Sides Of Khufu's Pyramid example essay topic

Today the Pyramids of Giza continue to be a mystery to researchers and admirers of ancient Egyptian architecture. One of these great pyramids, Cheops, is among the Seven Wonders of the World and is renowned for its geometric eccentricities. They were built around 4500 years ago and are located on the western bank of the Nile River in Al-Jizah, Egypt. These buildings took on original construction that was sought after an Egyptian sophistication of a concept called a "mastaba". A mastaba is a term that means "platform".

These platforms were built over royal tombs and eventually acquired numerous additions. Several stacked mastaba were incorporated together and tapered to form the pyramid shape. It is believed that this process took place over a 20 year time span. To begin, once the construction site was cleared and prepared, the blocks were transported into place. These blocks however, were made of stone and weigh around 2 tons each. Without the equipment and technology that we have today, one may ponder on how these massive building blocks were transported and set into place.

There are several theories describing methods that may have been used such as ramps and counterweights. Afterlife and respect for deceased pharaohs were important aspects in the Egyptian lifestyle. This was shown in the Pyramids of Giza by the use of various burial chambers, elaborate tombs, and human preservation for each pharaoh. Evidence of the importance of afterlife is provided by researchers' discoveries of artifacts including articles of clothing and various items found in the tombs. It was believed that death was not an important event in life. It was just a transition from one life to another.

Egyptians also had to perform many rituals before entering the pyramids. The great importance of these aspects is backed up by the grandeur of the pyramids and the amount of time it must have taken to build them. It challenges my mind to think of how these great buildings were constructed by the Egyptians so long ago. The use of precise measurements and careful architecture is outstanding for a civilization to practice in that period of time.

I felt that I needed to look into these ancient landmarks more deeply and research as much is I could about them. Writing this paper is a good way for me to better appreciate and understand the relationship between building structure and the use of geometry in the ancient Egyptian Pyramids of Giza. Geometry is probably one of the most important aspects in designing a building. When architects sit down at the drawing board, they must execute their knowledge and style of geometry in order to express their ideas. Architects formulate in their minds what they think a building should look like before it is drawn out.

The use of straight edges, measuring tools, and even computers help architects layout their designs very easily today. But around 2500 B.C., what did the architects and project planners use to design such a magnificent building such as a pyramid? The pyramids obviously take on a shape that most people are familiar with. Beginning with a square base, the pyramids have triangular sides which converge together forming a point at the very top of the building. These simple shapes may look average to a normal person but research has shown that they possess much more underlying meaning than one would think.

To give a more in depth, yet still basic description of the Pyramids of Giza, I will describe the Great Pyramid of Khufu, also known as Cheops. When this pyramid was built, it stood approximately 481 feet tall. As time went by, it lost about 30 feet off of the top due to natural weathering processes. At one time, this pyramid was deemed the tallest structure on earth until the 19th century. This pyramid was covered with stone casings which were used to smooth the outer surface. Some of these casings can be seen on the top of Khefre's Pyramid today.

The sides of the pyramid are each sloped at an angle of 51 degrees and 51 minutes. Each side also faces one of the four cardinal points of the compass: north, south, east, and west. If a horizontal cross-section of the pyramid was taken, the plan view would reveal a perfect square at any level. These sides were each about 751 feet long at the base. Amazingly accurate for that time period, the average error between side lengths is less than 0.1%. The Cheops used about 2 million blocks of stone for construction.

Each one of these stones weighs about two tons a piece. It is said that with the amount of stones in all three pyramids, a 10' x 1' wall could be built around France. The area of the pyramid could contain St. Peter's, the cathedrals of Florence and Milan, Westminster, and St. Paul's combined. The interior detail of the Cheops will be described to portray what it must have looked like inside.

The entrance is located on the north face. There are numerous corridors, galleries, and escape shafts leading to the King's burial chamber and other secret locations. In order to access the King's chamber, one must enter through the Great Gallery and an ascending corridor. This chamber is located in the heart or center of the pyramid. The King's sarcophagus or coffin is made of red granite as well as the interior walls of the chamber room.

The stones outlining the doorway have sharp edges that allow for minimal space between each adjoining stone. It is said that a playing card will not even fit between the stones because they are fit so well. This doorway was 10 feet long and 8 feet tall. The sarcophagus is aligned according to the compass directions, similar to the sides of the pyramid.

This coffin is only 1 centimeter smaller in dimensions than the doorway to the chamber. These building aspects indicate the intelligence of the ancient Egyptian civilization's builders. A more in depth description of the geometry will be discussed emphasizing this knowledge of how to relate buildings and geometry. The "royal master builder" was in charge of directing the planning that went into the construction prior to the actual building. This included physical orientation and the layout of the pyramid.

These responsibilities ultimately fell on the "vizier" who is the head of all royal works. The first step of the process was to draw up the plans for the pyramid. These steps were completed by specialists on papyrus paper. It seems that the complete plans were not drawn out completely before commencement of construction. As the project went on, additional plans were drawn on papyri or flat slabs of limestone.

Models of the pyramids were also made by the planners. These miniature limestone artifacts have been discovered in various Egyptian sites. After the planning stage was completed, each step of the pyramid construction was initiated with foundation rituals. Unlike many other religious structures around the world, the Pyramids of Giza required strict orientation to the cardinal points. In order to align the building with these points the Egyptians had to decipher North, South, East, and West.

This feat may have been carried out several different ways. The primary theory to how the Egyptians determined cardinal points has been through stellar arrangements and measurements. This involved building a small circular wall on the ground. These walls were probably built out of mud brick and had to be perfectly leveled off at the top. One man would stand inside the circle and site a circumpolar star as it rose. The spotting device used for this was a pole with a forked end.

The spotter would look through the fork and position the star where the fork starts. A second man standing on the perimeter of the circle would mark the spot on the wall where the star rose. Using a plumb line, a spot would also be marked on the bottom of the wall. When the star set, the process would be repeated. A measurement between the two spots would then determine true north from the center sighting pole. Another theory suggests that ancient Egyptians oriented their pyramids using the sun, wooden stakes, and ropes.

There is reference to this in ancient text referring to "the shadow" and the "stride of Ra". The sun rises and sets in equal but opposite angles to true north. Using a plumb line, a pole would have been set as vertical as possible. Then about three hours before noon, its shadow would be measured. This length was then used as the radius of a circle. The shadow shrinks back from the line and becomes longer as the sun rises throughout the day.

Eventually it reaches the circle again and forms an angle with the morning line. The bisection of this angle is then taken as true north. This method would have been less accurate than the stellar method but is a possible technique which may have been used. The use of perpendicular lines, circles, radii, and bisection of angles in the early planning stages of the pyramids relate the use of geometry to the buildings.

After the coordinates and orientation were completed, the next step was to lay out the ground plan. These methods varied from on pyramid to another, but the Great Pyramid of Khufu will be used once again as an example. To begin, a reference line was drawn along true north. Next, an exact square was created using precise right angles.

In the center of Khufu's pyramid there was a mass of natural rock protruding upward and was used as the core of the pyramid. This hindered the measuring process of the diagonals of the square making it impossible to check for accuracy. Scholars believe that the precise right angles could have been formed by any of three ways. The first method involves using and A-shaped set square.

This square would have been placed along the orientation line and a perpendicular line taken from the other leg of the square. The square would then be flipped and the other side would be measured. The exact 90 degree angle would then be calculated by determining the small amount of error of the angle between the two measurements. The discrepancy with this method is that there have been no discoveries of a large enough square to provide for an accurate right angle.

In the case of Khufu's Pyramid, the line would have been extended for some 754 feet. A second method that may have been used is that of Pythagorean's triangle. This method is widely used in geometric and mathematical problems today even. Basically, the triangle consists of three equal units on one side, four on the other, and 5 on the hypotenuse, also known as a "3-4-5" triangle. These sides form a perfect right angle.

At Khufu's pyramid, a series of holes along the orientation line were dug at 12 foot intervals. These were probably the positions that were used to measure each triangle. This method would have provided a much longer measurement for the perpendicular line that a set square would have. If the units used were any greater, the measurements would have been obstructed by the rock protrusion.

The third method that may have been used deals with intersecting arcs. In this method, two circles would have been made by rotating a string around two points on the orientation line. The intersection of the two circles would have then provided a right angle. This method is challenged by some, being that there could have been inaccuracies due to stretch in the string. However, the holes at Khufu's pyramid might have been used to draw these circles, so this method cannot be completely discarded. Elasticity could have even been eliminated by the use of a rigid object such as a rod, rather than a rope.

The use of these bisecting circles, setting squares, and Pythagorean's triangle all fall under common geometric techniques. These techniques reveal the Egyptians' desire to construct a perfect building by the use of geometry. The next phase of construction is building the platform. An orientation line was placed in a larger square by measuring off of the existing ground plan.

Post holes were dug at measured distances from the inner square in the bedrock and small posts were placed in them. And out reference line was then made by running string or rope through holes in the posts. Since the original reference lines would eventually be erases by building work, these outer lines were necessary. Various segments of the line could be removed to allow for materials to be brought in.

Measurements were then taken from the guide line as the platform materials were placed. This insured that the platform was in alignment with the initial floor plan. The platform of Khufu's pyramid was made out of what Tura h quality limestone slabs. Backing stones made of local limestone were used for leveling. Time shows that the platform's quality is important for withstanding the weakening forces of weathering. The builders of Khufu's pyramid proved to be more knowledgeable about this concept than those of other pyramids.

Some later pyramids were not built on solid bedrock or the platform was poorly constructed resulting in a building that did not survive for very long. Besides being perfectly square, the platform was required to be very level. In Khufu's pyramid, the platform was very much level, off by about one inch. Egyptians could have obtained this level either by the use of water or the use of the posts. It is thought by Dr. Charles Edwards that the ancient Egyptians might have constructed a mud enclosure around the platform and then filled it with water.

A grid of trenches would have then been cut a uniform depth below the water. If this cumbersome process was used, then the platform would " ve had to have been chiseled below the water surface. A more accepted theory describes channels being cut into the ground forming a grid within the platform, which was then filled with water. At the water's surface, the level would have been marked along the sides of the channels and then platform cut accordingly. Dr. Lerner, a renowned Egyptologists, refutes the theory of using water to level the platform because evaporation would have been a hindrance in accurate measuring. Khufu's pyramid is built on a sloping base, therefore the platform itself had to be leveled and not the bedrock underneath it.

Evidence shows that the builders were required to trim down the northwest corner of the platform and build up the opposite southeast corner in order to obtain proper leveling. Another leveling method may have used the posts that were placed to draw out the reference lines of the pyramid. These posts could have been cut to equal lengths or marked in order to provide a reference level. Considering these possible methods, the leveling techniques of the ancient Egyptian builders still remain unclear at this time. What is understood, however, is that the Egyptian builders of this time were very smart when considering how accurate and geometrically correct the pyramids were in Giza.

Getting into a more complicated mathematical approach, it has been brought up that the Great Pyramids follow a precise geometric concept involving circles and triangles. The use of these geometric shapes goes on to form important angles. The following will include diagrams to illustrate advanced geometric concepts: Diagram 16 shows how the length of and equal-sided pentagon can be calculated by using the Golden Ratio division along the circle's radius. This Golden Ratio takes on the value of 1.618 and is represented by the Greek letter "Phi".

To define this ratio would require extensive explanation. Basically, lengths of certain natural and geometric shapes can be related using this ratio of 1.618. Looking above, there is a right triangle COB. The side CO is set equal to 1 and is also the radius of the circle.

Side OB is calculated to equal. 618034 and angle OCT equals 31 degrees and 43 minutes. By subtraction, angle OBC equals 58 degrees and 17 minutes. This diagram is an important foundation for the following to come. In diagram 17, the same right triangle is circumscribed; however the hypotenuse of the triangle is now the radius instead of an adjacent side but is still equal to 1.

The angle opposite of side DE is 38 degrees and 10 minutes. Therefore, subtracting from 90 degrees, angle OED is equal to 51 degrees and 50 minutes. This is also the angle that the pyramid sides were sloped at in Giza. This is shown in the diagram below. Diagram 18 is a sketch of the Great Pyramid of Giza. It incorporates the 38 10' right triangle mentioned above.

If the pyramid cross-section in Diagram 18 is compared to the plan view of Diagram 17, it is obvious that side AB corresponds to. 78615, side AC corresponds to 1, and side BC corresponds to. 618034 of the radius. One may begin to wonder if the Egyptian builders were aware of these properties of the right triangle. There is too much of a relationship between the pyramids and geometry for it to be coincidental. Going more into depth with calculating important numbers, side BC is obviously equal to half of the length of one of the base sides.

So the perimeter of the base is equal to 8 x BC or. 618034 x 8 = 4.9443. The relative height of the pyramid is. 78615.

If this value is used as the circle's radius, then the circumference of the circle comes out to be 4.9443 as well. It can also be computed mathematically and shown visually that the Great Pyramids have the same perimeter when viewed in a horizontal plane as a square, as they do when viewed in a vertical plan as a circle which circumscribes a triangle. Another important fact is that the actual measured length one of the sides of Khufu's pyramid is 755.73 feet. Eight times this length or twice the perimeter is the exact length of one minute of latitude as measured off of the equator. The length of the side is equal to 1/8th of a minute of latitude and the height of the pyramid correlates to being 1/8th of 2Π . Were the similarities between geometry and the actual pyramid's measurements a coincidence or were they intentionally shown by the ancient Egyptian builders?

It is hard for one to think that this is mere coincidence once mathematical proofs illustrate a possible reasoning for the architectural design of the pyramids. It seems that the ancient Egyptian builders had some kind of knowledge of geometric design when drawing out the plans and constructing the Great Pyramids of Giza. I was able to discovery the various theories about how the pyramids were designed and constructed by researching various books and documentations on them. It is clear to see that there isn't a definite answer to how the Egyptians leveled, designed, and constructed the pyramids. All we can do is speculate and form theories and ideas. One can become more partial to one theory than to another.

After all, this is why the Great Pyramid is one of the Seven Wonders of the World. Whether it was intentional or not, the Egyptians created a worldly masterpiece of a monument. The intervention of geometry into the building of the pyramids provides for the early stages of architectural aspects. These days every building is geometrically sound. It is much easier for architects to design their buildings geometrically these days than it was for the builders in 2500 B.C. The Egyptians, overall, were important founding fathers of incorporating geometric complexities into the design of their buildings. Designing a building is a way of art; the architect expresses his or her ideas through their building.

By interpreting the historical aspects of the pyramids and comparing them to the building aspects of today, a similarity can be found. It may be taken to a higher level, but the weaving of geometry into buildings is evident both today and over 3000 years ago.