
penny
if you need any math articles i could contribute.
Ngilcach
** Thank you Penny, but I did not write it, Saillewen Darkwater, one of the
Elders in our tradition did. I will certainly pass on your appreciation.
<<..,especially the idea that standardized building materials removed geometry
from construction.>>
** This is an ancient Pagan idea that was destroyed by Xian thought. However,
the Oriental countries still maintained this lore, parts of which are now
re-surfacing. (Note the growth of fung-shui.)
There is a group in Seattle, Washington who is locating, mapping & building
power centers on ley lines. Check out their website at:
http://www.geo.org/index.htm
<<I have seen a similar thing happen in electronics in my lifetime.>>
** Could you expound on this idea in relation to Paganism? It was Pagans who
first studied & chronicled science, the arts, medicine, etc. Q-physics is now
validating that which *we* have known for years. Indeed, the circle is being
made complete.
<<However ,in regard to the greeks.
1) the log spiral is called the archimedian spiral because archimedes
discovered the construction and also the arclength ( by a form of
calculus,which
he called the method).
2) the pythagoreans discovered the irrational
numbers and irrational numbers are much discussed in Euclid.>>
** I will certainly forward this information to Saillewen (because this topic
is over my head). I am sure she will be pleased to receive it.
<<It was also known that the problems of angle trisection,circle squaring, and
cube duplication-the three famous geometry problems of antiquity-were inspired
by temple tasks and priestly requirements.>>
** I have read as much myself in:
- "Sacred Geometry", by Robert Lawlor. LCN 88-51328.
- "Mathematics for the Nonmathematician", by Morris Kline. ISBN 0-486-24823-2.
I highly recommend both these works. The former for its wonderful & concise
explanations & diagrams for the layman; the latter because of the length of
information regarding Pagan mathematics.
<<There is a nice article on Shinto (nature spirits) temple geometry in a
recent issue of scientific american.>>
** Not a magizine I normally pick-up ... but I will make a note of this.
<<.. if you need any math articles i could contribute.>>
** As I said earlier, if they relate to Pagan math, then by all means ...
share with us!
PSmith9626
the scientific american article might still be posted on the
scientific american website in the archives. It's fairly recent.
heart in hand
penny
georg
>
> Penny wrote <<I have seen a similar thing happen in electronics in my lifetime.>>
> ** Could you expound on this idea in relation to Paganism? It was Pagans who
> first studied & chronicled science, the arts, medicine, etc. Q-physics is now
> validating that which *we* have known for years. Indeed, the circle is being
> made complete.
Those that are not in touch with nature as much as they would/could
wish often connect with other cycles around them. The very literal
electricity that surrounds is alternating current- with 60 or 50
cycles per minute- and one can tune into this like a heartbeat. The
more one learns about electromagnetic fields, the more one can take a
few steps into playing with energy through visualization. If you can
raise a circle with just energy from the earth, imagine what you could
do if you could plug into the wall. Not recommended for the faint of
heart.
The idea of sacred geometry also reminds me of the Golden Square
concept for artistic composition, but that was a theory I never
understood well.
-georg
non ani sunt permittendi
PSmith9626
A great book on the golden section ,art and lots more is called
Dynamic Symmetry.
Its a classic and should be at dover .
Or perhaps amazon.com
best
penny
Ngilcach
would/could wish often connect with other cycles around them.>>
<< The very literal electricity that surrounds is alternating current <snip>
and one can tune into this like a heartbeat.>>
** For myself, this is strongly felt in gyms, malls, & other such locations
that employ large amounts of electronic equipment.
<<The more one learns about electromagnetic fields, the more one can take a few
steps into playing with energy through visualization.>>
** EMFs are fascinating to work with. Within our tradition, they are merely
an aspect of fire.
c...@luckymojo.com
ngil...@aol.com (Ngilcach) wrote:
> SACRED GEOMETRY -- WHAT IS IT?
>
> The term "sacred geometry" is used by archaeologists, anthropologists, and
> geometricians to encompass the religious, philosophical, and spiritual beliefs
> that have sprung up around geometry in various cultures during the course of
> human history. It is a catch-all term covering Pythagorean geometry and
> neo-Platonic geometry, as well as the perceived relationships between organic
> curves and logarithmic curves.
[snip long article]
> Prepared by Saillewen Darkwater, a teaching elder in Thistle Down Sword. She
> lives in Colorado Springs, Colorado.
>
> Forwarded with permission from the author.
Neither Saillewen Darkwater nor ngil...@aol.com (Ngilcach) wrote the
article posted here. I, catherine yronwode, wrote it. It is copyrighted
material. It was plagiarized from my web site. The Sacred Landscape. My name
was remioved from it. My copyright notice was removed from it. Not a word of
my writing was changed. Someone lied and claimed it was written by him or
her.
This is dishonest. It speaks ill of the intentions and morals of the so-called
"teaching elder" who did it. Such a person is nothing but low scum.
My original article -- with links to other pages at my site that did not come
through on the plagiarized post -- can be found at
http://www.luckymojo.com/sacreddefined.html
Thanks to Vincent Beall for alerting me to this and allowing me to set the
record staight.
catherine yronwode
Lucky Mojo Curio Co: http://www.luckymojo.com/luckymojocatalogue.html
The Lucky W Amulet Archive: http://www.luckymojo.com/luckyw.html
Sacred Sex: http://www.luckymojo.com/sacredsex.html
The Sacred Landscape: http://www.luckymojo.com/sacredland.html
Freemasonry for Women: http://www.luckymojo.com/comasonry.html
Comics Warehouse: http://www.luckymojo.com/comicswarehouse.html
check out news:alt.lucky.w for discussions on folk magic and luck
-----------== Posted via Deja News, The Discussion Network ==----------
http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own
georg
>
> Georg wrote << The very literal electricity that surrounds is alternating current <snip>
> and one can tune into this like a heartbeat.>>
> ** For myself, this is strongly felt in gyms, malls, & other such locations
> that employ large amounts of electronic equipment.
It is reassuring that I am not the only one who can feel this. I am
too accustomed to a myriad of strange looks. :)
> <<The more one learns about electromagnetic fields, the more one can take a few
> steps into playing with energy through visualization.>>
> ** EMFs are fascinating to work with. Within our tradition, they are merely
> an aspect of fire.
If one must be limited to 4 elements only, yes, I can see that
clearly. It certainly makes bans on Cold Iron intriguing indeed!
Ngilcach
locations that employ large amounts of electronic equipment.>>
Georg then said <<It is reassuring that I am not the only one who can feel
this. I am too accustomed to a myriad of strange looks.>>
I actually have my students test this themselves. Usually the most heavily
bombarded room in the house is the kitchen. Unplug everything then don a
blindfold. Have someone plug in an appliance. Allow yourself to feel/sense
where the energy is coming from. Even those folk who consider themself *dense*
to such vibrations are surprised that they can actually feel a difference.
I further stated that we consider EMCs as ".. an aspect of fire.. >>
Georg then added <<If one must be limited to 4 elements only, yes, I can see
that clearly.>>
** Notice I said that EMCs were an "aspect" of fire. We recognize & use seven
elements. And when you count all the combinations of the same - the number of
manifested forms of energy is rather extensive in its physical ramifications.
Which of course doesnt even go into the unmanifested forms of energy.
PSmith9626
The buzz one feels when touching a poorly grounded appliance is
sixty cycle ac.
Humans are electromagnetically sensitive.
I once felt an earthquake early this way. Most
people in the room felt the sense of dread.
This might amuse you:
earth----solid
water-----liquid
air--------gas
fire----electric plasma
Of course this is a mundane list of correspondence.
heart in hand
penny
I also see two colors in the ultra violet which may be more common than people
think.For example: People who have artifical cornea replacements see well into
the ultraviolet -many are confused-these days the doctors put an ultraviolet
screening dye into the artifical corneas.Thus normal genetic variation will
allow some people to see ultraviolet because their corneas have less or no
organic screening dye.
I wonder if astral sight has such a mundane component? Certainly not
the whole story.
Heart in Hand
penny
Dept. of Spam Management
>Georg then added <<If one must be limited to 4 elements only, yes, I can see
>that clearly.>>
>** Notice I said that EMCs were an "aspect" of fire. We recognize & use seven
>elements. And when you count all the combinations of the same - the number of
>manifested forms of energy is rather extensive in its physical ramifications.
>Which of course doesnt even go into the unmanifested forms of energy.
Seven? How interesting. I recognise and use 5 on a general basis, 9
where further specialization is required. Electrical and
electromagnetic energies fall under Storm. Yes, I can feel EMCs and
also X-rays, as well.
Ronnie
PSmith9626
Do you feel the strong nuclear, weak nuclear, and color forces as
well?
(smile)
penny