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North by Northwest: The Strange Case of Giza’s Misalignments By Glen Dash

The Giza Pyramids are aligned to cardinal points with uncanny accuracy. But many of Giza’s

other monuments share a strange, systematic alignment error.

The first impression made by a map of Giza is one of order. The bases of the pyramids appear

perfectly square and precisely aligned with cardinal points. Yet a closer look at some of Giza’s

other structures, including the Khentkawes monument and the Worker’s City, reveal something

different. Many seem to share a common pattern of misalignment; on a map they are rotated a

few degrees counter-clockwise from cardinal points (Figure 1). It is as if the Egyptians thought

that north was a little to the west of where it really was. Nor is the effect confined to Giza. We

find the same turn, north by northwest, at many other places in Egypt’s pyramid fields.

The Egyptians chose the orientation of their tombs, temples and civic buildings for both practical

and ceremonial reasons. They aligned many of their structures to the Nile. Others they built

along ridgelines. But often they chose cardinal directions for alignment.

The Great Pyramid of Khufu is oriented to cardinal points to better than seven minutes of arc, an

extraordinary achievement in the age before optical instruments. Only within the last few

hundred years have builders been able do better. By sighting on Polaris and using special star

charts known as ephemeris tables, a surveyor today can lay out a line accurate to better than 20

seconds of arc.

Such precision, though, is usually reserved for important structures such as highways or capitol

buildings. More ordinary structures do not require such precision. To lay out a residence or

office building, a builder might choose to align with an existing building, a nearby road, or a

natural feature such as a river. Absent those, the builder will need an instrument to provide

orientation. A magnetic compass is a common choice. However, a builder using a hand-held

compass can achieve an accuracy of no better than about two degrees of arc. That is, however,

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good enough to orient a building ascetically, allowing for sunlight to stream in at the right time

or the day.

The Egyptians faced similar choices. They appear to have reserved precision alignments for

royal structures. Other structures called for no such precision. However, the Egyptians had no

magnetic compasses to guide them. Instead, they probably used the sun.

All it takes to determine true north, and therefore all cardinal points, is an upright stone or pole.

Even today, scouts are taught the “shadow method.” The method uses a rod set vertically in the

ground. As the day passes, one marks the tip of the shadow as it moves in an arc along the

ground. The next step is to fix a string to the base of the rod and draw a circular arc across the

shadow pattern. The circular arc will cross the shadow arc at two points (Figure 2). Draw a line

through these points and it will run east-west. Bisect the line and draw a ray to the base of the

rod and that line will run north-south. Do this with care, and you can achieve an accuracy of

better than one-half of one degree. (Ghilani 2004)

If we were to leave our vertical rod standing over the course of the seasons and track the

movement of the sun on the winter solstice, summer solstice and the equinox, it would create the

pattern shown in Figure 3. On the summer solstice, with the sun high in the sky, the tip of the

shadow’s trace forms a curve pointing away from the rod. On the winter solstice the opposite is

true. In between, on the equinoxes, the shadow traces out a straight line. Over a precisely leveled

surface, this line runs almost exactly east-west.

We get an error however when we try the same experiment over sloping ground (Figure 4). We

see the results most clearly on the equinox (Figure 5). Over west-to-east sloping ground our

formerly east-west line now runs from the southwest to the northeast.1

Giza’s mastabas, the Khentkawes monument, and its associated town sit on a limestone plateau

that dips from northwest to southeast at an average angle of six degrees. The west-to-east

1 As we show in the figure, it is only the west-to-east slope that causes the error. A north-to-south slope simply changes the length of the shadows.

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component averages about three degrees. For most structures, all the Egyptians may have wanted

was a general orientation to the sun. Since accuracy beyond a few degrees was not necessary,

they did not have to level the bedrock first to use the shadow method. The result was a slight

rotation of the structure.

We see such rotations in Figure 6. The Khentkawes monument and Khentkawes Town are

rotated counter-clockwise by a little more than three degrees. However, Khentkawes was the

tomb of a Queen and the Egyptians usually oriented a queen’s tombs more precisely. The reason

Khentkawes did not end up with a precise orientation may have been due to its history. Mark

Lehner believes that Khentkwes was originally a “quarry cube,” a section of the plateau

channeled out on four sides in order to prepare it for further quarrying into building blocks. A

lowly quarry cube did not call for precise alignment. It was later converted into a tomb for a

Queen. When Khentkawes Town was built, it was aligned to the Khentkawes monument and

shared its misalignment.

The main thoroughfares of the Worker’s City, Main Street and North Street, are also rotated

slightly, but only by a degree or so. The Workers City was not built on bedrock like Kehntkawes

Town but on the ancient Nile floodplain. Flood deposits have a natural leveling effect on the

features they cover. Even so, they still exhibit some a dip towards the river because of run off.

Thus, the Worker’s City exhibits a slight dip to the east which resulted in a one degree or so

rotation off cardinal points.

The Wall of the Crow runs six and one half degrees north of east. This deviation is so great that

it probably was not a product of the shadow method. While we do not know the exact purpose of

the wall, it may have functioned in part as a flood diversion dam. As such, it may have been

deliberately built to parallel to the course of the Central Wadi, rather than being oriented to the

sun or aligned with the rest of the Worker’s City.

The slope of the land immediately west of the Nile is predominately to the east. Likewise the

land to the east of the river dips to the west. This offers us a way to test our hypothesis. While

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structures built to the west of the Nile exhibit a counter-clockwise rotation, those to the east of

the river should be rotated clockwise.

Most Old Kingdom settlements and cemeteries lie to the west of the Nile. A few, though, do lie

to the east. Helwan, for example, is a cemetery that lies opposite Saqqara on the Nile’s east bank.

At Giza, mastabas close to the Great Pyramid are aligned with it and to cardinal points. Mastabas

farther away tend to exhibit rotations. At Helwan, there is no pyramid for the Egyptians to have

used for alignment, so tombs do exhibit rotations off cardinal points. For the most part these are

clockwise, opposite that at Giza, as our theory would predict. (Jeffreys 1994: 143). (Figure 7)

I believe that the Egyptians used the shadow method the way today’s builders use a compass. A

compass is not a precision instrument. If more precision is required, the builder can use a total

station and sight Polaris at night. The Egyptians had similar options. When precision was

required they could sight on the stars. But where precision was not required, and there were no

local landmarks to align with, they probably used the shadow method. The evidence of that is in

the errors they left behind.

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References

Belmonte, J. A. and Shaltout, M., eds.

2009 In Search of Cosmic Order: Selected Assays on Egyptian Archaeoastronomy, 1st

ed., Cairo: Supreme Council of Antiquites Press.

Ghilani, C.

2004 “Astronomical Observations,” Astronomical Observation Handbook,

Pennsylvania State University,

http://surveying.wb.psu.edu/sur351/CelestialCoords/ASTRO.pdf, Accessed 25

August.

Jeffreys, D. and Tavares, A.

1994 “The Historic Landscape of Early Dynastic Memphis,” Mitteilungen des

Deutschen Archaeologischen Instituts Abteilung Kairo, vol. 50.

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Figure 1: A map of Giza. While on a broad scale Giza looks orderly and rectilinear, upon closer examination

many of its structures exhibit counter-clockwise rotations.

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Vertical Rod

Shadow

Pattern Formed by Tip ofthe Rod's Shadow as theSun Moves East to West

Circle Drawn from Base ofVertical Rod InterceptsShawdow Line at Two

Points

Intersection Marks East-West Line

North-South

Rod Height 2mNorth (m)

Wes

t (m

)

Base of Vertical Rod

Figure 2: The shadow method for finding north. A vertical pole produces an arc shaped shadow as the sun

moves from east to west. To find east-west, a second, circular arc is drawn from the base of the pole crossing the shadow arc at two points. North-south is perpendicular to that line. (WS=Winter Solstice)

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Shadow Method Pattern for Giza

(Over Level Ground)Base of Vertical Rod

Rod Height 2mNorth (m)

Wes

t (m

)

Figure 3: Seasonal shadows. The pattern produced by the shadow method varies with the seasons.

(WS=Winter Solstice, SS=Summer Solstice)

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Shadow Method Pattern for Giza(Over Level Ground)Base of Vertical Rod

Rod Height 2mNorth (m)

Wes

t (m

)

North (m)

Wes

t (m

)

Rod Height 2m

Shadow Method Pattern for Giza(Over 3 Degree West-East Slope)

Figure 4: The effect of a west-east slope. This slope causes the tip of the shadows to rotate counterclockwise. Deriving north using the shadow method will result in an error. (WS=Winter Solstice, SS=Summer Solstice)

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Rod Height 2m

Shadow Method at Equinox for Giza

North (m)

Wes

t (m

)

Figure 5: The equinox and the slope. Results on the equinox illustrate the effects that slopes have on the

shadow method. A north-south slope does not change the results; true north is still derived correctly using the shadow method. However, deriving north from data accumulated over a west-east slope will result in an

error.

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Figure 6: Different features at Khentkawes and the Workers City have differing rotational magnitudes. The

angle off cardinal directions is shown in red.

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Figure 7: Rotations at Helwan. Helwan lies on the east bank of the Nile, where the prevailing slope is opposite that at Giza, resulting in a clockwise rotation of features.

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APPENDIX

We can compute the elevation and azimuth of the sun at any time and any location from the

following formulas:

Φδ+Φδ−=θ sinsincoscos)cos(sin hs

ss

hθ

δ−=ϕcos

cos)sin(cos

Where:

φs = Solar Azimuth Angle

θs = Solar Elevation Angle

h = Solar Hour Angle

δ = Solar Declination

Φ = Local Latitude

Our latitude at Giza is 30 degrees north. The solar declination depends on the time of year. On

the solstices it is equal in magnitude to the Earth’s tilt, 23.5 degrees. In the winter, it is equal to

minus 23.5 degrees and the summer, plus 23.5 degrees. On the equinoxes, the declination is

zero. The hour angle is the time as expressed by the position of the sun in the sky. At solar

noon, the sun is at its zenith, or at an hour angle of 180 degrees (midnight is zero degrees). Thus,

at solar noon on the winter solstice at Giza, h= 180 degrees, δ=-23.5 degrees and Φ=30 degrees,

so our elevation angle is:

°=θ=θ

−+=θ−+−−=θ

5.365948.sin

)5)(.399.()866)(.917)(.1(sin)30sin()5.23sin()30cos()5.23cos()180cos(sin

s

s

s

s

Our azimuth is:

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00cos)5.36cos(

)5.23cos()180sin(cos

=ϕ=ϕ

−−=ϕ

s

s

s

Likewise, one hour earlier, the solar hour angle will be at 165 degrees resulting in a solar

elevation of 34.6 degrees and an azimuth of -16.8 degrees (west of north).

Referring to Figure A1, a gnomon with a height a will produce a shadow of length:

s

arθ

=tan

Where:

a = Gnomon height in meters

r = Length of shadow in meters

Thus if our gnomon is 2 meters high, the shadow at noon on the winter solstice at Giza will be

2.7 meters long. The shadow one hour before noon (solar hour angle = 165 degrees) would have

been slightly longer, 2.9 meters.

As we show in Figure A1, on a horizontal Cartesian grid whose origin is at the base of the

gnomon, the tip of the shadow will fall at:

s

s

ryrx

ϕ=ϕ=

cossin

Thus one hour before noon on the winter solstice at Giza, the tip of the gnomon’s shadow would

fall at x=-.84 meters and y = 2.78 meters.

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If the surface is not level, then, in effect, the height of the gnomon changes as a function of the

position of the tip of the shadow on our Cartesian grid. If the slope in the west-to-east direction

is α, then we can compute the position of the tip of the shadow by re-computing the effective

height of the gnomon as shown in Figure A2. We would add to the height of the gnomon an

additional effective height c:

α−= tanxc

The new gnomon height would then be:

α−=+= tanxacaaeff

The length of the shadow would now be:

αϕ+θ=

αϕ+θ=

αϕ−=θ

αϕ−=θ

θαϕ−

=

ϕ=θ

α−=

θ=

tansintan

tansintan

tansintan

tansintan

tantansin

sintan

tantan

ss

ss

ss

ss

s

s

s

s

s

eff

ar

ra

ra

rra

rar

rx

xar

ar

Once again, the tip of the shadow will fall at:

s

s

ryrx

ϕ=ϕ=

cossin

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Similarly, if we consider the case where the surface slopes both west-to-east and north-to-south,

using the same analysis we find:

βϕ+αϕ+θ=

tancostansintan sss

ar

Where:

α= West-to-east slope

β= North-to-south slope

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Figure A1

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Figure A2