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Calculating the pyramid angle: seked-measurements

Sources for the seked

There are no written sources on this question from the Old Kingdom, but it does feature in the largest surviving mathematical manual (British Museum EA 10057-10058 and fragments Brooklyn Museum 37.1784E). The manual dates about 1550 BC and is copied, according to its opening lines, from an earlier document, probably of the Middle Kingdom (from the language and vocabulary).

The seked

In the manual the word seked denotes the horizontal deplacement for each given unit of height. In Middle Kingdom measurements, the standard unit for linear measurements is the cubit (equivalent to about 52 cm): each cubit was divided into 7 palms of about 7.5 cm each, and each palm was further divided into 4 finger-breadths. For the seked, the cubit is taken as the reference height, and the seked is given in palms and fingers.

Examples

the slope of the Bent Pyramid at Dahshur | |

the slope of the red pyramid at Dahshur |

5 1/2 = Great Pyramid, Khufu | 5 1/4 = pyramid of Khafre

In her doctoral dissertation, Corinna Rossi defines the motive stimulating pyramid construction by the simple and convincing formula 'maximum height for minimum width'. In the seked, the Egyptians are formulating the same principle in numerical form, aiming for the minimum seked possible with the materials and technology at their disposal. When the scale of pyramids is reduced drastically, it allows a much sleeper slope, giving the characteristic profile of New Kingdom (about 1550-1069 BC) and later pyramids (note though the retention of the gentler slope in Memphite New Kingdom pyramidia, perhaps under the influence of the Old Kingdom monuments still dominating the landscape in that part of Egypt). The formula also explains the selection of the Middle Kingdom and later obelisk, where a single shaft of granite raises the pyramid-cap over the temple enclosure walls.

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