North by Northwest: The Strange Case of Giza s Misalignments

Transcription

1 North by Northwet: The Strange Cae of Giza Mialignment By Glen Dah The Giza Pyramid are aligned to cardinal point with uncanny accuracy. But many of Giza other monument hare a trange, ytematic alignment error. The firt impreion made by a map of Giza i one of order. The bae of the pyramid appear perfectly quare and preciely aligned with cardinal point. Yet a cloer look at ome of Giza other tructure, including the Khentkawe monument and the Worker City, reveal omething different. Many eem to hare a common pattern of mialignment; on a map they are rotated a few degree counter-clockwie from cardinal point (Figure 1). It i a if the Egyptian thought that north wa a little to the wet of where it really wa. Nor i the effect confined to Giza. We find the ame turn, north by northwet, at many other place in Egypt pyramid field. The Egyptian choe the orientation of their tomb, temple and civic building for both practical and ceremonial reaon. They aligned many of their tructure to the Nile. Other they built along ridgeline. But often they choe cardinal direction for alignment. The Great Pyramid of Khufu i oriented to cardinal point to better than even minute of arc, an extraordinary achievement in the age before optical intrument. Only within the lat few hundred year have builder been able do better. By ighting on Polari and uing pecial tar chart known a ephemeri table, a urveyor today can lay out a line accurate to better than 20 econd of arc. Such preciion, though, i uually reerved for important tructure uch a highway or capitol building. More ordinary tructure do not require uch preciion. To lay out a reidence or office building, a builder might chooe to align with an exiting building, a nearby road, or a natural feature uch a a river. Abent thoe, the builder will need an intrument to provide orientation. A magnetic compa i a common choice. However, a builder uing a hand-held compa can achieve an accuracy of no better than about two degree of arc. That i, however, 1

2 good enough to orient a building acetically, allowing for unlight to tream in at the right time or the day. The Egyptian faced imilar choice. They appear to have reerved preciion alignment for royal tructure. Other tructure called for no uch preciion. However, the Egyptian had no magnetic compae to guide them. Intead, they probably ued the un. All it take to determine true north, and therefore all cardinal point, i an upright tone or pole. Even today, cout are taught the hadow method. The method ue a rod et vertically in the ground. A the day pae, one mark the tip of the hadow a it move in an arc along the ground. The next tep i to fix a tring to the bae of the rod and draw a circular arc acro the hadow pattern. The circular arc will cro the hadow arc at two point (Figure 2). Draw a line through thee point and it will run eat-wet. Biect the line and draw a ray to the bae of the rod and that line will run north-outh. Do thi 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 tanding over the coure of the eaon and track the movement of the un on the winter oltice, ummer oltice and the equinox, it would create the pattern hown in Figure 3. On the ummer oltice, with the un high in the ky, the tip of the hadow trace form a curve pointing away from the rod. On the winter oltice the oppoite i true. In between, on the equinoxe, the hadow trace out a traight line. Over a preciely leveled urface, thi line run almot exactly eat-wet. We get an error however when we try the ame experiment over loping ground (Figure 4). We ee the reult mot clearly on the equinox (Figure 5). Over wet-to-eat loping ground our formerly eat-wet line now run from the outhwet to the northeat. 1 Giza mataba, the Khentkawe monument, and it aociated town it on a limetone plateau that dip from northwet to outheat at an average angle of ix degree. The wet-to-eat 1 A we how in the figure, it i only the wet-to-eat lope that caue the error. A north-to-outh lope imply change the length of the hadow. 2

3 component average about three degree. For mot tructure, all the Egyptian may have wanted wa a general orientation to the un. Since accuracy beyond a few degree wa not neceary, they did not have to level the bedrock firt to ue the hadow method. The reult wa a light rotation of the tructure. We ee uch rotation in Figure 6. The Khentkawe monument and Khentkawe Town are rotated counter-clockwie by a little more than three degree. However, Khentkawe wa the tomb of a Queen and the Egyptian uually oriented a queen tomb more preciely. The reaon Khentkawe did not end up with a precie orientation may have been due to it hitory. Mark Lehner believe that Khentkwe wa originally a quarry cube, a ection of the plateau channeled out on four ide in order to prepare it for further quarrying into building block. A lowly quarry cube did not call for precie alignment. It wa later converted into a tomb for a Queen. When Khentkawe Town wa built, it wa aligned to the Khentkawe monument and hared it mialignment. The main thoroughfare of the Worker City, Main Street and North Street, are alo rotated lightly, but only by a degree or o. The Worker City wa not built on bedrock like Kehntkawe Town but on the ancient Nile floodplain. Flood depoit have a natural leveling effect on the feature they cover. Even o, they till exhibit ome a dip toward the river becaue of run off. Thu, the Worker City exhibit a light dip to the eat which reulted in a one degree or o rotation off cardinal point. The Wall of the Crow run ix and one half degree north of eat. Thi deviation i o great that it probably wa not a product of the hadow method. While we do not know the exact purpoe of the wall, it may have functioned in part a a flood diverion dam. A uch, it may have been deliberately built to parallel to the coure of the Central Wadi, rather than being oriented to the un or aligned with the ret of the Worker City. The lope of the land immediately wet of the Nile i predominately to the eat. Likewie the land to the eat of the river dip to the wet. Thi offer u a way to tet our hypothei. While 3

4 tructure built to the wet of the Nile exhibit a counter-clockwie rotation, thoe to the eat of the river hould be rotated clockwie. Mot Old Kingdom ettlement and cemeterie lie to the wet of the Nile. A few, though, do lie to the eat. Helwan, for example, i a cemetery that lie oppoite Saqqara on the Nile eat bank. At Giza, mataba cloe to the Great Pyramid are aligned with it and to cardinal point. Mataba farther away tend to exhibit rotation. At Helwan, there i no pyramid for the Egyptian to have ued for alignment, o tomb do exhibit rotation off cardinal point. For the mot part thee are clockwie, oppoite that at Giza, a our theory would predict. (Jeffrey 1994: 143). (Figure 7) I believe that the Egyptian ued the hadow method the way today builder ue a compa. A compa i not a preciion intrument. If more preciion i required, the builder can ue a total tation and ight Polari at night. The Egyptian had imilar option. When preciion wa required they could ight on the tar. But where preciion wa not required, and there were no local landmark to align with, they probably ued the hadow method. The evidence of that i in the error they left behind. 4

5 Reference Belmonte, J. A. and Shaltout, M., ed In Search of Comic Order: Selected Aay on Egyptian Archaeoatronomy, 1t ed., Cairo: Supreme Council of Antiquite Pre. Ghilani, C Atronomical Obervation, Atronomical Obervation Handbook, Pennylvania State Univerity, Acceed 25 Augut. Jeffrey, D. and Tavare, A The Hitoric Landcape of Early Dynatic Memphi, Mitteilungen de Deutchen Archaeologichen Intitut Abteilung Kairo, vol

6 Figure 1: A map of Giza. While on a broad cale Giza look orderly and rectilinear, upon cloer examination many of it tructure exhibit counter-clockwie rotation. 6

7 Vertical Rod Shadow Pattern Formed by Tip of the Rod' Shadow a the Sun Move Eat to Wet Wet (m) Bae of Vertical Rod North-South Interection Mark Eat-Wet Line North (m) Rod Height 2m Circle Drawn from Bae of Vertical Rod Intercept Shawdow Line at Two Point Figure 2: The hadow method for finding north. A vertical pole produce an arc haped hadow a the un move from eat to wet. To find eat-wet, a econd, circular arc i drawn from the bae of the pole croing the hadow arc at two point. North-outh i perpendicular to that line. (WS=Winter Soltice) 7

8 Bae of Vertical Rod Shadow Method Pattern for Giza (Over Level Ground) Wet (m) North (m) Rod Height 2m Figure 3: Seaonal hadow. The pattern produced by the hadow method varie with the eaon. (WS=Winter Soltice, SS=Summer Soltice) 8

9 Bae of Vertical Rod Shadow Method Pattern for Giza (Over Level Ground) Wet (m) North (m) Rod Height 2m Shadow Method Pattern for Giza (Over 3 Degree Wet-Eat Slope) Wet (m) North (m) Rod Height 2m Figure 4: The effect of a wet-eat lope. Thi lope caue the tip of the hadow to rotate counterclockwie. Deriving north uing the hadow method will reult in an error. (WS=Winter Soltice, SS=Summer Soltice) 9

10 Shadow Method at Equinox for Giza Wet (m) North (m) Rod Height 2m Figure 5: The equinox and the lope. Reult on the equinox illutrate the effect that lope have on the hadow method. A north-outh lope doe not change the reult; true north i till derived correctly uing the hadow method. However, deriving north from data accumulated over a wet-eat lope will reult in an error. 10

11 Figure 6: Different feature at Khentkawe and the Worker City have differing rotational magnitude. The angle off cardinal direction i hown in red. 11

12 Figure 7: Rotation at Helwan. Helwan lie on the eat bank of the Nile, where the prevailing lope i oppoite that at Giza, reulting in a clockwie rotation of feature. 12

13 APPENDIX We can compute the elevation and azimuth of the un at any time and any location from the following formula: in θ = co( h)co δcoφ + in δin Φ co ϕ in( h)coδ = coθ Where: φ = Solar Azimuth Angle θ = Solar Elevation Angle h = Solar Hour Angle δ = Solar Declination Φ = Local Latitude Our latitude at Giza i 30 degree north. The olar declination depend on the time of year. On the oltice it i equal in magnitude to the Earth tilt, 23.5 degree. In the winter, it i equal to minu 23.5 degree and the ummer, plu 23.5 degree. On the equinoxe, the declination i zero. The hour angle i the time a expreed by the poition of the un in the ky. At olar noon, the un i at it zenith, or at an hour angle of 180 degree (midnight i zero degree). Thu, at olar noon on the winter oltice at Giza, h= 180 degree, δ=-23.5 degree and Φ=30 degree, o our elevation angle i: Our azimuth i: in θ in θ in θ θ = co(180) co( 23.5) co(30) + in( 23.5) in(30) = (1)(.917)(.866) + (.399)(.5) =.5948 =

14 coϕ = in(180) co( 23.5) co(36.5) co ϕ ϕ = 0 = 0 Likewie, one hour earlier, the olar hour angle will be at 165 degree reulting in a olar elevation of 34.6 degree and an azimuth of degree (wet of north). Referring to Figure A1, a gnomon with a height a will produce a hadow of length: a r = tan θ Where: a = Gnomon height in meter r = Length of hadow in meter Thu if our gnomon i 2 meter high, the hadow at noon on the winter oltice at Giza will be 2.7 meter long. The hadow one hour before noon (olar hour angle = 165 degree) would have been lightly longer, 2.9 meter. A we how in Figure A1, on a horizontal Carteian grid whoe origin i at the bae of the gnomon, the tip of the hadow will fall at: x = r in ϕ y = r co ϕ Thu one hour before noon on the winter oltice at Giza, the tip of the gnomon hadow would fall at x=-.84 meter and y = 2.78 meter. 14

15 If the urface i not level, then, in effect, the height of the gnomon change a a function of the poition of the tip of the hadow on our Carteian grid. If the lope in the wet-to-eat direction i α, then we can compute the poition of the tip of the hadow by re-computing the effective height of the gnomon a hown in Figure A2. We would add to the height of the gnomon an additional effective height c: The new gnomon height would then be: c = x tan α a eff = a + c = a x tan α The length of the hadow would now be: aeff r = tan θ a x tan α r = tan θ x = r in ϕ a r in ϕ r = tan θ tan θ tan θ a r = tan θ r = tan θ + in ϕ a + in ϕ tan α a r in ϕ = r a = in ϕ r tan α tan α tan α tan α Once again, the tip of the hadow will fall at: x = r in ϕ y = r co ϕ 15

16 Similarly, if we conider the cae where the urface lope both wet-to-eat and north-to-outh, uing the ame analyi we find: r = tan θ + in ϕ a tan α + co ϕ tanβ Where: α= Wet-to-eat lope β= North-to-outh lope 16

17 Figure A1 17

18 Figure A2 18