Archaeoastronomy, sacred geometry, more! is a Public Group with 435 members.
 Archaeoastronomy, sacred geometry, more!

 Public Group,
 435 members
Delian Problem
 Does he mention the Delian Problem, that of doubling the cube Dominick?I've pasted an (pretty poor) outline belowMikePS I don't get Livio Stecchini. He can obviously do sums and has great ideas, but his work  at least that I've seen  is so sloppy  mistakes everywhere, no references, castles of speculation built on sand... Does anyone agree?In a message dated 21/10/06 03:42:28 GMT Daylight Time, scorpio_eagle2002@... writes:Mike, et al:
Still, IMHO, the best work ever done on the subject of the socalled
Cheops Pyramid is embodied in "Secrets of the Great Pyramid" by
Peter Tompkins, which contains a fascinating appendix on ancient
weights & measuring systems by Livio Stecchini (and Stecchini's
conclusions therein re: Akhenaten and his "religious revolution"
should make all sit up and take notice!)Geometry > Solid Geometry > Polyhedra > Cubes
Geometry > Geometric Construction
Cube Duplication
Cube duplication, also called the Delian problem, is one of the geometric problems of antiquity which asks, given the length of an edge of a cube, that a second cube be constructed having double the volume of the first. The only tools allowed for the construction are the classic (unmarked) straightedge and compass.The problem appears in a Greek legend which tells how the Athenians, suffering under a plague, sought guidance from the Oracle at Delos as to how the gods could be appeased and the plague ended. The Oracle advised doubling the size of the altar to the god Apollo. The Athenians therefore built a new alter twice as big as the original in each direction and, like the original, cubical in shape (Wells, 1986, p. 33). However, as the Oracle (notorious for ambiguity and doublespeaking in his prophecies) had advised doubling the size (i.e., volume), not linear dimension (i.e., scale), the new altar was actually eight times as big as the old one. As a result, the gods remained unappeased and the plague continued to spread unabated. The reasons for the dissatisfaction of the gods under these circumstances is not entirely clear, especially since eight times the volume of original altar was a factor of four greater than actually requesting. It can therefore only be assumed that Greek gods were unusually ticklish on the subject of "altar"ations being performed to their exact specifications.Under these restrictions, the problem cannot be solved because the Delian constant (the required ratio of sides of the original cube and that to be constructed) is not a Euclidean number. However, the impossibility of the construction required nearly 2000 years, with the first proof constructed by Descartes in 1637. The problem can be solved, however, using a Neusis construction.
Equally informative and packed with excellent material are his
companion volumes:
Mysteries of the Mexican Pyramids
and
The Magic of Obelisks
Respectfully,
Dominick  I didn't find any reference by him.
What would you like to know?
An excellent starting point is:
http://wwwgroups.dcs.stand.ac.uk/~history/HistTopics/Doubling_the_cube.html
If link doesn't work it, try google's cache:
http://72.14.205.104/search?q=cache:57LLG2iNNSwJ:wwwgroups.dcs.stand.ac.uk/
~history/HistTopics/Doubling_the_cube.html+doubling+the+cube
+math&hl=en&ct=clnk&cd=1
Only story it does not relate is that this was not the only one: also story of doubling the
king's throne (cube).
Chris
 In sacredlandscapelist@yahoogroups.com, mikebispham@... wrote:
>
>
> Does he mention the Delian Problem, that of doubling the cube Dominick?
>
> I've pasted an (pretty poor) outline below
>
> Mike
>
> PS I don't get Livio Stecchini. He can obviously do sums and has great
> ideas, but his work  at least that I've seen  is so sloppy  mistakes
> everywhere, no references, castles of speculation built on sand... Does anyone agree?
>
>
>
>
> In a message dated 21/10/06 03:42:28 GMT Daylight Time,
> scorpio_eagle2002@... writes:
>
> Mike, et al:
>
> Still, IMHO, the best work ever done on the subject of the socalled
> Cheops Pyramid is embodied in "Secrets of the Great Pyramid" by
> Peter Tompkins, which contains a fascinating appendix on ancient
> weights & measuring systems by Livio Stecchini (and Stecchini's
> conclusions therein re: Akhenaten and his "religious revolution"
> should make all sit up and take notice!)
>
>
>
> Geometry > Solid Geometry > Polyhedra > Cubes
> Geometry > Geometric Construction
>
>
> Cube Duplication
>
>
>
> Cube duplication, also called the Delian problem, is one of the geometric
> problems of antiquity which asks, given the length of an edge of a cube, that a
> second cube be constructed having double the volume of the first. The only
> tools allowed for the construction are the classic (unmarked) straightedge and
> compass.
>
> The problem appears in a Greek legend which tells how the Athenians,
> suffering under a plague, sought guidance from the Oracle at Delos as to how the
> gods could be appeased and the plague ended. The Oracle advised doubling the
> size of the altar to the god Apollo. The Athenians therefore built a new alter
> twice as big as the original in each direction and, like the original, cubical
> in shape (Wells, 1986, p. 33). However, as the Oracle (notorious for
> ambiguity and doublespeaking in his prophecies) had advised doubling the size
> (i.e., volume), not linear dimension (i.e., scale), the new altar was actually
> eight times as big as the old one. As a result, the gods remained unappeased and
> the plague continued to spread unabated. The reasons for the dissatisfaction
> of the gods under these circumstances is not entirely clear, especially
> since eight times the volume of original altar was a factor of four greater than
> actually requesting. It can therefore only be assumed that Greek gods were
> unusually ticklish on the subject of "altar"ations being performed to their
> exact specifications.
>
> Under these restrictions, the problem cannot be solved because the Delian
> constant (the required ratio of sides of the original cube and that to be
> constructed) is not a Euclidean number. However, the impossibility of the
> construction required nearly 2000 years, with the first proof constructed by
> Descartes in 1637. The problem can be solved, however, using a Neusis construction.
>
>
>
> Equally informative and packed with excellent material are his
> companion volumes:
>
> Mysteries of the Mexican Pyramids
>
> and
>
> The Magic of Obelisks
>
> Respectfully,
> Dominick
>   In sacredlandscapelist Dominick wrote:
> Mike, et al:
>
> Still, IMHO, the best work ever done on the subject of the so
> called Cheops Pyramid is embodied in "Secrets of the Great
> Pyramid" by Peter Tompkins, which contains a fascinating appendix
> on ancient weights & measuring systems by Livio Stecchini (and
> Stecchini's conclusions therein re: Akhenaten and his "religious
> revolution" should make all sit up and take notice!)
 In sacredlandscapelist Mike wrote:
> Does he mention the Delian Problem, that of doubling the cube
> Dominick?
>
> I've pasted an (pretty poor) outline below
At the end of this post I touch upon a possible variant of the
Delian problem in relationship to the King Chamber
> Under these restrictions, the problem cannot be solved because
> the Delian constant (the required ratio of sides of the original
> cube and that to be constructed) is not a Euclidean number.
> However, the impossibility of the construction required nearly
> 2000 years, with the first proof constructed by Descartes in
> 1637. The problem can be solved, however, using a Neusis
> construction.
Here the Delian constant is not the cube root of 2 or 2^(1/3) 
the inverse ratio of the edge of a cube to the edge of the cube
double its volume, it is the cube root of the square of 2 or 2^(2/3)
 the inverse ratio of the area of the face of a cube to the area of
the face of the cube double its volume. Hence, the area of the
square which equals 440 square cubits to the area of the square
which equals 280 square cubits equals 1.57142 = 22 / 14 or around
one half Pi which is very roughly equal to 2 ^ ( 2 / 3 ) the Delian
Constant. The square root of 2 is more directly incorporated in
the monument since the area of the horizontal cross section of the
pyramid at the height of the floor of the king chamber equals half
the area of the square base of the pyramid, and the distance from
the apex of the monument to the level of the floor of the king cham
ber is 198 cubits, so that the ratio of the height of the monument
to the distance from the height to the floor of the king chamber
equals 280 / 198 = 1.41414... ~ 1.41421... = Square Root of 2. And
of course it is one of those interesting things that if your divide
280 + 280 / 198 king chamber cubits into the mean volumetric radius
of the earth you get 43200.1  the number of seconds in one half day
accurate to .1 seconds.
6370973.27862 /.5240528 cubit in meters /(280+280/198) = 43200.1
http://www.jqjacobs.net/astro/cosmo.html
> PS I don't get Livio Stecchini. He can obviously do sums and has
> great ideas, but his work  at least that I've seen  is so
> sloppy  mistakes everywhere, no references, castles of
> speculation built on sand... Does anyone agree?
I would not be quite so severe in judgment Mike. At times it does
appear that he reasons from an a priori conviction that the ancient
Egyptians, and the ancient world in general, knew the dimensions of
the Earth, including its flattening, to a high degree of accuracy,
and incorporated that knowledge into the lengths of their basic
units of measure, and in the case of the ancient Egyptians, into the
dimensions of their greatest monument. Stecchini is not unique. It
is said that Newton thought the ancient Egyptians knew the dimensions
of the earth, and asked the astronomer John Greaves to obtain the
length of the cubit of the Great Pyramid so that Newton might
evaluate his theory of gravitation using the data of the dimensions
of the earth he thought in that cubit. I think I remember that
Greaves even included the essay of Newton on the cubit of the He
brews and the cubit of Memphis as an appendix to his work Pyramido
graphia. Newton, however, was more a bibliomaniac than a pyramido
maniac, and in that work, more concerned with determining the cubit
of the temple in Jerusalem. For a brief, and misleading overview see
http://en.wikipedia.org/wiki/Pseudoscientific_metrology
Then compare the essay of Stecchini on the Roman and Egyptian foot:
http://www.metrum.org/measures/romegfoot.htm
It is important to distinguish his meticulous historical research
from some of the hypotheses and theories he based on it. None
theless, they do not diminish the value of his work, if it is a
field one is interested in. One can always refer to his sources
and judge for oneself. The last quarter of the link above con
tains an illuminating discussion of pyramidites and their errors.
The first link above on pseudoscientic metrology is so misleading,
it characterizes Petrie as a practitioner of it, when he is in fact
one of the fathers of scientific archaeology, metrology, Egyptology,
and surveying. His father was a pyramidite, and friends with Smyth,
then astronomer royal of Scotland, and one of the foremost pyramid
ites of his time, being the author of Our Inheritance in the Great
Pyramid.
The current Director of the Gizeh Plateau, who along with Lerner
has very little patience for modern pyamidites like Hancock and
Graham, writes in his introduction to a 1990 abridgement of the
Pyramids and Temples of Gizeh by Petrie
"Petrie's work inside the Great Pyramid and at the other pyramids at
Giza, concentrated on the taking of measurements of the structures.
These measurements were so accurate that even today they are still
used by many Egyptologists. At the same time with the advance of
scientific practices more detailed information has been extracted.
The base of the Great Pyramid was resurveyed by Cole in 1926, by J.
Dorner and by Lehner and Goodman in 1984. Recently, Maragioglio and
Rinaldi did a visual survey of the pyramids of Giza and took measure
ments of many elements inside the Great Pyramid." p.98 ( 1990 )
Maragioglio and Rinaldi, note in their work L'Architettura Delle
Piramidi Menfite, Parte IV, La Grande Piramide di Cheope, 
( Tipagrafia Canessa  Rapallo, 1965 ), page 5,
"...in caso di quasi identità...fra le nostre misure e quelle date
dal Petrie nella sua opera « The Pyramids and Temples of Giza »
abbiamo spesso data la preferenza a quelle dell'archeologo inglese
in quanto ottenute con strumenti di alta precisione."
"in case of small differences...in the measurements we took and
those given by Petrie in his volume « The Pyramids and Temples
of Giza » we have often given preference to his measurements
as they were obtained with very high precision instruments"
Notwithstanding the remarks of Hawass, no one has measured the
interior of the monument to the degree of precision of Petrie. I
would be happy if a team of surveyors went in with the best laser
instruments and measured the Grand Gallery, the Passages, the
Chambers, the Niche, and the Sarcophagus. But you work with what
is available. The measures of Petrie work almost too perfectly.
It is difficult to think he did not apply the highest standards
in his work, but in science, unless there are at least two sets of
data which corroborate each other, one may be erecting a theory
on the basis of erroneous data. The external survey of Petrie
has been corroborated by Cole, and no doubt by Lerner, although
I have yet to review his GPS data, and the measures of Maragioglio
and Rinaldi of the internal features of the monument corroborate
Petrie to the lesser degree of precision that they measured. His
measures work so precisely that it is necessary to measure at least
to his degree of precision or greater(measures taken to 1 hundredth
of an inch accuracy, and the divisions of the rulers machined to one
thousandth of a inch accuracy ) to verify the precision geometry of
the monument that I have discovered through his measures.
However, it is just as unscientific to dismiss out of hand a know
ledge of geodesy and/or astronomy on the part of our ancestors
without examining all available evidence, as it is to assert such a
knowledge on the basis of faulty survey data, complex geometrical
speculation, and a priori historical knowledge, e.g.  the testa
mony of Greece and Israel affirms the wisdom of the Egyptians, there
fore they must have known how to measure the beetle ball of Kephra,
by Ra Heru! The academy tends to swing like a pendulum at times, con
servatively retreating from hypotheses it embraced uncritically on
the basis of little or no evidence at one time, and then tentatively
reexamining earlier hypotheses when it gets over its knee jerk re
actions. Egypt is a case in point. All the fanfare of the ages led
historians to believe the ancient Egyptians possessed advanced math
ematics, astronomy, and geodesy, at least to the level of the civil
ization of the Babylonians. However, as I mentioned in an earlier
post, when at last the few crumbling remains of the hieroglyphic
record of the knowledge of the ancient Egyptians in those areas began
to be discovered and translated after Champollion, scholars were
gravely disappointed. For Mathematics, the most extensive record
we have is that of the Rhind Papyrus, and another few of similiar
content. Although it demonstrates the ancient Egyptians knew how
to reckon well with fractions, and calculate volumes of truncated
pyramids, it does not demonstrate an advanced knowledge of algebra,
or geometrical reasoning ( a la Euclid ). An argument has been made
that their method of reckoning fractions hindered their advances in
mathematics. The Rhind Papyrus has been speculated to be a col
lection of schoolboy mathematical exercises, so there is not alot
of justification for concluding that it represents the limit of math
ematical knowledge of the Egyptians. In Essay 24, Nonliterary
Texts, of his Middle Egyptian, An Introduction to the Language and
Culture of Hieroglyphs, James P. Allen, Curator of Egyptian Art at
the Metropolitan Museum of Art in New York, writes
Middle Egyptian mathematical treatises are represented by four
papyri and two wooden tablets. Of the most important is
the Rhind Mathematical Papyrus, which contains a table of division
of 2 by odd numbers from 3 to 101 and a series of 84 problems in
arithmetic and plane and solid geometry. Page 360
The Papyrus dates from the reign of the Hyksos Pharoah Apophis, ca.
1560 bce, 1000 year after Khufu, although apparently copied from an
older writing dating to the reign of Amenemhat III, ca. 18441797
bce. The Hyksos Pharoahs were invaders. Three times as many major
medical papyri have come down to us as major mathematical papyri.
Egypt is known to have been a profoundly conservative culture, and
at times it gives the appearance that its earlier dynasties
surpassed its later dynasties in knowledge. Their are several
difficulties over and above the paucity of textual evidence in
determining the scientific knowledge of the ancient Egyptians:
1) The bulk of the written testamony of their civilization has sur
vived through the fact that it was engraved or painted on material
able to withstand the decay of time: monuments, and stone and
wooden stele. However, it is almost invariably the case that these
monuments and stele are either commemorative of acts of the Pharoah
accomplished during his reign, or of the deeds of a deceased person
in life, or are religious in nature  spells to guard the soul of
the deceased in the afterlife, or accounts of cosmogony and the life
of the gods. So although there are references to a more extensive
body of scientific, linguistic ( i.e. hieroglyphic ), geometical,
mathematical, astronomical, and medical knowledge ( for instance the
42 Books of Thoth ), very few scientific papyri have survived the
debris of their civilization, and the accidental burning of the
Library of Alexandria by the Romans did not help. A fair number of
literary papyri primarily from the Middle Kingdom, in the genre of
wisdom literature and stories, have also come down to us, but it is
the architectual art of their monuments, for which Egypt is argu
ably most celebrated, so much so that it caused Hegel to write
in his Philosophy of History:
It is the distinguishing feature of the Egyptian spirit, that it
stands before us as this mighty taskmaster. It is not splendour,
amusement, pleasure, or the like that it seeks. The force which
urges it is the impulse of selfcomprehension; and it has no other
material to work on, in order to teach itself what it is, to realize
itself for itself, than this working out its thoughts in stone;
and what it engraves on stone are its enigmasthese hieroglyphs.
They are of two kinds: hieroglyphs proper, designed rather to ex
press language, and having reference to subjective conception; and
a class of hieroglyphs of a different kind, viz., those enormous
masses of architecture and sculpture, with which Egypt is covered.
While among other nations history consists of a series of events
as, e.g., that of the Romans, who century after century, lived only
with a view to conquest, and accomplished the subjugation of the
worldthe Egyptians raised an empire equally mightyof achievements
in works of art, whose ruins prove their indestructibility, and which
are greater and more worthy of astonishment than all other works of
ancient or modern time.
( Great Books of the Western World, Volume 46, page 255 )
One need not agree with the dialectical method of Hegel and the
subject object dyad ceaselessly sublimating itself which lies at
the foundation of his philosophy, to agree that when it comes to
the art of the monument the Egptians exerted themselves to the limit.
Also contrary to his assessment it is not true the Egyptians were not
a fun loving people. The question Dan raises of enduring art versus
fashionable art finds one index of an answer in the perennial popu
larity of ancient Egypt in the imagination of mankind.
2) Another difficulty in determining the level of scientific know
ledge of the ancient Egyptians is related to their identification of
divine science and physical science, Thoth being the patron God of
both sciences. In our culture certain kinds of knowledge is class
fied, knowledge related to state security, for instance, or know
ledge which gives a business a competitive edge in the marketplace.
Did the priests of Egypt jealously guard their knowledge? What gave
a priest power in ancient Egypt is the idea that they embodied the
knowledge of the gods of their country, & were the actual represent
atives of the gods, and at least in the case of the Pharoah, embodied
the god itself. It is striking that Khufu is considered to be the
first Pharoah who explictly identified himself with the sun Ra in his
life, and his sons are the first Pharoahs whose cartouches carry the
additional title Son of Ra. Whereas all other pyramids place the
crypt either at, below, or a little above the base of the pyramid,
his crypt is placed high up in the structure of the monument, which
from one perspective is an earlier form of the heresy of Akhenaton,
the absolute identification of the Pharoah with Ra in life, rather
than merely upon death, as the pyramid is a form of the ancient
Ben Ben stone of Heliopolis, and symbolizes Ra himself. Now if it
is true that the entire Horizon of the earth, and the celestial
bodies which move through the Horizon, were conceived to be divine
in the mind of ancient Egypt, the actual motion of the Horizon, and
the measure of that motion, will measure the motion of divinity, &
the embodying of the measured motion in a monument will embody
heaven on earth, & divinize Pharoah through his entombment in that
Horizon, and its perfect reflection of the movements of Heaven and
Earth. When we remember than Khufu is celebrated in the historical
imagination of ancient Egypt as having sought out the knowledge of
the secret passageways of Tahuti, to incorporate it into his Akhet,
or Horizon, and we reflect that Tahuti is the scribe of the gods,
and the measurer & weigher of Heaven and Earth, it is not too great
of a leap of the imagination to think that Khufu, and his priests
and architects incorporated their astronomical and mathematical
knowledge in the monument. The question arises, why is it not
incorporated through direct lineal measurements, or if one is able
to demonstrate that some of it is incorporated directly, why is it
that a knowledge of elementary solid geometry is necessary to dem
onstrate other of this knowledge? First is should be remarked that
even it one is able to demonstrate a very precise measure of the
movements of the heavens incorporated in the monument, that fact
of being able to precisely measure the time and space of these
movements does not demonstrate a knowledge of heliocentric theory,
or celestial mechanics. The knowledge of the actual fact that the
heavens precess through time is a case in point. Given the length
of the Egyptian civilization and the period of time that they ob
served the heavens, it is impossible that they did not notice that
the heavens precess, as they do so at around one minute a year, or
1.39686 degrees per century at our era, and using some recent pre
cessional formulae, at around 1.369 degrees per century at the era
of Khufu. However, to measure the rate of precession is extremely
difficult. One can measure it in a given time span, and say that
the rate of precession changes at such a rate, but a true under
standing of the rate of precession requires an understanding of
the nonlinear dynamics of the solar system, and the mathematics
of a nonrigid earth. Of all local astronomic constants the rate
of precession is one subject to the greatest uncertainty. However,
it is extremely easy to measure the actual fact of precession if
you are a regular observer of the heavens, as each spring equinox
the stars which arise before the sun will be different, & will be the
stars of the zodiac which are one arc minute behind the stars which
arose in the previous year. The Egyptians were fully capable of ob
serving the heavens to a minute degree of accuracy. The Great Pyramid
itself is oriented to true north to that degree of accuracy, & Petrie
argues that its foursquare form demonstrates an ability to measure to
12 seconds of accuracy. To know why the heavens precess requires a
knowledge not only of the actual rotation and revolution of the earth
around the sun, it also requires a knowledge of the revolution of the
moon around the earth, and a knowledge of the gravitational effects
of the sun and moon on the oblate spin of the earth, meaning, at the
least it requires a knowledge of elementary Newtonian mechanics. Re
cent research suggests that one possible way that ancient Egyptians
oriented the Great Pyramid is through using a plumb line centered on
two polar stars circling the center of the Heavens. If indeed that
was how they went about orienting their monuments, it would soon be
come apparent to them as they built through the decades, and even
centuries, that the center of the Heavens did not remain fixed. In
fact, using an extremely simple formula, it is possible to derive
centennial precession at the era of Khufu, through the slope of his
monument. The 9 / 10 rise run on the edge works best, although the
14 / 11 rise run on the face is almost as accurate. Using the date
2528 bc as the date of the death of Khufu cited most often in the
literature, current precessional formulae calculate centennial pre
cession for that era at 1.369 degrees. If one uses a face slope of
the monument as if it was built using a 9 / 10 rise run on its as
cending edge, then its slope in radians raised to the inverse power
of Pi equals centennial precession of that era accurate to the time
of the building of the monument
.9048518739 ^  PI = Radian Slope Khufu ^  PI = 1.36904 ~
Number of Degrees Centennial Precession circa Khufu 2528 bce.
If you would like to calculate it yourself, first determine the
radian slope of Khufu. Using a perimeter inradius of 220 cubits,
the perimeter circumradius equals Square Root ( 220² + 220² ), the
height will equal 9 / 10 * Square Root ( 220² + 220² ), the ratio
of the height and the perimeter inradius will equal the tangent of
the radian slope, and the radian slope equals the Inverse Tangent
of 9 / 5 / Square Root 2 = 0.90485187394...
To calculate precession circa era Khufu use the precessional formula
(1.396291666... + 0.0006180555...T)°
given at http://www.jqjacobs.net/astro/astrofor.html#periodicity
where T is the number of tropical centuries from 1900.00, although
in our case we will be subtracting .0006180555 times the number
of centuries before 1900.00. Notice that the constant of precession
.0006180555 is quite close to 1 / GoldenRatio / 1000. Solving for T
the number of centuries prior to 1900 at which precession equals
.9048518739 ^  PI° = 1.36904°:
(1.396291666...  0.0006180555 * T)° = .9048518739 ^  PI°
and T = 44.09411418... centuries or 4409.411418 years. Subtracting
this number of years from 1900 to determine what year centennial pre
cession equaled Radian Slope Khufu ^  PI° we get
1900 ce  4409.411418 = 2509.411418... bce
Now that is pretty darn remarkable. The date of the death of Khufu
is usually given as 2528 bce, it is often stated that it took 20
years to build the Great Pyramid, so that by these figures, the
radian slope of Khufu raised to the inverse power of PI equals
centennial precession in degrees at the year the architects and
masons were laying the foundations of the monument. Compare this
method of deriving precession from Khufu with that of Stecchini. Of
course, there are many uncertainties. The two main king lists from
which the dynasties of Egypt are dated have yielded at least two
major competing dating schemes. The date I give is the one most
often given, for instance in the Oxford Atlas of Egypt, and in the
Complete Pyramids of Lerner. The precessional formula is open to
uncertainty. In fact it contradicts the division of the perimeter
into celestial sphere obliquity of the current era accurate to 1/3
arc second. Why does the face slope and the circumference / diameter
ratio yield centennial precession era Khufu, and the celestial
sphere / perimeter ratio yield obliquity ecliptic our era? It would
appear that the perimeter of the Great Pyramid is predictive of the
tilt of the axis of the earth  i.e. that it predicts a given time.
It is true that we live in a time that is absolutely critical, and
the decisions we make in this era in regards to carbon dioxide
emissions, deforestation, depletion of aquifiers, soil conservation,
active forestation, equilibrium economics, species extinction,
overfishing of the oceans, ozone holes, et cetera will determine
specific consequences, so it is absolutely appropriate that the
celestial sphere / perimeter ratio of the Great Pyramid, the
greatest monument and tomb in the solar system, and perhaps the
universe, a monument signifying death and resurrection, is the
perfect analogy for the creation of a new thinking and new logic, a
logic of life and not death, a logic of the living and not the dead,
a logic of peace and not war, a logic of the actual existence of
heaven on earth, of earth in heaven, a logic of transcendental
consciousness identical matter, a logic of absolute exteriority and
the surface of existence, the infinitely flat surface of the four
dimensional hypersphere of universal expansion, a logic of the
surface of the earth, a logic of the word of God made flesh.
But given the current imprecision in our ability to actually cal
culate the rate of precession through time, limited by both a small
empirical set of temporal data, and the complexity of the equations,
we really have no way of knowing if that was the actual rate of pre
cession of that era, let alone, given the other fundamental geodetic
and astronomic features of the monument more directly related to its
mathematical structure, why such a simple equation should allow its
slope  apparently what it is for other reasons  to be so easily
related to the period of precession  after all this would appear to
suggest that the slope which relates the length of the polar axis
and a circle of equatorial latitude mediated through the period of
the rotation of the earth  meaning the height and perimeter slope
of the Great Pyramid  possesses a real mathematical relationship
to the period of the precession of the earth. That would need
to be demonstrated through celestial mechanics for it to be anything
other than a coincidence, which brings us to the third point of why
it is difficult to know what if any knowledge of astronomy et cetera
is incorporated in the Great Pyramid: coincidental occurances of
numbers, and jumping to conclusions from these coincidences, all of
which has contributed to the general disrepute any research into
the subject has fallen these days.
3) When one reads the works of Smyth, Edwards, and the Edgar brothers
and the crazy theories, and wild speculations they contain, they make
folk like Stecchini and Schwaller De Lubicz look like sober scholars.
Nothing has done more damage than a faithful attempt to identify the
length of the passages of the pyramid measured in an imaginary pyra
mid inch derived from an equally imaginary sacred cubit of the He
brews with the timeline of major events in the Bible ( whether real
or imaginary events ) & human history. You do not even need to re
sort to this imaginary pyramid inch to engage in predictive pyramid
ology. Petrie measured the edge of the base of the north face to the
center axis of the pyramid  meaning the point below the apex, which
is also the distance of the point below the apex of the roof of the
Queen Chamber, & the distance of the point below the step of the
Grand Gallery, from the northermost edge of the Great Pyramid  to
be 4534 British Imperial Inches. Well, if you take the commonly
accepted date of the death of Khufu  2528 bce  and subtract it from
4534, you get 2006 the current year. Anything world shaking occur
in this year? The most historical event that I perceive which has
occurred this year is that for the first time in history the fact
that global warming is actually occurring has finally begun to be
admitted by the American power structure. Biblically, 2006 is the
year of Creation that Noah died. Remember him? The guy that rode
out the flood and preserved species on the universal ark? Now these
numbers may be able to be used to catalyse the popular religious
imagination to act if embodied in a creative work of sufficient
appeal, but otherwise, they have little or no value outside of the
role myth can play in cultivating unscientific and irrational
understanding.
There is alot more than can be said on these matters, but I am a
little tired, and I will save it for another occasion. I will sim
ply conclude by noting that the value of folk like Stechinni and
Tompkins, is they awaken the imagination to the possibility of a
specific fact of history. The truth or falsity of that fact, if it
exists, and it is not merely a nebulous fancy and wishful think
ing, can only be examined on the basis of extremely accurate scient
ific measurements. That is the first conditio sine qua non. Once
one has these measurements one can examine the basic geometrical
facts of the monument and compare them to the facts of the science
of geometry, astronomy, and geodesy, and see if they warrant the
conclusion that the authors of the monument incorporated any such
knowledge in it. The analogy of the blind watchmaker is not exact.
The geometry ought to be as evident as if one were examining geome
trical figures drawn on paper. The difference is this: on paper
one is able to explicitly see the geometrical problem and its sol
ution, and one is often given a few words to help demonstrate the
proof of its solution. In a monument like the Great Pyramid, the
proof is often not quite as explicit. Yet though implicit in the
geometry of its architecture, it ought to be as transparent as the
geometry drawn on paper. After all, what is the difference between
seeing a geometrical problem and its solution in two dimensions, and
seeing one built in stone in three dimensions? Let us take a very
simple example  one of the basic geometrical facts of the King
Chamber. I have already pointed out that it exactly doubles the
volume of the Queen Chamber  accurate to 1/2 millimeter. I think
it is more improbable that the volume of the King Chamber is double
the volume of the Queen Chamber by accident than that it is by de
sign. One could perchance calculate the odds. It by itself demon
strates no great measure of intentional design in the monument. How
ever, let us look a little closer at the geometry of the King Cham
ber. The King Chamber cubit is derived from the hypothesis of a 10
by 20 cubit width and length of the base of its walls. This is a
hypothesis proven by the fact that the cubit so derived is within
.002 inches of the mean cubit of ancient Egypt derived from a variety
of its monuments  20.632 versus 20.63 inches. Knowing then that the
architects designed the chamber to be 10 by 20 cubits at its base, we
ask ourselves, given the height of the walls of the King Chamber 
235.2 inches, or in king chamber cubits  11.399767... cubits  why,
if at all, did the architects design the height to be that many cu
bits? The easy answer, since the King Chamber was built after the
Queen Chamber, is that number of cubits allows the volume of the
chamber to equal 2280 cubic cubits, or twice the volume of the 1140
cubic cubits of the Queen Chamber. However, to settle on just this
does not do justice to the possibility that the architect may have
had a level of intelligence above the average bear. And indeed,
the simplest volumetric analysis of the chamber ( & one need not in
voke the idea of De Lubicz, correct as it may be, that the ancient
Egyptians were fond of volumetric thinking ) demonstrates the King
Chamber solves not the Delian Problem, but a two dimensional variant
of it: given two squares equal in area to each other  here the
actual dimensions of the floor of the King Chamber  10 cubits in
width by 20 cubits in length  two squares each 10 by 10  how high
must the chamber be so that its volume equals that of a cube whose
circumradius is that height? It turns out that there is one height
which perfectly cubes the volume of a 10 by 20 hexahedron so that
the volume of the cube equals the volume of the hexahedron, and the
height of the hexahedron equals the circumradius of the cube, and
that height is 11.397535...
Since the actual height of the walls of the King Chamber equals
11.399767 cubits, it is one millimeter more than the height the
chamber would need to be for its volume to equal the volume of
a cube whose circumradius ( half its space or body diagonal ) is
the height of the King Chamber. Now again, it is not this actual
fact itself of the King Chamber which by itself demonstrates the
intentional incorporation of elementary solid geometry into its
design. It is only through the demonstration of the existence
of a series of such simple geometrical facts in the monument that
the existence of intelligence designing it there acquires the force
of a demonstration, as it outweighs the improbability of such beau
tiful and simple geometrical facts existing there by accident.
 John  One way to get the cube root of 2 Delian Constant
http://en.wikipedia.org/wiki/Doubling_the_cube
from the volume of the King Chamber ( KC ) is as follows:
Let the Volume of the Chamber equal 10 by 20 by 11.4 cubic cubits or
2280 cubic king chamber cubits. The area of a sphere of volume 2280
cubic cubits is 837.736341, and if we divide it by half the area of
the floor of the king chamber  100 square cubits  we get a unit
free number
Area of Sphere of Volume of KC / Area of 1/2 Floor of KC = 8.37736341
8.37736341 is the Volume of a Sphere of Radius 1.259910...
which is quite close to the Delian Constant of 1.2599210...
Or if we want to get a little ridiculous, let the height of the GP
equal 280 cubits, and its base equal 440, then its Apothem will equal
356.0898..., and the Inverse of the Cosine of its Face Slope in rad
ian or the Secant of its Slope will equal
1 / Cosine .904827089... = 1.618590346... = 356.0898... / 220
and instead of letting the sphere be equal in volume to that of the
King Chamber  2280 cubic cubits  let the sphere be equal in volume
to a cube of edge length 13 + one tenth of the Secant of the Radian
Slope of Khufu:
13 + Secant .904827089... / 10 = 13 + .1618590346...
If you divide the length of the apothem of a pyramid of height 280 &
base 440 by the sum of the length of its perimeter and the length of
the indiameter of its perimeter, you get .1618590346...
So let the Sphere be equal in volume to
( 13 + .1618590346... )³ cubic king chamber cubits
then the king chamber cubit surface area of that sphere divided by
half the king chamber cubit area of the floor of the king chamber 
100 square cubits  equals 8.37758021264, and the radius of a sphere
of volume 8.37758021264 equals
1.2599210400 which equals the cube root of 2 or
1.2599210498 and the Delian constant accurate to nine digits!
 John  A generalization of the geometry of the king chamber from end of post
7000 and beginning of post 7007 :
The area of the sphere equal in volume to a cuboid of width 1, length 2,
and height equal to the circumradius of a cube equal in volume to the
cuboid, equals the volume of a sphere of radius 1.2598 ~ 1.2599 = 2 ^ (
1 / 3 ) = Delian constant
Another way to the Delian constant through the GP:
The length of the ascending edge of a pyramid of height 280 cubits and
base 440 cubits is 4.18569 egyptian rods, and the length of the radius
of a sphere whose volume equals the number of rods in the 4 ascending
edges of the pyramid is
1.58700.. ~ cube root of 4 = 1.58740... =
the ratio of the area of the face of a cube to the area of the face of
the cube half its volume, and the square of the Delian constant   In sacredlandscapelist
>
...
>
> One way to get the cube root of 2 Delian Constant
>
100 square cubits equals 1 land cubit in ancient Egypt.
> So let the Sphere be equal in volume to
>
> ( 13 + .1618590346... )³ cubic units
>
> then surface area of that sphere divided by 100 square units 
> equals 8.37758021264, and the radius of a sphere of volume
> 8.37758021264 equals
>
> 1.2599210400 which equals the cube root of 2 or
>
> 1.2599210498 and the Delian constant accurate to nine digits!
Mike  notice that the commonly agreed height of the GP  280 cubits 
and its inradius  220 cubits  yields an apothem of
Square Root ( 280² + 220² ) = 356.0898762952... cubits
and hence its apothem divided by the length of a side of its square
base equals
356.0898762... / 440 = .809295173....
This ratio is close to .809295992... =
Square Root ( 2 × 2 ^ ( 2 / 3 )  2 × 2 ^ ( 1 / 3 ) )
the square root of ( twice the area of the face of a cube double the
volume of a unit cube minus twice the length of the edge of the cube
double the volume of the unit cube ) a number which figures promin
ently in one of the most celebrated solutions to the doubling of the
cube  Archytas' Duplication of the Cube
http://plato.stanford.edu/entries/archytas/
http://www.ms.uky.edu/~carl/ma330/projects/dupcubfin1.html
Archytas determined the length of the Delian constant from the inter
section of a cone, a cylinder, and torus. Using modern notation the
Delian constant equals the solution of three simultaneous equations:
Cone: x² = y² + z²
Cylinder: 2x = x² + y²
Torus: ( x² + y² + z² )² = 4x² + 4y²
Here the length of the edge of the cube to be duplicated equals unity
( See the second link for details ) When x = the cube root of two or
the Delian constant 1.259921... in the above equations z equals
Square Root ( 2 × 2 ^ ( 2 / 3 )  2 × 2 ^ ( 1 / 3 ) )
= .80929599291... ~ = 0.809295173... 356.089876295... / 440 =
Apothem of Great Pyramid / Base of Perimeter of Great Pyramid
As the second link notes "The human eye has a difficult time reading
a 3D figure, so we will take a slice of this figure to make it easy
to see...You need to slice it at the appropriate z = constant or y =
constant plane in order to "see" that the xcoordinate of the inter
section is 2 ^( 1 /3 ). Of course you can only approximately see
that ...By playing with the solutions to the equations, you can work
out that (.80929599291... ) is the exact value for z."
 J  Hi, John
You have a note to Mike in this post. Because of other priorities,
Mike has signed off the list for the time being. I have forwarded your
post on to him. If you want to contact him directly his email is Mike
Bispham@....
Dan
jjdepompeo wrote:
> In sacredlandscapelist
>
>
>>One way to get the cube root of 2 Delian Constant
>>
>>
>
>...
>
>
>
>>So let the Sphere be equal in volume to
>>
>>( 13 + .1618590346... )³ cubic units
>>
>>then surface area of that sphere divided by 100 square units 
>>equals 8.37758021264, and the radius of a sphere of volume
>>8.37758021264 equals
>>
>>1.2599210400 which equals the cube root of 2 or
>>
>>1.2599210498 and the Delian constant accurate to nine digits!
>>
>>
>
>100 square cubits equals 1 land cubit in ancient Egypt.
>
>Mike  notice that the commonly agreed height of the GP  280 cubits 
>and its inradius  220 cubits  yields an apothem of
>
>Square Root ( 280² + 220² ) = 356.0898762952... cubits
>
>and hence its apothem divided by the length of a side of its square
>base equals
>
>356.0898762... / 440 = .809295173....
>
>This ratio is close to .809295992... =
>
>Square Root ( 2 × 2 ^ ( 2 / 3 )  2 × 2 ^ ( 1 / 3 ) )
>
>the square root of ( twice the area of the face of a cube double the
>volume of a unit cube minus twice the length of the edge of the cube
>double the volume of the unit cube ) a number which figures promin
>ently in one of the most celebrated solutions to the doubling of the
>cube  Archytas' Duplication of the Cube
>
>http://plato.stanford.edu/entries/archytas/
>http://www.ms.uky.edu/~carl/ma330/projects/dupcubfin1.html
>
>Archytas determined the length of the Delian constant from the inter
>section of a cone, a cylinder, and torus. Using modern notation the
>Delian constant equals the solution of three simultaneous equations:
>
>Cone: x² = y² + z²
>
>Cylinder: 2x = x² + y²
>
>Torus: ( x² + y² + z² )² = 4x² + 4y²
>
>Here the length of the edge of the cube to be duplicated equals unity
>( See the second link for details ) When x = the cube root of two or
>the Delian constant 1.259921... in the above equations z equals
>
>Square Root ( 2 × 2 ^ ( 2 / 3 )  2 × 2 ^ ( 1 / 3 ) )
>
>= .80929599291... ~ = 0.809295173... 356.089876295... / 440 =
>
>Apothem of Great Pyramid / Base of Perimeter of Great Pyramid
>
>As the second link notes "The human eye has a difficult time reading
>a 3D figure, so we will take a slice of this figure to make it easy
>to see...You need to slice it at the appropriate z = constant or y =
>constant plane in order to "see" that the xcoordinate of the inter
>section is 2 ^( 1 /3 ). Of course you can only approximately see
>that ...By playing with the solutions to the equations, you can work
>out that (.80929599291... ) is the exact value for z."
>
> J
>
>
>
>
>
>Topics suitable for discussion in this elist can be found at:
>http://www.luckymojo.com/sacredland.html
>
>To UNsubscribe, send email to:
>sacredlandscapelistunsubscribe@yahoogroups.com
>
>Yahoo! Groups Links
>
>
>
>
>
>
>
Thanks Dan. I mentioned his name in passing as he indicated his researches led him to think that the proportional external geometry of the GP indicates a concern with the Delian problem on the part of its architect. I stumbled on the Archytas z coordinate and thought to share it. It still may be that it is possible to directly derive a precise Delian constant from the simplicity of the proportional geometry of the GP ... BTW if anyone followed up on any of those links in my last post, the geometrical drawing in the first link http://plato.stanford.edu/entries/archytas/#Cube has a 3D animation analogue of it at ( of all places! )http://www.larouchepac.com/pages/economy_files/2004/041014_anim_eco_graphics.htm
Click on the control and drag it to rotate it about any axis. It is easy to see how the cylinder and torus are generated ( respectively ) from the original horizontal circle ABDZ at the stanford link, and the rotation of the semicircle erected perpendicular to it on the diameter AED. The center 'hole' of the torus is the point A. A question: is an infintesimal point a hole, and is a torus with such a center actually a torus? :) It is more difficult to see why the curve of the cone is necessarily generated from the rotating triangle. Also I did not review or check the math at the second site ( the one that uses the maple program ) so I am not able to vouch for it.
John
> Hi, John
>
> You have a note to Mike in this post. Because of other priorities,
> Mike has signed off the list for the time being. I have forwarded your
> post on to him. If you want to contact him directly his email is Mike
> Bispham@...
>
> DanOr more simply and accurately where
Pyramid Apothem = 356.089876... cubits = Square Root ( 280² + 220² ) = Square Root ( Height² + Inradius² )
Pyramid Side = 440 Cubits
Delian constant = 1.25992104... = cube root of 2 = Edge of Cube of Volume Two = D
Edge of Unit Square of Volume One = 1
( Apothem ÷ Side )² ÷ 2 = D²  D = ( D  1 ) × D = ( D  1 ) ÷ ( 1 ÷ D )
the D variable on the right side of the equation equals 1.259921 ~ 1.25992104... = Delian constant accurate to 7 places
Apothem² ÷ 2 / Area of Square Base of Pyramid = 1.259921²  1.259921 = ( 1.259921  1 ) ÷ ( 1 ÷ 1.259921 )
1/2 the area of the square of the altitude of each face triangle of the Great Pyramid is in the same proportion to the area of the square of its base as the difference between the delian constant and unity is to the quotient of unity and the delian constant
Thanks to Mike for suggesting the idea!
 John
In sacredlandscape list jdepompeo wrote:
> Or more simply and accurately where
> Pyramid Apothem = 356.089876... cubits = Square Root ( 280² + 220² ) = Square Root ( Height² + Inradius² )
> Pyramid Side = 440 Cubits
> Delian constant = 1.25992104... = cube root of 2 = Edge of Cube of Volume Two = D
> Edge of Unit Square of Volume One = 1
> ( Apothem ÷ Side )² ÷ 2 = D²  D = ( D  1 ) × D = ( D  1 ) ÷ ( 1 ÷ D )
> the D variable on the right side of the equation equals 1.259921 ~ 1.25992104... = Delian constant accurate to 7 places
> Apothem² ÷ 2 / Area of Square Base of Pyramid = 1.259921²  1.259921 = ( 1.259921  1 ) ÷ ( 1 ÷ 1.259921 )
> 1/2 the area of the square of the altitude of each face triangle of the Great Pyramid is in the same proportion to the area of the square of its base as the
> difference between the delian constant and unity is to the quotient of unity and the delian constant
>Thanks to Mike for suggesting the idea!
Even more simply, where the rise / run of Khufu = 280 / 220 = 14 / 11,
( 1 + 14^{2} / 11^{2 }) / 2^{3} = D ( D  1 )
the actual value of D = 1.2599206... = ( 22 + 1118^{1/2} ) / 44 = 1 + ( 1 + rise^{2} / run^{2 }) / 2^{3} / 2^{1/3} = 2^{1/3}
 John