Lodin, Marie, Anisha og June

Myp Report information

Background to the Project

What we are trying to achieve for this unit, is to create a model of the Pyramids at Giza, using maths relevant to the topic, such as Trigonometry, Pythagoras, and basic maths skills.

Historical Information

Information about the Pyramids

The Pyramids of Giza were built when Egypt was one of the biggest and strongest civilization. The massive size represents the role of a pharaoh in the society, as well as the peak of Egyptian civilization. The pyramid was starting to be built at 2550 BC and its purpose was to be King Khufus grave. It was made in an astonishing 30 years! Beside the great pyramid lies three queens pyramids, the middle one made for his mother, Queen Hetepheres. Approximately 2.3 million blocks of stone (averaging about 2.5 tons each) had to be cut, transported and assembled to build Khufu's Great Pyramid. Greek historian Herodotus said the pyramids took 20 years to build (without 10 years for a way to the nile from the pyramid. Together it would be the stated 30 years), as well as a workforce of 100 000 men. New research shows however that only 20 000 built the pyramids, especially during the seasonal flooding. The workers were not slaves like a common belief, but workers that would in other times work on the now flooded plains.

Khufu Pyramid

The second biggest pyramid is the Khafre pyramid, and was built along with the sphinx built for Khafre (2558-2532), Khufu's son. The sphinx was the largest statue in the ancient world measuring 73.2m long and 20.1 meter tall. In the 18th dynasty (c. 1500 B.C.), the sphinx would later be worshiped itself, due to its resemblance to the local god Horus.

Khafre Pyramid

The southernmost pyramid at Giza was built for Khafre's son Menkaure (2532-2503 B.C.) and bears the same name. It is the smallest of the three major pyramids, with a height of ‘only’ 66.45m. It is a precursor of smaller pyramids that would be built in the old kingdom. Outside the Menkaure there are three smaller pyramids, two of them not being finished while the middle one being the grave of an unknown egyptian.

Menkaure and the queen pyramids.

Why the Pyramids are placed where they are

The placement of the Great Pyramid of Giza is believed to be the precise center of gravity of the earth, at 30 degrees north and 31 degrees east. This might seem coincidental, but with other factors included, it probably was very consciously placed. In order to create such an accurate placement (which is only displaced by a few feet), the Egyptians must have had an equally accurate ellipsoid (a 3D shape built up of at least one perfect circle and ellipses (eg. oval)) model of the earth than the most accurate scale to this day, the WGS84. This is also a mathematical factor which lead to its development and placement.

Mathematics, Social and other factors who lead to the pyramids Mathematical Issues

http://www.petrospec-technologies.com/Herkommer/pyramid/pyramid.htm

How did the egyptians manage do things so accurate with the things they had?

Alot of people dont know

Social Issues

How where the pyramids build?

Not slaves, but workers.

http://www.pbs.org/wgbh/nova/ancient/who-built-the-pyramids.html

Other

The pyramids were built to protect the body of the deceased pharaoh. These massive tombs were constructed to withstand the elements of time and were intended to last forever. Most Ancient Egyptians planned for their death and the pharaoh was no exception. His death was much more elaborate than the typical Ancient Egyptian and considered an important event; this process was tied to the rising and setting of the sun.

It was believed that while alive, the pharaoh represented Horus and upon his death he represented Osiris. During his state as Osiris, he would set the sun, while the new pharaoh, his son, in the image of Horus, would raise the sun. This process continued for hundreds of years and this is why it was important the pharaoh be protected eternally to avoid a cosmic disturbance.

The pharaoh also believed that his death was an extension to a journey towards eternal life. In order to become a “being” of the afterworld, it was important the pharaoh’s physical body be safeguarded and recognizable by his spirit, this in turn, lead to the process of mummification. The process itself consisted of being embalmed then wrapped in fine linen. Once the process of mummification was complete, the pharaoh was buried with his most prized possessions such as jewelry, funerary statues, and items that would aid him in his afterlife.

http://www.kingtutone.com/pyramids/beginning/

Myths and Mysteries - June

1: The great pyramid itself deviates only 16 minutes off of true north.

2: The casing stones, or surface stones, were about 100 inches (2.5 m), weighing nigh on 15 tons - And their six sides had near-perfect right angles.

3: The mortar used to hold the stones cannot be reproduced today, and still stands as strong as stone thousands of years later.

4: The cornerstone sockets are made in a ball-and-socket fashion, to resist earthquakes and heat expansion.

5: The Great Pyramid could have functioned as an observatory while it was being constructed.

6: During Summer Solstice, looking from the Sphinx, the sun sets exactly in the middle between the pyramids of Khefrem and Khufu

7: During Winter Solstice, looking from the entrance to the great pyramid, the sun rises on the base of the Sphinx’s head, one the left side, then traces around its head, to fall on the right side of the base of the head.

8: The greatest deviation from a perfect right angle at the base of the Great Pyramid (From here and onwards name TGP) is 2 centimeters.

9: TGP lies exactly at 31 degrees latitude and longitude.

10: The Sphinx has lines of water erosion - the last time there was water at its level, was approximately 10 000 BC

11: The sphinx’s body is built to have a perfect size for a lion’s head - the human head is far too small, however

12: The pyramids would have been perfectly aligned with Orion’s belt at 10 500 BC - around the same time the Sphinx would have been eroded.

13: TGP, built in 3000 BC, still stands perfect, while the pyramids built only 500 years after are crumbling.

14: The height of TGP is 1/1 000 000 000 of the distance from the sun.

15:If you take the length of the perimeter of TGP, then divide it by the height times 2, the number will correspond to pi up to the fifteenth digit.

16: The Egyptians kept very close records of everything they did; what and when they built something, when wars were being fought, and what kings they had - yet there are no records of the three great pyramids being built.

17: In the king’s chamber of one of the pyramids, there was found a fungus that had never before been seen on earth.

18: Geographically, TGP lies in the center of the world concerning land area.

19: Due to Earth’s “wobble”, Polaris would, at the time, not have been overhead to mark true north.

http://www.outerworlds.com/likeness/aliens/aliens.html

http://www.world-mysteries.com/mpl_2_4.htm

You and your team

- Method used to map the site: We printed out a 1:5000 scaled map of the area, and measured where the pyramids were located in relation to the Great Pyramid, by first measuring vertically and then horizontally, from the lower-left corner of the pyramids to the to the upper-right corner of the pyramids.

- How it was scaled down: the actual dimensions of the pyramids were found via the internet. They could then be divided by 1000, to scale it down, although as it was still in meters, we had to multiply it by 100. Therefore, all the calculations were divided by 10.

- Why did we choose the scale: We first chose to use the scale 1:500, to make them very big and stand out. However, we soon found out that this was too big, and would be very difficult to display. Therefore, the scale was changed to 1:1000, which would be more ideal in terms of location and the production of the pyramids.

- How sizes was calculated: After we had scaled down the base of the pyramids, and realized that the angle between the slant height and the base was approx. 53 degrees for all the pyramids, we were able to use trigonometry to find the slant height. COS was used to find the hypotenuse, in this case the slant height, by using half the base, from edge to the perpendicular height, multiplied by COS 53 degrees. Another method that could also be used was pythagoras, although we used trig. However, for the Great Pyramid we found an image that included the dimensions of it, which meant that the trig was not needed for this one.

- How this info was used to help produce accurate models: By using the slant height measurement, after having done the calculations, and the base measurements we were able to draw the net of the pyramids on thick black paper. This was done by first creating a square, which would indicate the base of the pyramid, followed by a line segment created perpendicularly to the middle of the base using the slant height measurement. It was then created into a triangle, by drawing two lines from the top of the line, to the square vertex.

- Actual Dimensions and calculations:

The Great Pyramid

Real

Base: 230m one side

height: 146 m

Slant height: 186 m

Scaled

Base: 23cm

Height: 14.6 cm

Slant height: 18.6 cm

Khefren

Real

Base: 214.5m

Height: 143.5m

Scaled

Base: 21.5 cm

Height: 14.35 cm

Slant height: 17.9 cm

Menkaure

Real

Base: 105m

Height: 65.5m

Scaled

Base: 10.5 cm

Height: 6.55 cm

Slant height: 8.7 cm

Queens G1a + c

We had first found dimensions of the queens pyramids located beside the Great Pyramid, but wr were to make the queens pyramids located south of Menkaure. Therefore we had to find these dimensions instead. We were not able to find them, so we used a scaled map, 1:5000, to find the base length, and from there used trig to find the slant height.

Scaled

G1a & G1c

Base: 3.5 cm

Height: 2.32 cm

Slant height: 2.91 cm

G1b

Base: 4 cm

Height: 2.65 cm

Slant height: 3.32 cm