Seked
Encyclopedia
The seked was an ancient Egyptian unit for the measurement of the slope of an inclined surface. The system was based on the Egyptian's linear measure known as the royal cubit
. The royal cubit was subdivided into seven palms and each palm was further divided into four digits. The inclination of measured slopes was therefore expressed as the number of palms and digits moved horizontally for each royal cubit rise.
Information on the use of the seked in the design of pyramids has been obtained from two mathematical papyri; the Rhind Mathematical Papyrus
in the British Museum and the Moscow Mathematical Papyrus
in the Museum of Fine Arts. Although there is no direct evidence of its application from the archaeology of the Old Kingdom, there are a number of examples from the two mathematical papyrii, which date to the Middle Kingdom that show the use of this system for defining the slopes of the sides of pyramids, based on their height and base dimensions. The most widely quoted example is perhaps problem 56 from the Rhind Mathematical Papyrus
.
The most famous of all the pyramids of Egypt is the Great Pyramid
of Giza built around 2,550 B.C.. Based on the surveys of this structure that have been carried out by Flinders Petrie and others, the slopes of the faces of this monument were a seked of 5 1/2, or 5 palms and 2 digits [See figure above], which equates to a slope of 51.84° from the horizontal, using the modern 360 degree system. This slope would probably have been accurately applied during construction by way of 'A frame' shaped wooden tools with plumb bobs, marked to the correct incline, so that slopes could be measured out and checked efficiently.
Furthermore, according to Flinders Petrie's survey data in "The Pyramids and Temples of Gizeh" the mean slope of the Great Pyramid's entrance passage is 26° 31' 23" ± 5". This is less than 1/20th of one degree in deviation from an ideal slope of 1 in 2, which is 26° 33' 54".
This equates to a seked of 14, and is generally considered to have been the intentional designed slope applied by the Old Kingdom builders for internal passages.
is thought to have had sides that sloped by 51.842° or 51° 50' 35", which is a seked of 5 palms.
The Great Pyramid scholar Professor I.E.S Edwards considered this to have been the 'normal' or most typical slope choice for pyramids. Flinders Petrie also noted the similarity of the slope of this pyramid to that of the Great Pyramid at Giza, and both Egyptologists considered it to have been a deliberate choice, based on a desire to ensure that the circuit of the base of the pyramids precisely equalled the length of a circle that would be swept out if the pyramid's height were used as a radius. Petrie wrote "...these relations of areas and of circular ratio are so systematic that we should grant that they were in the builder's design".
The seked of a pyramid is described by Richard Gillings in his book 'Mathematics in the Time of the Pharaohs' as follows:
Cubit
The cubit is a traditional unit of length, based on the length of the forearm. Cubits of various lengths were employed in many parts of the world in Antiquity, in the Middle Ages and into Early Modern Times....
. The royal cubit was subdivided into seven palms and each palm was further divided into four digits. The inclination of measured slopes was therefore expressed as the number of palms and digits moved horizontally for each royal cubit rise.
Information on the use of the seked in the design of pyramids has been obtained from two mathematical papyri; the Rhind Mathematical Papyrus
Rhind Mathematical Papyrus
The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...
in the British Museum and the Moscow Mathematical Papyrus
Moscow Mathematical Papyrus
The Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes...
in the Museum of Fine Arts. Although there is no direct evidence of its application from the archaeology of the Old Kingdom, there are a number of examples from the two mathematical papyrii, which date to the Middle Kingdom that show the use of this system for defining the slopes of the sides of pyramids, based on their height and base dimensions. The most widely quoted example is perhaps problem 56 from the Rhind Mathematical Papyrus
Rhind Mathematical Papyrus
The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...
.
The most famous of all the pyramids of Egypt is the Great Pyramid
Great Pyramid of Giza
The Great Pyramid of Giza is the oldest and largest of the three pyramids in the Giza Necropolis bordering what is now El Giza, Egypt. It is the oldest of the Seven Wonders of the Ancient World, and the only one to remain largely intact...
of Giza built around 2,550 B.C.. Based on the surveys of this structure that have been carried out by Flinders Petrie and others, the slopes of the faces of this monument were a seked of 5 1/2, or 5 palms and 2 digits [See figure above], which equates to a slope of 51.84° from the horizontal, using the modern 360 degree system. This slope would probably have been accurately applied during construction by way of 'A frame' shaped wooden tools with plumb bobs, marked to the correct incline, so that slopes could be measured out and checked efficiently.
Furthermore, according to Flinders Petrie's survey data in "The Pyramids and Temples of Gizeh" the mean slope of the Great Pyramid's entrance passage is 26° 31' 23" ± 5". This is less than 1/20th of one degree in deviation from an ideal slope of 1 in 2, which is 26° 33' 54".
This equates to a seked of 14, and is generally considered to have been the intentional designed slope applied by the Old Kingdom builders for internal passages.
Pyramid slopes
Many of the smaller pyramids in Egypt have varying slopes, however, like the Great Pyramid of Giza, the pyramid at MeidumMeidum
Located about 100 km south of modern Cairo, Meidum or Maidum is the location of a large pyramid, and several large mud-brick mastabas.-Pyramid:...
is thought to have had sides that sloped by 51.842° or 51° 50' 35", which is a seked of 5 palms.
The Great Pyramid scholar Professor I.E.S Edwards considered this to have been the 'normal' or most typical slope choice for pyramids. Flinders Petrie also noted the similarity of the slope of this pyramid to that of the Great Pyramid at Giza, and both Egyptologists considered it to have been a deliberate choice, based on a desire to ensure that the circuit of the base of the pyramids precisely equalled the length of a circle that would be swept out if the pyramid's height were used as a radius. Petrie wrote "...these relations of areas and of circular ratio are so systematic that we should grant that they were in the builder's design".
The seked of a pyramid is described by Richard Gillings in his book 'Mathematics in the Time of the Pharaohs' as follows:
"The seked of a right pyramid is the inclination of any one of the four triangular faces to the horizontal plane of its base, and is measured as so many horizontal units per one vertical unit rise. It is thus a measure equivalent to our modern cotangent of the angle of slope. In general, the seked of a pyramid is a kind of fraction, given as so many palms horizontally for each cubit of vertically, where 7 palm equal one cubit. The Egyptian word 'seked' is thus related to our modern word 'gradient'."
Further reading
- Verner, Miroslav, "The Pyramids – Their Archaeology and History", Atlantic Books, 2001, ISBN 1-84354-171-8
- Arnold, Dieter. "Building In Egypt: Pharaonic Stone Masory", 1991. Oxford: Oxford University Press
- Jackson, K & J. Stamp. "Pyramid : Beyond Imagination. Inside the Great Pyramid of Giza"BBC Worldwide Ltd, 2002, ISBN 978-0563488033
- Sekeds and the Geometry of the Egyptian Pyramids - Information about the use of sekeds in the construction of Egyptian pyramids
- Sekeds and the Geometry of the Great Pyramid - Information about the use of seked in the construction of the Great Pyramid of Giza