PART 2 - THE GOLDEN SECTION

The Greek letter phi (φ) signifies the golden section, also known as the divine proportion of 1.618 to one. The mathematical formula for φ is the square root of five plus one, divided by two. If a line is divided with the ratio of φ between the longer segment and the shorter segment, the ratio between the whole line and the longer segment is also φ. Given a length of one for the shorter segment, the length of the longer segment is φ and the length of the entire segment is φ plus one. φ plus one is also equal to φ²:

1.618 + 1 = 2.618 and
1.618 x 1.618 = 2.618

Given a length of one for the longer segment, the shorter segment is 1/φ (.618) and the length of the entire segment is φ. 1/φ is also equal to φ minus one:

1 ÷ 1.618 = .618 and
1.618 minus 1 = .618

There is also an extremely close mathematical relationship between φ and π expressed as φ² x 6 = π x 5 (2.618 times 6, divided by 5, equals 3.1416). There is an ongoing dispute about whether or not there was an ancient awareness of φ and π. In the ancient world, φ existed naturally in the proportions and growth rates of plant and animal species, as well as spirals ranging from microscopic to flowering plants and seashells to the spiral arms of the galaxy. The π ratio between the diameter and the circumference of a circle also existed naturally in the ancient world. It has also been shown that φ and π existed in many man-made ancient buildings, including the great pyramid of Giza.

The ancient Egyptian royal cubit, equal to 20.625 English inches, was used to build the great pyramid. The height of the pyramid is 280 cubits and the baselength of the sides at ground level is 440 cubits. The ratio between the height and two baselengths of the pyramid is an accurate expression of π (880/280 = 3.1428). The slant height of the pyramid is 356 cubits. The ratio between the baselength of the pyramid and the two slant heights that form the pyramid triangle is an accurate expression of φ (712/440 = 1.618).

4,745 x 1.618 = 7,677

Ninety miles northeast of Angkor Wat are the Angkor temples at Prassat Preah Vihear. Preah Vihear is 4754 miles from the Great Pyramid. The line of ancient sites crosses over the Great Pyramid and Angkor Vihear.

Angkor Prassat Preah Vihear

Twenty five miles northwest of the city of Nazca is a figure known as the Hummingbird. The Hummingbird is 7,692 miles from the Great Pyramid. The line of ancient sites also crosses over the Hummingbird.

The Hummingbird

The relationship between the distances from Angkor Vihear to the Great Pyramid and from the Great Pyramid to the Nazcan Hummingbird is also a precise expression of φ:

4,754 x 1.618 = 7,692

Because the Hummingbird and Angkor Vihear are antipodal sites, with a distance between them of one-half of the circumference of the earth, two Golden Section relationships between these three sites are shown by the circumference of the earth along the line of ancient sites:

4,754 x 1.618 = 7,692

4,754 + 7,692 = 12,446, and

7,692 x 1.618 = 12,446

These Golden Section relationships may also be diagramed on a straight line:

The line of ancient sites is a line, from the perspective of the first illustration in Part One, and it is a circle, from the perspective of the azimuthal projection above. The line and the circle are found in the greek letter φ and the number 10. Zero and one are also the first two numbers and the only two numbers in the binary code.

The φ relationships between these sites are reflected repeatedly in the first 500 Fibonacci numbers. The first three prime numbers, 2, 3 and 5, approximate the intervals along the circumference of 20%, 30% and 50%, between these three sites. This same percentage of the circumference relationship, accurate to three digits, is found in Fibonacci numbers 137-139:

The next prime Fibonacci number after #137 is #359. The distances between these sites, in miles, is reflected by Fibocacci numbers 359-361, accurately to five digits:

The Great Pyramid is aligned with Machu Picchu, the Nazca lines and Easter Island along a straight line around the center of the Earth, within a margin of error of less than one tenth of one degree of latitude.

Other sites of ancient construction that are also within one tenth of one degree of this line include:

• Perseopolis, the capital city of ancient Persia

• Mohenjo Daro, the ancient capital city of the Indus Valley

• the lost city of Petra.

The Ancient Sumerian city of Ur and the temples at Angkor Wat are within one degree of latitude of this line.

The alignment of these sites is easily observable on a globe of the Earth with a horizon ring. If you line up any two of these sites on the horizon ring, all of the sites will be right on the horizon ring. 3-D world atlas software programs can also draw this line around the Earth.

Start on the Equator, at the mouth of the Amazon River, at 49° 17' West Longitude; go to 30° 18' North Latitude, 40° 43' East Longitude, in the Middle East, which is the maximum latitude the line touches.

Then go to the Equator at,

• 130° 43' East Longitude, near the Northwest tip of New Guinea

• then to 30° 18' South Latitude, 139° 17' West Longitude, in the South Pacific

• then back to 49° 17' West Longitude, at the Equator

The circumference of this line around the center of the Earth is 24,892 miles. Along this line, the great circle distance from,

• the Great Pyramid to Machupicchu is 7,487 miles, 30.0% of the circumference.

• Machupicchu is 2,564 miles from Easter Island, 10.3%.

• Easter Island is 10,096 miles from Angkor Wat, 40.6%.

• Angkor Wat is 2,490 miles from Mohenjo Daro, 10.0%.

• Mohenjo Daro is 2255 miles from from the Great Pyramid, 9.1%.

In addition to calculating the distances between these sites as a percentage of the circumference of the Earth, the distances may also be calculated in degrees of the 360° circumference, by multiplying the percentage by 3.6.

For example, the Great Pyramid is 108° away from Machupicchu.

Angkor Wat and Angkor Thom were constructed at a time when 72 temples were built across the Angkor Plain. The Angkor temple at Prassat Preah Vihear, 90 miles Northeast of Angkor Wat, is within one tenth of one degree of the line. Like Machupicchu, the temple at Prassat Preah Vihear was built on the edge of a mountaintop. The first temples built around Angkor are near the city of Rolous, Southeast of Angkor Wat.

The temples near Rolous are also thought to have been built on foundations constructed at a much earlier time.

Halfway between Angkor Wat and the Great Pyramid is the Indus Valley, the city of Mohenjo Daro, and the unexcavated city of Ganweriwala, which is East of Mohenjo Daro, and thought to be just as large. Both of these sites are on the line between Angkor and the Great Pyramid. The Indus Valley is also antipodal to Easter Island.

It is an interesting coincidence concerning these two sites, opposite each other on Earth, that of the few ancient written languages of the world that remain undeciphered, two are Indus Valley Script and Rongorongo, the written language of ancient Easter Island. The world's first known written languages, Egyptian Hieroglyphics and Sumerian Cuneiform, were also developed along this line of ancient sites. The Jewish, Christian, Muslim, Hindu, Brahman and Buddhist religions, as well as ancient Egyptian and Peruvian religions, were also developed along this line.

Anatom Island is the southernmost Island in the new Republic of Vanuatu, formerly known as the New Hebrides. Anatom Island is exactly halfway between Easter Island and Angkor Wat 5,048 miles each way, or 20.3% of the circumference of the Earth. Stone ruins on Anatom Island once housed the largest missionary church in the southern hemisphere.

The line crosses over the source and the mouth of the Amazon, the mouth of the Nile, the mouth of the Tigris-Euphrates, the Indus River and the Bay of Bengal near the mouth of the Ganges.

The line also crosses over a number of areas of the world that are largely unexplored or unexcavated, including the Sahara Desert, the Brazilian Rainforest, the highlands of New Guinea, and underwater areas of the North Atlantic Ocean, the South Pacific Ocean and the South China Sea.

For example, the midway point between the Great Pyramid and Machupicchu is in the North Atlantic Ocean, less than one degree south of the Cape Verde Islands. This is also the midway point between Easter Island and the Indus Valley.

Although the Cape Verde Islands were found to be uninhabited when they were rediscovered in 1460 A.D., maps and geographical descriptions for the past 2000 years have shown this location to be the home of ancient island civilizations, including maps showing this location to be the site of Atlantis. In Plato's account of Atlantis, there was a mountainous region north of the city.

Are the higher elevations of those mountains now the Cape Verde Islands?

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PART 2 - GOLDEN SECTION SITES - ANGKOR, THE GREAT PYRAMID & NAZCA

Angkor Wat is 4,745 miles from the Great Pyramid and the Great Pyramid is 7,677 miles from Nazca.

This is a precise expression of ?, the Golden Section: 4,745 x 1.618 = 7,677

Because the distance from the Hummingbird to Angkor Vihear is one-half of the circumference of the earth, two Golden Section relationships between these sites are shown by the circumference of the earth along the line of ancient sites:
 

These Golden Section relationships may also be diagramed on a straight line
 

The line of ancient sites is a line, from the perspective of the illustration in Part One, and it is a circle, from the perspective of the illustration in Part Six.

The line and the circle are found in the Greek letter phi and the number 10. Zero and one are also the first two numbers and the only two numbers in the binary code.

The phi relationships between these sites are reflected repeatedly in the first 500 Fibonacci numbers. The first three prime numbers, 2, 3 and 5, approximate the intervals along the circumference of 20%, 30% and 50%, between these three sites.

This same percentage of the circumference relationship, accurate to three digits, is found in Fibonacci numbers 137-139:

The next prime Fibonacci number after #137 is #359.
The distances between these sites, in miles, is reflected
by Fibocacci numbers 359-361, accurately to five digits.

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PART 3 - LINES THROUGH THE EARTH

Since the circumference of the Earth is also 3.1416 times the diameter of the Earth, the straight line distance from Easter Island to Machupicchu times pi² equals the circumference of the Earth.

The angle formed by the lines from Easter Island to Machupicchu, and to the Indus Valley, is 72°. The angle formed by the lines from Easter Island to Machupicchu, and to the Great Pyramid, is 54°.
 

The ratio of the base to the slant height of the Great Pyramid is exactly two times the ratio of the base to the height of the triangle formed by through the earth straight lines connecting the Great Pyramid, Angkor and Easter Island.

Because the distance between the Great Pyramid and Angkor is very nearly 20% of the circumference, they are very nearly 72° apart, along the circle.

Because the distance from the Great Pyramid to Easter Island is very nearly 40% of the circumference, and the distance from Angkor to Easter Island is very nearly 40% of the circumference, the Great Pyramid and Angkor are both very nearly 144° away from Easter Island, along the circle.

The number 72, and to a lesser extent the numbers 54, 108, and 144, have been associated with the designs of these sites, particularly at the Great Pyramid and Angkor. The ratio of the height and the perimeter of the Great Pyramid, to the size of the Earth, is a multiple of 72.

The number of temples built around Angkor is 72, and the number 54 is reflected in the numbers of statuary in the temples at Angkor. The use of these numbers is also prevalent in ancient writings and folklore surrounding these sites. The number 54 is itself a factor of 72, in that 72 plus ½ of 72, or 36, equals 108, which divided by two equals 54.

The number 72 is also associated with the astronomical phenomenon known as precession, because 72 years is the length of time it takes for the constellations to move one degree due to precession. This has been offered as an explanation for the use of these numbers, suggesting that the builders of these sites were aware of the precession of the equinoxes.

In the 2nd century B.C., the Greek mathematician, Archimedes, wrote an article entitled The Sand Reckoner, in which he cited earlier Greek mathematicians (like Archimedes, they had studied in Alexandria and Heliopolis) who had calculated that the Sun occupied 1/720 of the circle of the constellations. This may be an additional, or alternative, explanation for the prevalence of the number 72, and its multiples and factors, found in these sites.

In any event, the existence of these numbers in the geometric relationships between these sites is complementary to the use of these numbers in their internal designs.

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