Project GeoArch helped the students to explore various facts about geometry, its integration with art and architecture by analysing and comparing different geometrical patterns found in the monuments, pyramids, temples and towers in Egypt, France and India.

2. Glimpse of Egyptian Architecture
 Ancient Egyptian architecture of Ancient Egypt, one of the most
influential civilizations throughout history, which developed a vast
array of diverse structures and great architectural monuments along
the Nile, including pyramids and temples.
Characteristics were very unique.
 Due to the scarcity of wood, the two predominant building materials
used in ancient Egypt were sun-baked mud brick and stone, mainly
limestone, but also sandstone and granite in considerable quantities.
From the Old Kingdom onward, stone was generally reserved
for tombs and temples, while bricks were used even for royal
palaces, fortresses, the walls of temple precincts and towns, and for
subsidiary buildings in temple complexes. The core of
the pyramids consisted of locally quarried stone, mud bricks, sand or
gravel. For the casing stones were used that had to be transported
from farther away, predominantly white limestone from Tura and
red granite from upper Egypt.

3. • Thus, our understanding of ancient Egyptian architecture is based mainly on
religious monuments, massive structures characterized by thick, sloping walls
with few openings, possibly echoing a method of construction used to obtain
stability in mud walls. In a similar manner, the incised and flatly modelled
surface adornment of the stone buildings may have derived from mud wall
ornamentation.

5. • There are many such legendary
examples of this unique
architectural style.

8. THE MYSTERY OF THE GREAT SPHINX
 Facing the rising sun, the Great Sphinx is located on
the Giza plateau, about 10 km west of Cairo, on the west
bank of the Nile River. Later Egyptian rulers worshipped it
as an aspect of the sun god, calling it Hor-Em-Akhet
(“Horus of the Horizon”). The Sphinx sits in part of the
necropolis of ancient Memphis, the seat of power for the
pharaohs, a short distance from three large pyramids – the
Great Pyramid of Khufu (Cheops), Khafre (Chephren) and
Menkaura (Mycerinus).

9.  The monument is the largest surviving sculpture from the
ancient world, measuring 73.5 m in length and in parts 20
m in height. Part of the uraeus (sacred cobra which
protected from evil forces), the nose and the ritual beard
are missing; the beard is now displayed in the British
Museum. The extensions at the side of the head are part of
the royal head cloth. Although the head of the Sphinx has
been badly affected by thousands of years of erosion, traces
of the original paint can still be seen near one ear. It is
thought that originally the Sphinx’s face was painted dark
red. A small temple between its paws contained dozens of
inscribed steel placed by the Pharaohs in honour of the
Sun god.

10. 240ft long and 66ft high.
Uses a proportional Symmetry from front &
the rear side.
It is located with the 3 pyramid in decreasing
order of side at its back and 6 small pyramids.
This pyramid system uses brainy mathematics
of Pythagoras theorem to check out the
constellations and location of the pole star.

13. THE MYSTERY OF THE KARNACK !
 The Karnack Temple Complex, comprises a vast mix of
decayed temples, chapels, pylons, and other buildings.
Building at the complex began during the reign
of Senusret in the Middle Kingdom and continued into
the Ptolemaic period, although most of the extant
buildings date from the New Kingdom. The area around
Karnak was the ancient Egyptian Ipet-isut ("The Most
Selected of Places") and the main place of worship of the
eighteenth dynasty Theban Triad with the god Amun as
its head. It is part of the monumental city of Thebes.
The Karnack complex gives its name to the nearby, and
partly surrounded, modern village of El-Karnack, 2.5
kilometres (1.6 miles) north of Luxor.

15. MATH MIRACLES

16. This is the largest precinct of the temple complex, dedicated to Amun-Re (a form of Amun), the
chief deity. There are several colossal statues including the figure of Pinedem I which is 10.5
meters tall. The sandstone for this temple, including all the columns, was transported from
Gebel Silsila 100 miles south on the Nile river. It also has one of the largest obelisks weighing
328 tonnes and standing 29 meters tall.
The Great Hypostyle Hall
• The Great Hypostyle Hall of Karnak, within the
Karnak temple complex, in the Precinct of Amon-Re,
is one of the most visited monuments of Ancient
Egypt.
• The hall covers an area of 50,000 sq ft (5,000 m2).
The roof, now fallen, was supported by 134 columns
in 16 rows; the 2 middle rows are higher than the
others (being 33 feet (10 m) in circumference and 80
feet (24 m) high).

17. THE HALL WAS NOT BEGUN BY
HOREMHEB, OR AMENHOTEP III
AS EARLIER
SCHOLARS HAD THOUGHT.
IT WAS BUILT
ENTIRELY BY SETI I WHO
COVERED THE
NORTHERN WING OF THE
HALL WITH INSCRIPTIONS.
DECORATION OF THE
SOUTHERN WING
WAS COMPLETED BY
RAMESSES II.
RAMESSES CHANGED SETI'S NAMES
TO HIS OWN ALONG THE MAIN
EAST-WEST
AXIS OF THE HALL AND PART OF THE
NORTH-SOUTH PROCESSIONAL ROUTE.

18. • Karnack is an open air museum and the largest ancient religious site
in the world. It is one of the top tourist magnets to Egypt.
• Karnack consists of huge pillars, towering columns, massive avenues
of sphinxes, and an obelisk that stands 97-feet tall and weighs 323-
tons.
• The 54,000 square feet Great Hypostyle Hall is large enough for the
Cathedral of Notre Dame to fit in comfortably.
• A whopping 30-pharaohs contributed to the construction of Karnak -
and this list includes the crazy Queen Hatshepsut.

20. THE MYSTERY OF THE BENT PYRAMID !
 The Bent Pyramid is an ancient Egyptian pyramid located at the
royal necropolis of Dahshur, approximately 40 kilometres south
of Cairo, built under the old Kingdom Pharaoh Sneferu (c. 2600 BC).
A unique example of early pyramid development in Egypt, this was
the second pyramid built by Sneferu.
 The Bent Pyramid rises from the desert at a 54-degree inclination,
but the top section (above 47 meters) is built at the shallower angle
of 43 degrees, lending the pyramid its very obvious 'bent'
appearance.

21.  Archaeologists now believe that the Bent Pyramid represents a
transitional form between step-sided and smooth-
sided pyramids (see Step pyramid). It has been suggested that due
to the steepness of the original angle of inclination the structure
may have begun to show signs of instability during construction,
forcing the builders to adopt a shallower angle to avert the
structure's collapse.[2] This theory appears to be borne out by the
fact that the adjacent Red Pyramid, built immediately afterwards
by the same Pharaoh, was constructed at an angle of 43 degrees
from its base. This fact also contradicts the theory that at the
initial angle the construction would take too long because Sneferu's
death was nearing, so the builders changed the angle to complete
the construction in time. In 1974 Kurt Mendelssohn suggested the
change of the angle to have been made as a security precaution in
reaction to a catastrophic collapse of the Meidum Pyramid while it
was still under construction.

22.  The Bent Pyramid has two entrances, one fairly low down on the north
side, to which a substantial wooden stairway has been built for the
convenience of tourists (though so far the pyramid is not open to
tourists, although plans have been proposed to open it).The second
entrance is high on the west face of the pyramid. Each entrance leads
to a chamber with a high, corbelled roof; the northern entrance leads
to a chamber that is below ground level, the western to a chamber built
in the body of the pyramid itself. A hole in the roof of the northern
chamber (accessed today by a high and rickety ladder 15 m (50 ft) long)
leads via a rough connecting passage to the passage from the western
entrance.
 The western entrance passage is blocked by two stone blocks which
were not lowered vertically, as in other pyramids, but slid down 45°
ramps to block the passage. One of these was lowered in antiquity and
a hole has been cut through it, the other remains propped up by a piece
of ancient cedar wood.The connecting passage referenced above
enters the passage between the two portcullises.

23. SATELLITE PYRAMID
 An axonometric projection of the inside of the satellite pyramid
 The satellite pyramid
 Entrance of the satellite pyramid
 A satellite pyramid, built to house the Pharaoh's Ka, is located 55 metres south of the Bent
Pyramid.The satellite pyramid originally measured 26 metres in height and 52.80 metres in
length, with faces inclining 44°30'.The structure is made of limestone blocks, relatively thick,
arranged in horizontal rows and covered with a layer of fine limestone from Tura.The burial
chamber is accessible from a descending corridor with its entrance located 1.10 metres above the
ground in the middle of the north face. The corridor, inclined at 34°, originally measured 11.60
metres in length.A short horizontal passage connects the corridor with an ascending corridor,
inclined at 32° 30', leading up to the chamber.
 The design of the corridors is similar to the one found in the Great Pyramid of Giza, where the
Grand Gallery takes up the place of the ascending corridor.The corridor leads up to the burial
chamber (called this despite that it most probably never contained any sarcophagus).The
chamber, located in the center of the pyramid, has a corbel vault ceiling and contains a four
metres deep shaft, probably dug by treasure hunters, in the southeast part of the chamber.
 Like the main pyramid, the satellite had its own altar with two stele located at the eastern side.

24. The pyramid has a very unique dimension
and approximations.......

26.  The Giza pyramid complex is an archaeological site on
the Giza Plateau, on the outskirts of Cairo, Egypt.
 This complex of ancient monuments includes the
three pyramid complexes known as the Great Pyramids, the
massive sculpture known as the Great Sphinx, several
cemeteries, a workers' village and an industrial complex. It is
located in the Libyan Desert, approximately 9 km (5 mi) west of
the Nile river at the old town of Giza, and about 13 km (8 mi)
southwest of Cairo city centre.
 The pyramids, which have historically loomed large as emblems
of ancient Egypt in the Western imagination were popularised
in Hellenistic times, when the Great Pyramid was listed
by Antipater of Sidon as one of the Seven Wonders of the
World. It is by far the oldest of the ancient Wonders and the only
one still in existence.

27.  Mathematical Encoding in the Great Pyramid
 The Great Pyramid of Giza, the oldest and sole
surviving Wonder of the Ancient World, has attracted
the interest of philosophers, savants, and travelers
for at least four millennia. Some of this interest has
centered on the question of whether the ancient
Egyptian culture possessed and encoded certain
mathematical concepts in the pyramid’s proportions
and measurements.

28.  I have analyzed the Great Pyramid’s geometry without considering the issues of
what mathematical knowledge ancient Egyptian architect-engineers did or did not
authoritatively possess. If ancient designers did encode something in the Great
Pyramid, I believe they would have wanted the structure’s very geometry to carry
the message.
 Perhaps the Great Pyramid is not a tomb but a tome—open to all those who
possess geometrical understanding, much in the same spirit of what Plato’s
Academy inscribed millennia later on its door, “Let no one ignorant of geometry
enter.”
 The Great Pyramid’s elevation encodes three of the most important constants in
mathematics: π, Φ, and e. The slope angle of 51°51’ (51.85° in decimal form) comes
from measurements taken off the remaining casing stones, according to detailed
survey data from Flinders Petrie and J.H. Cole .
 You can verify these encodings with the following equations. The accuracy
percentage for each equation is given in parenthesis.
 4 x 51.85° / 76.30° = e (99.998%)
 tan 51.85° = 4 / π (99.99%)
 cos 51.85° = 1 / Φ (99.95%)
 sin 51.85° = 4 / πΦ (99.94%)

29.  The constant e, known as Euler’s number, wasn’t discovered until
1618. Consider the possibility that this was merely rediscovery. We
are generally so accepting of the myth of linear progress that it is
easy to forget that sometimes knowledge is lost and not rediscovered
for a very long time, if ever. For example, it has been said that the
plumbing system in the palace of Knossos, Crete built in 1900 BCE
wasn’t matched until circa 1900 CE in England.
 Another way of looking at the Great Pyramid is by analyzing its
relative proportions. If the base measures unit length then a square
with an equal unit edge can be drawn as shown below. A circle drawn
from the center with radius equal to the height of the pyramid has a
circumference equal to the square’s perimeter. Thus, the Great
Pyramid “squares the circle.” Its form approximates the solution
sought by ancient geometers, who for many generations endeavored
to square the circle by length (squaring a circle by area is a related
problem). Note: exact squaring of the circle is impossible due to the
transcendental qualities of π (proved by Lindemann in 1882).
 In addition to relative proportions, I wondered if the Great Pyramid
might encode anything in absolute units? To find out, I drew its
elevation in a computer aided design program in actual size,
according to J.H. Cole’s surveyed values (I used 755.775 feet for the
mean base edge length).

30.  I rediscovered many layers of encoding in the Great Pyramid by taking its
precise measure and looking for patterns. If you inscribe an equilateral
triangle inside, this is what immediately pops out: the green triangle’s
edge length is 555.5 feet, or 6666 inches. This distance happens to match
the height of the Washington Monument and the length of St. Paul’s
Cathedral but that is another story (to learn more see my video series,
Secrets In Plain Sight). The key pattern to observe is the repeating digits,
even though it goes against what we think we know about the English
foot’s provenance.
 Another method of analyzing the pyramid geometry is with a square
rotated 45° with respect to the ground plane. The largest such square that
fits inside the pyramid’s elevation encodes two circles across its
diagonal, having circumferences equaling exactly 666.6 feet each,
allowing that the slope angle of 51°51’ has an accuracy of only 99.8%.
 If you circumscribe the pyramid’s elevation triangle and then measure the
circle’s diameter, this equals 777.7 feet (99.97% accurate with a slope of
51°51’ and base length of 755.775 feet). In addition, the inscribed circles
shown below in red and blue have circumferences of 777 feet and 7777
inches, respectively. Sevens abound!
 This next image depicts a single inscribed circle having a diameter of
365.242 feet inside the pyramid’s elevation that echoes Earth’s tropical
year of 365.242 days. The circle’s exact diameter is 2.2 feet longer than
this measurement, but this small delta might reference the Earth’s
atmosphere, which is like an onionskin at this scale, roughly equaling the
thickness of the diagram’s lines.

31. ARCHITECTURAL DIAGRAMS.......
• ARCHITECTURAL DIAGRAMS............

33.  Made By : SAGAR
 Class : IX
 Roll. – No: 29