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Applying Solar Geometry to Understand the Foundation Rituals of ‘Old Kingdom’ Egyptian Pyramids a
Richard Kittler & Stanislav Darula
a
a
Institute of Construction and Architecture, Slovak Academy of Sciences , 9 Dubravska Road, SK-84503, Bratislava, Slovak Republic Published online: 09 Jun 2011.
To cite this article: Richard Kittler & Stanislav Darula (2008) Applying Solar Geometry to Understand the Foundation Rituals of ‘Old Kingdom’ Egyptian Pyramids, Architectural Science Review, 51:4, 407-412 To link to this article: http://dx.doi.org/10.3763/asre.2008.5145
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doi:10.3763/asre.2008.5145
Architectural Science Review Volume 51.4, pp 407-412
Applying Solar Geometry to Understand the Foundation Rituals of ‘Old Kingdom’ Egyptian Pyramids Richard Kittler* and Stanislav Darula Institute of Construction and Architecture, Slovak Academy of Sciences, 9 Dubravska Road, SK-84503 Bratislava, Slovak Republic *Corresponding Author: Tel: +421 2 59309267; Fax: +421 2 54773548; Email:
[email protected] Received 3 July 2008; accepted 1 October 2008
Abstract: The republication of the interesting book by Rossi (2003, 2007) has provoked questions about the orientation of Egyptian pyramids from the architectural science point of view. The main concern of this paper is to explain several hypothetical possibilities for orientation of the first step Egyptian pyramid. The oldest monumental step pyramid complex, at Saqqara, was built by Imhotep in the 3rd millennium BC, 3rd Dynasty of the ‘Old Kingdom’. Pre-construction involved a foundation ritual, during which the pyramid’s south orientation and plan were determined by ‘stretching the cords’, a site design procedure partially illustrated on stone engravings. Though helpful, these illustrations incompletely depict the method of pyramid plan design and orientation. Due to rhombic signs on mastaba foundation stones, some Egyptologists claim that geometry was the primary method used for pyramid plan design and orientation. However, due to the solar equinox festive day and the pharaoh’s presence at the festival, it is possible that the solar equinox was the day chosen for the ceremony. During the equinox, the gnomon shadow would directly indicate the E-W cardinal points. It is plausible, therefore, that solar geometry, at the equinox, was chosen to establish the primary orientation of the pyramid complex. Sundials based on the equinox sun-shadow were commonly used during the 3rd century BC. This paper discusses methods that could have been available to determine the orientation of pyramids.
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Keywords: Cardinal points, Gnomon, Orientation, Pyramids, Solar geometry
Introduction
Studies of the relationship of architecture and mathematics to the design of Old Kingdom pyramids commonly refer to a pre-construction planning and design ceremony or foundation ritual. The ceremony of Old Kingdom pyramids involved a procedure called “the Stretching of the Cords”. However, the method used to set the orientation of the first stepped pyramid remains speculative. Nevertheless, there are several clues that suggest the use of methods based on solar geometry. Vitruvius (13 BC, 1970), in his first of ten textbooks for architects, documents a philosophy on building orientation and thus reinforces the case for the ancient application of solar geometry to understanding the planning and orientation of the great pyramids.
Historical Background and Developments in Determining Time and Orientation
Since the dawn of civilisation, daytime activities have needed nighttime rest. Annual periods of gathering fruit or raising and harvesting crops have depended on diurnal and annual
sunlight changes. There is no record of who first realized that there are two equinox days within a year, when the lengths of day and night are the same, exactly twelve hours each. In the equinox days, the sunrise precisely indicates the East cardinal point, the sun zenith is South, while the sunset determines the West cardinal point. These directions are especially valuable for orientation in desert regions where there may be no other orientation points or objects in view. In two-dimensional geometry, it is quick and simple to draw a circle in sand, using a tent peg at its centre and a ‘cord’ or rope with another peg to move it around with respect and proportional to its centre. However, it is much more difficult even to imagine even the two-dimensional result of tracking the shadow thrown by the three-dimensional sun path intercepted by a vertical pole inserted at the centre of a circle. Such a ‘solar time and geometry’ based orientation and clock would mark sunrise and sunset and, when exactly 90º from the noon sun zenith, define equal time intervals on its primitive clock face. The sun zenith occurs at noon and, at that instant, the basic
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N-S cardinal points in any northern location, and the shortest sun shadow indicates north. Mesopotamian and Egyptian priests conducted relatively simple geometric experiments with cords and pegs, or gnomon sticks, with defined right triangles to acquire knowledge and build practical tools. The first wooden rectangular forms to manufacture clay bricks, and wooden triangles with a 90º angle to indicate a cubic or prismatic cut in stone blocks for luxury buildings or pyramids, were such practical tools. Some primitive sundials were constructed using first gnomon sunshadow observations. Sumerian priests identified equinox days exactly and symbolically used the double-faced god Ismu, with one profile facing day and the other facing night. This symbolic “dual” god is documented on a cylindrical seal of the Mesopotamian scripter Addu from the 23rd century BC (Zamarovsky, 1984). Such tasks were certainly solved in the ancient Sumer town kingdoms where a remarkable sexagonal time and space measurement system was introduced in the 3rd millennium BC. Even a standardization was established in the united Sumer in its capital Ur by King Dungi I around 2650 BC (Paturi, 1993). There was extraordinary coordination of the investigation of the yearly sun-path and moon image changes – a solar year of 360 days, equal to 12 month (regular moon changes) of 30 days each, with a day having two times each of 12 hours for day and night, with a circular orientation scheme of four times 90º, equal to 360º encircling the horizon, and a 180º sky vault in the vertical section. Indeed, the overall advantage of the number 360, which is divisible into many smaller units represented by integers, has been applied throughout the history of civilisation. In the same period, in Egypt, a simple calendar with a solar year having 360 plus 5 festive days was used probably under the influence of yearly Nile flood cycles. Accidently both capitals, Sumerian Ur and Egyptian Mennofer/Memphis, were located close to the 30º and 31º geographical latitude experiencing approximately the same annual sun-path changes. It was also known that by using arbitrary units for two sides of a 90º triangle, a definition of both the angles and proportions of the sides, in ratios, could be obtained without having knowledge of trigonometric relationships. The purpose, its symbolism and the technology underlying the stretching of the cords in the ceremony of the monumental Egyptian pyramids remains speculative. However, it is more than 4500 years since Imhotep, the court architect and builder for the Pharaoh Djoser of the 3rd Dynasty, 3rd Millennium BC, designed and built the first step pyramid in Saqqara, around 2650 BC, and appended to it a huge south ceremonial court. If Imhotep is characterised as the first stone builder and architect (Zamarovsky, 1986), then he certainly understood ancient geometry, stonecutter experience and tools as well as the simplest sundial principles. Of course, Imhotep was also familiar with the older ‘stretching of the cord’ foundation ceremony inherited from the last king of the 2nd Dynasty, Chasechemvey/Khasekhemuy. A description of the ‘stretching of the cord’ procedure was The gnomon is the part of a sundial that casts the shadow. Gnomon (γνωμων) is an ancient Greek word meaning “indicator”, “one who ʹ discerns” or “that which reveals.” [Editor]
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attempted, but not satisfactorily explained, by Engelbach (1934), from engravings of the process found on at least four stone engravings (Rossi, 2003, 2007; Shaltout & Belmonte, 2005). It is also quite possible that the equinox day dedicated to the Egyptian goddess Maat was the chosen ritual day for the foundation ceremony and therefore she was the patron of the basic ritual orientation knowledge used during the foundation ceremony. In some pictures describing the ‘stretching of the cords’, some Egyptologists perceive (interpret) the Pharaoh and the priest representing the goddess Seshet holding dual sticks tied together with a cord. The Egyptologists do not perceive any person with a single stick placing this gnomon in vertical position. In the oldest scene (e.g., Figure 74 in Rossi, 2007) on the left side, a person is kneeling and seems to hold a water level to measure the horizontal plane for the gnomon shadow or the foundation level. Many Egyptologists noticed a perfect orientation of Egyptian Old Kingdom pyramids with their sides facing the cardinal points. The Great Pyramid of Cheops built in 2550 BC at Giza was aligned to the N-S direction with a precision of less than 3 arc minutes; some authorities suggest that astronomical orientation was applied (e.g., Spence, 2000). Such a ‘magic’ precision might be either accidental or recently adjusted, but it certainly is not a reason to assume or deduce an astronomical orientation of all pyramids. The hieroglyphic structure of the script and the drawing of figures in side-views, and projections in plan or elevation, indicate that basic descriptive geometry was understood and commonly applied in designs and construction drawings especially in the case of huge pyramids or temples. However, in Rossi’s (2007) descriptions of the foundation ceremony there is no explanation how pyramid orientation to cardinal points was achieved.
The Egyptian Foundation Respecting the Cardinal Points
Ceremony
Magdolen (2000b) published an interesting study of the core problem concerning the pyramid orientation to cardinal points. His assumptions of the gnomonic were based on three possibilities including: 1) The possible daily available noon method when the gnomon shadow has the shortest length indicating the North direction seems to be only approximate and not very trustworthy. This method is using a common experience that the shortest sun shadow takes place at noontime shining exactly from the south cardinal point. However, during the sun culmination, the differences in solar altitudes are only minute, so the sun shadow is a poor indicator for orientation purposes. 2) The sunrise and sunset or East-West method with extreme length shadows on an equinox day can be applied only in an absolutely flat land with a free, wide horizon where the gnomon tip shadow should point to E and W in infinity, which makes it very problematic and unpractical. 3) The so-called “Indian method” is utilising the symmetry of the sun-path around noon, indicated on an arbitrary
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Applying Solar Geometry to Egyptian Pyramids
Applying Solar Geometry to Egyptian Pyramids
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circle by morning and afternoon gnomon tip shadow. As a result, their azimuth angles have the same extent from the N-S axis, as seen in Figure 1. This orientation method is based on the assumption that at the same time difference from noon time both solar altitudes are the same and also azimuth displacement are same, that is, this method relied on the symmetry of morning and afternoon sun shadows with the N-S symmetry axis. This third method was also described in detail by Vitruvius in year 13 BC in his Book I, Chapter VI for the definition of wind stream orientation in urban design but was probably used for any orientation purposes for a long time as documented by the ancient Greek Tower of the Winds in Athens. Vitruvius stressed the importance of the straight gnomon recommended to be made of bronze and the levelling of a flat marble plate to hold the gnomon in exactly a vertical position. This method is also called rhombic. Foundation stones with graffito drawings Figure 1: Alignment to cardinal points in an arbitrary building found by Czech Egyptologists in Abusir document the Old site, method 3. Kingdom grave orientations quite persuasively (Verner, Figure 1. Alignment to cardinal points in an arbitrary building site, met 1997). However, there are at least two interesting further possibilities for the more sophisticated pyramid orientation connected with the ritual foundation ceremony on the equinox day based on current time measurements and sundial design used in ancient Sumer and Egypt. These are: 4) The equinox shadow-line method, that is using the fact that during the whole equinox day the gnomon tip shadow follows the straight line in the E-W direction while its noon shortest shadow is 3 units long thrown by a 5 unit tall gnomon in northern Egypt where pyramids were built, as shown in Figure 2. The ratio of the shadow s and the gnomon height g in fact defines in the right triangle both the geographical latitude of Ur as well as the solar altitude on the equinox day at noontime, i.e., at 12 o’clock. 5) The sundial method is similar to method 4, but has the additional advantage that time intervals measured by common sundials, as seen in Figure 3. During the equinox day, the sun path can be divided in equal hourly time steps with an hour angle of 15°. This could help to organise the foundation ceremony, which could suit to arrange the Pharaoh’s presence as well as the preliminary procedures of the “Stretching of Cord” prepared by the priests. Probably the Pharaoh did not want to get to the pyramid building site too early in the morning and the alignment of the pyramid sides to cardinal points had to end soon after noon as further steps of the foundation ritual, e.g., the plan measurements, levelling and setting of the corner stone blocks etc., had to be done before sunset. The equinox day as a ritual date had also the advantage of avoiding the summer heat especially at noon time in Egypt. To study the local sun-shadows using a vertical gnomon and define the equinox day, which is on the 21st March and 22nd September world-wide, was an interesting task in all ancient cultural centres. In ancient Egypt, the daughter of the main sun god Re, goddess Maat, represented justice and dual equality, Figure 2: The orientation of Egyptian Old Kingdom pyramids i.e., dividing equally any whole into two halves and therefore using an equinox line shadow. originally the equinox day was dedicated to her too.
Figure 2. The orientation of Egyptian Old Kingdom pyramids after equinox lin
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mentioned by Vitruvius in 13 BC.
Thus, it can be also
assumed the equinox day was taken as a ritual main pyramid corners, where the gnomon stick that was placed in an exactly vertical position and day for the main pyramid corners, where the gnomon stick was placed in an exactly vertical position and foundation ceremony performed by the Pharaoh. He decided
a circle with the three-unit was drawn. Because sun-shadow endmain pointspyramid were main pyramid corners, where gnomon stickthree-unit was radius placed in an exactly vertical position andof the the building sitethe and athe circle with the radius was drawn. Because theone sun-shadow end points were corners, where
the gnomon stick was placed in an exactly vertical position and a circle with the three-unit radius was drawn. Because with thisalong W-E adirection to it.towards Either the point orwas the centre of two section points had to the sun-shadow end points were gradually moving from the gradually moving from the West straight line the touch circle a rope with this W-E direction to it. Either the touch point oraligned the centre of two section points had to West along a straight line towards the circle a rope was aligned thethe N-Stouch direction should beof exactlysection 90º to points the E-Western stretched cord. Now, it with this W-E direction toindicate it. indicate Either pointthat or the to the N-S direction thatcentre should exactly to thehad E-Western stretched cord. Now, it or the withbetwo this W-E90ºdirection to it. Either the touch point centre of two was section points hadthe to right indicate assumed that either the “sacred triangle rule” applied to itcheck angle,the i.e.N-S direction indicate the N-S directioncan thatbe should be exactly to the stretched cord. can be assumed that90º either theE-Western “sacred triangle was Now, applied the right angle, i.e. that should rule” be exactly 90º totocheck the E-Western stretched cord. forming the 12triangle unit cord into a triangle with three sides ofi.e. 3, 4 and 5 length units, or can be assumed that either theforming “sacred rule” to check thebe right angle, Now, itexactly can assumed that “sacred rule” the 12 unit cordwas intoapplied a triangle with exactly three sides of 3,either 4 and 5the length units,triangle or was applied to check the right angle, i.e. forming the 12 unit two other longer cords with equal length had to check their intersection which has to forming the 12 unit cord simply into simply a triangle exactly ofequal 3, 4 and 5 length twowith other longerthree cordssides with length had tounits, checkor their intersection which cord into a triangle with exactly three sides of 3,has4 to and 5 length units, orofsimply two other longer cords with be on with the exact noon S-N line. The existence the 3-4-5 triangle 2000a) andequal length had simply two other longer cords equal length had to line. check their intersection which to(Magdolen, be on the exact noon S-N The existence of the 3-4-5has triangle (Magdolen, 2000a) and to check their intersection which has to be on the exact noon Sof existence the ellipseofbythea 12-unit cord is(Magdolen, indicated also by Rossi (2003) but not in relation be on the exact noon S-Ndrawing line.drawing The 3-4-5 triangle 2000a) and Ncord line. existence ofRossi the 3-4-5 2000a) of the ellipse by a 12-unit is The indicated also by (2003) triangle but not in(Magdolen, relation and drawing of the ellipse by a 12-unit cord is indicated also with the gnomon or any orientation task. However, it could be utilised for drawing the sundrawing of the ellipse by a 12-unit cord is indicated by Rossitask. (2003) but not in relation with the gnomon or anyalso orientation could for drawing the sunby RossiHowever, (2003) itbut notbeinutilised relation with the gnomon or any path ellipsetask. on a However, fictitious sky hemisphere on the upper side ofsunthe gnomon (asbe shown in for drawing the with the gnomon or any orientation it could beorientation utilised for the task. However, it gnomon could path ellipse on a fictitious sky hemisphere on drawing the upper side of the (asutilised shown in sun-path ellipse on a fictitious sky hemisphere on the upper 3). If the equinox day isside given the by the s / g =(as 3/5 ratio then the short axis of the ellipse path ellipse on a fictitiousFigure skyFigure hemisphere on the upper in then s / g shown = 3/5 (as 3). If the equinox day isofgiven by the the gnomon ratio axis3). of the ellipse side gnomon of showntheinshort Figure If the equinox day is given by axis the s/g =3/5 ratio then 2 the 2 short axis of the ellipse s / g = Figure 3). If the equinox bday is given by the 3/5 ratio then the short of the ellipse = 3, while the sky hemisphere radius r is equal to the ellipse axis a = 3 + 52 = 2 34 = 3 + 5r is= equal is equal to sky the ellipse axis a =radius 34 = to the ellipse b = 3, while the sky hemisphere bradius = 3,rwhile the hemisphere 2 2 axis = 3 + 5 = 34 == 25.831 and the focus from the ellipse radius r is equal theellipse ellipsecentre axis aahas = b = 3, while the sky hemisphere 5.831 and the focus fromtothe a distance e = a ! b22 = 2 34 ! 9 = 5. Thus 5.831 and the focus from the ellipse centre a distance ee == a ! b = 34 ! 9 = 5. Thus centre hashasa distance Thus applying the 3/5 ratio in a plan circle a cord tied to pegs placed 2 same 2 a ! b = 34 ! 9 = 5 . 5.831 and the focus fromapplying the ellipse distance e = circle a cord tied to pegs placed Thus in the two focuses by thecentre samehas 3/5 aratio in a plan applying the same 3/5 ratio in a plan circle a cord tied to placeda in theround two focuses in the two focuses bypegs turning peg couldbybe drawn the sun-path ellipse, as is r1 + r2 = 2 a . turning a pegcircle round couldtied be drawn the sun-path ellipse, as it it is applying the same 3/5 ratio inturning a plan a cord placed in the twoellipse, focuses a peg round couldtobepegs drawn the sun-path asby it is r + r2 = 2 a . Moreover, once the dual 1equinox days were identified within Moreover, once the dual as equinox within the year, in thisany method is i.e. not only turning a peg round could be drawn the sun-path ellipse, it is r1 +days r2 =were 2this a . identified the year, method is applicable Moreover, once the dual equinox days were identified within the year, this locale, method is in the North Egypt centres around the 30° latitude but also in applicable in anydays locale, i.e.identified not onlywithin in the North Egypt centres the 30° latitude but Moreover, once the dual equinox this methodaround is around applicable in anywere locale, i.e. not only in the the year, North Egypt the (ca 30°24°), latitude the Upper Egypt fromcentres Asuan/Phillae in but Karnak/Luxor centres (ca 26°) orKarnak/Luxor Abydos/This (ca 27°). alsonot in the Upper Egypt from Asuan/Phillae (ca 24° ), in centres (ca 26°No ) orspecial ratios s/g applicable in any locale, i.e. only in the North Egypt centres around the 30° latitude but also in the Upper Egypt from Asuan/Phillae (ca 24° ), in Karnak/Luxor centres (ca 26° ) or were thus needed to orient any graves or buildings to E-W and Abydos/This (ca 27°). No special ratios s / g were thus needed to orient any graves or also in the Upper Egypt fromAbydos/This Asuan/Phillae (ca27°). 24° ),No inspecial Karnak/Luxor 26° ) or N-S directions. s centres / g were(ca (ca ratios thus needed to orient any graves or In fact, Vitruvius in his Book IX, Chapter 7 mentioned the buildings to E-Ws /and N-S thus directions. g were Abydos/This (ca 27°). No special ratios needed to orient any graves or to E-W and hours. N-S directions. Figure 3: Sun path diagram for equinoxbuildings days using sundial ratio 3/5 for the geographical latitude of Alexandria for his In fact, Vitruvius in his Book analemma IX, Chapter 7rule mentioned the ratio the 3/5 for the determining sun-path during the whole buildings to E-W and N-S directions. In fact, Vitruvius in his Book IX, Chapter 7 mentioned the ratio 3/5 for the year, thus indicating such an ancient knowledge (Kittler & latitude of Alexandria for analemma rule determining the sun-path during the In fact, Vitruvius geographical in his Book IX, Chapter 7of mentioned thehis ratio 3/5 for the geographical latitudesundial Alexandria for his2004). analemma rule to determining thethat sun-path the Darula, It has be noticed whileduring the sun-path on 3. Sun path diagram for equinox days using hours. In the equinox day at latitude noon, i.e., exactly atfor thehis 12analemma hour in true the fictitious sky hemisphere geographical of Alexandria rule determining the sun-path during the follows the sun movement from E solar time, the sun is directly shining from the South direction through S to W cardinal points during equinox days, the sunand its shadow points to the North cardinal point, while the shadows have a reverse trend with the morning end 9 pointing to 9 shadow is the shortest within that day. The shadow length s at the W direction and indicating N at noon. 9 noon relative to the gnomon height g in Ur or Alexandria, i.e., Questionable is the Astronomic Orientation on approximately the geographical latitude = 31° was s/g = of Pyramids or other Reasons for their S-N 3/5 corresponding to = 30°58´ and thus the solar altitude γs = 90° - = 59°02´, valid precisely at exact noon on an equinox Orientation day. Due to current, more sophisticated knowledge of astronomy Figure 3 also illustrates a sundial drawn to be turned into the and the magnetic field of the Earth, some speculations how plane of the solar meridian using the s/g = 3/5 ratio in section and why the pyramid orientation to cardinal point has been and the sun-shadow traces in plan. The end points of the sun- achieved. Because elemental solar geometry may have been shadow are sliding during an equinox day along a straight line restricted to priests and not recorded, some current researchers parallel with the E-W direction. and specialists have deduced several plausible or implausible Sun-shadow measurements and sundials with vertical sticks, ways and means that could be applied. Many archeoastronomers later called gnomon, were also known and probably the equinox wonder whether the pyramid orientation was not set to some day proportion of a 3 unit shadow to a 5 unit gnomon height star on the North horizon or even try to date the pyramids after was already applied after the Sumerian/Chaldean tradition the star placement occurring during the Old Kingdom (see
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gradually moving from the West along a straight towards the circle a rope was aligned a circle with the three-unit radius was drawn. sun-shadow endline points gradually movingBecause from thethe West along a straight line were towards the circle a rope was aligned
Figure
Applying Solar Geometry to Egyptian Pyramids
Applying Solar Geometry to Egyptian Pyramids
coming in peace”). Is it only a fiction deduced from fantastic interpretations of recent research hypotheses, or, could it be as true as the same assumption of the astronomic pyramid orientation? Nevertheless, exposing the main pyramid side and court in Saqqara to the maximal influence of the Sun as the mightiest god Re, Horus or Aton respected the positive Egyptian intentions and religious beliefs (Zamarovsky, 1986).
Conclusions
In the early centuries of civilization, architectural and building design employed simple rules, knowledge of descriptive geometry and measurable proportions respecting available tools and traditional knowledge gained from practical experience. The development of monumental buildings, especially pyramids and temples, represented huge investments. In the Old Kingdom, such monumental development required the Pharaoh to approve, and a Foundation Ritual or Ceremony involving “stretching of the cords”. Such a ceremony was standard from the 2nd Dynasty. Imhotep, as Pharaoh Djoser´s architect and builder, was certainly obliged to employ such a Ceremony for the oldest step pyramid at Saqqara. Sundials measuring time, sun-shadow and sun-path changes with everyday noon culminations were the best orientation indications to the S-N direction and the gnomon shadow could be locally utilised on any building site. Ancient stone pictures show a stretched cord alignments with the noon sun-shadow trace, ground level checking by water flow and verticality by gnomon all known to be needed for the ceremony. The secret part of the ‘stretching of the cord’ was probably only the knowledge that the properties of a right angle triangle and the shadow ellipse by the peg rotation rule. This is probably the secret of the Egyptian Foundation Ceremony known to only privileged priests, authorities and the Pharaoh. Such a simple task with simple tools enabled construction of the desired plan with its S-N orientation. All accomplished by wooden pegs or sticks, without any sophisticated instruments,
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Spence, 2000). However, no architect or builder can expect the foundation ceremony to take place during nighttime or imagine the Pharaoh to perform it in darkness by stretching some cords. This fact was recently stressed also by an Egyptian author, who stated that archeoastronomers “did not take into consideration the simple fact that today or in the ancient past, architects and site-engineers, never do construction-survey activities at night, i.e. in darkness, which is contrary to the construction norms” (Aboulfotouh, 2007, p. 24). It is no wonder that even some experts who were convinced about the astronomical orientation of pyramids have recently started to question their beliefs (see Spence, 2003). There is continued speculation about how the orientation of pyramids to the cardinal points was achieved. Architectural science questions are raised through modern understanding of the influence of the Earth’s magnetic field on the true North direction. In addition, because knowledge of elementary solar geometry may have been restricted to priests, and not recorded, some current researchers and specialists sites (eg, Valentovich, 1997) have ‘reasoned’ several plausible or implausible hypotheses on the ancient technology employed. There are high levels of magnetic influence at 30º N latitude and 30º E longitude, which exactly coincides with the pyramid sites (Valentovich, 1997). Was the presence of favourable magnetic fields a priority reason of pyramid orientation and/or location? Such magnetic field conditions in the Pharaoh’s burial chamber were positive but to the contrary, a 45º rotation of the pyramid plan to SE-NW or SW-NE orientations could have a harmful effect, as illustrated in Figure 4. Did Imhotep know and try to utilize such magnetic field knowledge, as imagined in the Columbian video series called “The Eye of Horus” directed by Markun (2000)? However, it has also been suggested that the Saqqara pyramid was designed by Imhotep as a crystal complex used to concentrate huge quantities of cosmic energy (“the wise man
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North North North North
East East
West West
South South
West West
East East
South South
Figure 4: If the pyramid sides are orientated to N-S and E-W, cosmic radiation is concentrated at the pyramid axis, while under the side orientation to 45º, radiation is redirected to the ground.
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to align the future building plan with some particular stars seen on the night sky vault seems to be very unpractical. Readers of Rossi’s (2007) book could have noticed that an important influence of solar geometry is missing in her studies. Egyptians were certainly influenced not only by the religious belief in the Sun powers, they also realized the practical aspects of diurnal and annual sun-path changes which defined time and orientation in space, not only on the flat land but also on the sky vault during day and night. Geometric relationships, symmetry and angular extensions expressed by the ratio of sides in a triangle based on gnomon investigations were certainly an important basis of early mathematics. When applied to architecture, this knowledge of orientation in relation to cardinal points has facilitated construct development from simple geometry to the complex, symbolic, built forms of ceremonial and monumental structures. Of course, no one can define the exact secret procedure of the Foundation Ceremony and the basis of the “Stretching of Cord” that was used by Imhotep, but, as argued in this paper, most probably geometric rules were employed and, it should be realised that in ancient Rome, two thousand years after the Egyptian pyramids were built, the first concise architectural textbook by Vitruvius stated the traditional geometric principles and rules ‘already inherited’ and that should be learned by all architectural students.
Acknowledgement
The authors are grateful for the support of this study that has been provided by the Slovak Grant Agency VEGA under the grant 2/0060/08.
References
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Aboulfotouh, H. (2007). The relativistic tilts of Giza pyramids’ entrance-passages. Mediterranean Archaeology and Archeometry, 7(1), 23-37. Engelbach, R. (1934). A foundation scene of the second dynasty. Journal of Egyptian Archaeology, 20, 183-185.
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Kittler, R., & Darula, S. (2004). Analemma, the ancient sketch of fictitious sunpath geometry: Sun, time and history of mathematics. Architectural Science Review, 47(2), 141-144. Magdolen, D. (2000a). The solar origin of the “sacred triangle” in ancient Egypt? Studien zur altägyptischen Kultur, 28, 207-217. Magdolen, D. (2000b). On the orientation of Old Kingdom royal tombs. Archiv orientální, Suplementa IX, Prague, 68, 491- 498. Malkun, F., & Haus, A. (2000). El ojo de Horus, 5, Saqqara, El Complejo de Cristal. Video series Colombia. Prague: Egipto, Spectrum. Paturi, F.R. (1993). Kronika techniky (Slovak translation of Chronik der Technik. Dortmund: Harenberg Verlag, 1988). Bratislava: Fortuna Print. Rossi, C. (2003, 2007). Architecture and Mathematics in Ancient Egypt. London: Cambridge University Press. Shaltout, M., & Belmonte, J.A. (2005). On the orientation of Egyptian temples I. Journal for the History of Astronomy, 36, 273298. Spence, K. (2000). Ancient Egyptian chronology and the astronomical orientation of pyramids. Nature, 408, 320-324. Spence, K. (2003). Are pyramids oriented after stars? In The Seventy Great Mysteries of Ancient Egypt (Slovak translation by V. Krupa, 2004). London: Thames & Hudson. Valentovich, M. (1997). Súboj žiarení (The duel of radiation). Bratislava: Svornosť Publications. Verner, M. (1992). Abusir II: Baugraffiti der Ptahschepses Mastaba. Praha: Univerzita Karlova. Vitruvius, M.P. (13 BC, printed 1487). De architectura libri X. Rome (Vitruvius: The Ten Books on Architecture, English trans by M.H. Morgan). Cambridge, Mass: Harvard University Press, 1914, and New York: Dover, 1960. (Also Vitruvius: Of Architecture. English trans by F. Granger. London: Heinemann, 1970). Zamarovsky, V. (1984). Na Počiatku bol Sumer (At the beginning was Sumer). Bratislava: Mladé letá Publications. Zamarovsky, V. (1986). Bohovia a Králi Starého Egypta (Gods and Kings of Ancient Egypt). Bratislava: Mladé letá Publications.
