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Applying Solar Geometry to Understand the

Foundation Rituals of ‘Old Kingdom’ Egyptian

Pyramids

Richard Kittler a & 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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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, 3}rd_{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.}

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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Architectural Science Review Volume 51, Number 4, December 2008 408

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,1_{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 sun-shadow 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 1 The gnomon is the part of a sundial that casts the shadow. Gnomon (γνωμων) is an ancient Greek word meaning “indicator”, “one who discerns” or “that which reveals.” [Editor]

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., Figure74 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

Ceremony

Respecting the Cardinal Points

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 ʹ

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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 found by Czech Egyptologists in Abusir document the Old Kingdom grave orientations quite persuasively (Verner, 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 22}nd

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, i.e., dividing equally any whole into two halves and therefore originally the equinox day was dedicated to her too.

Figure 1. Alignment to cardinal points in an arbitrary building site, method 3.

Figure 2. The orientation of Egyptian Old Kingdom pyramids after equinox line shadow.

Figure 1: Alignment to cardinal points in an arbitrary building site, method 3.

Figure 2: The orientation of Egyptian Old Kingdom pyramids using an equinox line shadow.

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In the equinox day at noon, i.e., exactly at the 12 hour in true solar time, the sun is directly shining from the South direction and its shadow points to the North cardinal point, while the shadow is the shortest within that day. The shadow length s at noon relative to the gnomon height g in Ur or Alexandria, i.e., on approximately the geographical latitude

Figure 3. Sun path diagram for equinox days using sundial hours.

= 31° was s/g = 3/5 corresponding to

Figure 3. Sun path diagram for equinox days using sundial hours.

= 30°58´ and thus the solar altitude γ_s = 90° -

Figure 3. Sun path diagram for equinox days using sundial hours.

= 59°02´, valid precisely at exact noon on an equinox day.

Figure 3 also illustrates a sundial drawn to be turned into the plane of the solar meridian using the s/g = 3/5 ratio in section and the shadow traces in plan. The end points of the sun-shadow are sliding during an equinox day along a straight line parallel with the E-W direction.

Sun-shadow measurements and sundials with vertical sticks, later called gnomon, were also known and probably the equinox day proportion of a 3 unit shadow to a 5 unit gnomon height was already applied after the Sumerian/Chaldean tradition

mentioned by Vitruvius in 13 BC. Thus, it can be also assumed that the equinox day was taken as a ritual day for the foundation ceremony performed by the Pharaoh. He decided the building site and one of the main pyramid corners, where the gnomon stick was placed in an exactly vertical position and a circle with the three-unit radius was drawn. Because the sun-shadow end points were gradually moving from the West along a straight line towards the circle a rope was aligned with this W-E direction to it. Either the touch point or the centre of two section points had to indicate the N-S direction that should be exactly 90º to the E-Western stretched cord. Now, it can be assumed that either the “sacred triangle rule” was applied to check the right angle, i.e. forming the 12 unit cord into a triangle with exactly three sides of 3, 4 and 5 length units, or simply two other longer cords with equal length had to check their intersection which has to be on the exact noon S-N line. The existence of the 3-4-5 triangle (Magdolen, 2000a) and drawing of the ellipse by a 12-unit cord is indicated also by Rossi (2003) but not in relation with the gnomon or any orientation task. However, it could be utilised for drawing the sun-path ellipse on a fictitious sky hemisphere on the upper side of the gnomon (as shown in Figure 3). If the equinox day is given by the s/g =3/5 ratio then the short axis of the ellipse

b = 3, while the sky hemisphere radius r is equal to the ellipse

axis a =

9 main pyramid corners, where the gnomon stick was placed in an exactly vertical position and a circle with the three-unit radius was drawn. Because the sun-shadow end points were gradually moving from the West along a straight line towards the circle a rope was aligned with this W-E direction to it. Either the touch point or the centre of two section points had to indicate the N-S direction that should be exactly 90º to the E-Western stretched cord. Now, it can be assumed that either the “sacred triangle rule” was applied to check the right angle, i.e. forming the 12 unit cord into a triangle with exactly three sides of 3, 4 and 5 length units, or simply two other longer cords with equal length had to check their intersection which has to be on the exact noon S-N line. The existence of the 3-4-5 triangle (Magdolen, 2000a) and drawing of the ellipse by a 12-unit cord is indicated also by Rossi (2003) but not in relation with the gnomon or any orientation task. However, it could be utilised for drawing the sun-path ellipse on a fictitious sky hemisphere on the upper side of the gnomon (as shown in Figure 3). If the equinox day is given by thes /g=3/5 ratio then the short axis of the ellipse

=

b 3, while the sky hemisphere radius r is equal to the ellipse axis a = 32+52= 34=

5.831 and the focus from the ellipse centre has a distance e = 2 2 34 9 5. = ! = ! b

a Thus

applying the same 3/5 ratio in a plan circle a cord tied to pegs placed in the two focuses by turning a peg round could be drawn the sun-path ellipse, as it isr r ₂a

2 1+ = .

Moreover, once the dual equinox days were identified within the year, this method is applicable in any locale, i.e. not 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 Abydos/This (ca 27°). No special ratioss /gwere thus needed to orient any graves or buildings to E-W and N-S directions.

In fact, Vitruvius in his Book IX, Chapter 7 mentioned the ratio 3/5 for the

geographical latitude of Alexandria for his analemma rule determining the sun-path during the

= 5.831 and the focus from the ellipse centre has a distance e =

=

5.831 and the focus from the ellipse centre has a distance e = 2 2 34 9 5. = ! = ! b

a Thus

applying the same 3/5 ratio in a plan circle a cord tied to pegs placed in the two focuses by turning a peg round could be drawn the sun-path ellipse, as it isr r ₂a

2 1+ = .

Thus applying the same 3/5 ratio in a plan circle a cord tied to pegs placed in the two focuses by turning a peg round could be drawn the sun-path ellipse, as it is

=

5.831 and the focus from the ellipse centre has a distance e = a2! b2 = 34!9=5.Thus applying the same 3/5 ratio in a plan circle a cord tied to pegs placed in the two focuses by turning a peg round could be drawn the sun-path ellipse, as it isr r ₂a

2 1+ = .

Moreover, once the dual equinox days were identified within the year, this method is applicable in any locale, i.e. not 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 Abydos/This (ca 27°). No special ratios s/g were thus needed to orient any graves or buildings to E-W and N-S directions.

In fact, Vitruvius in his Book IX, Chapter 7 mentioned the ratio 3/5 for the geographical latitude of Alexandria for his analemma rule determining the sun-path during the whole year, thus indicating such an ancient knowledge (Kittler & Darula, 2004). It has to be noticed that while the sun-path on the fictitious sky hemisphere follows the sun movement from E through S to W cardinal points during equinox days, the sun-shadows have a reverse trend with the morning end pointing to the W direction and indicating N at noon.

Questionable is the Astronomic Orientation

of Pyramids or other Reasons for their S-N

Orientation

Due to current, more sophisticated knowledge of astronomy and the magnetic field of the Earth, some speculations how and why the pyramid orientation to cardinal point has been achieved. Because elemental solar geometry may have been restricted to priests and not recorded, some current researchers and specialists have deduced several plausible or implausible ways and means that could be applied. Many archeoastronomers wonder whether the pyramid orientation was not set to some star on the North horizon or even try to date the pyramids after the star placement occurring during the Old Kingdom (see

Figure 3. Sun path diagram for equinox days using sundial hours.

Figure 3: Sun path diagram for equinox days using sundial hours.

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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

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,

19 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.

North West South East North West _East 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.

19 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.

North West South East North West East South 13:50

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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.

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