The Architect's Plan

There have been a limited number of surveys over the years which have attempted to measure the exterior and interior details of the Red Pyramid at Dahshur. The most notable of these include: 1) J. S. Perring's survey of the late 1830's; 2) W.M.F. Petrie's partial surveys of the early 1880's and of 1887; 3) the Maragioglio and Rinaldi partial survey of the early 1960's; and 4) the 1980 and 1997 surveys of Josef Dorner.¹

Of these, only the Perring and Dorner surveys undertook to determine the original location of the exterior opening of the pyramid's entrance passage relative to the pyramid's north base edge. Although these two surveys are in nearly exact agreement in regard to the dimensions and relative locations of the pyramid's interior passageways and chambers, they do not agree as to the original length and slope of the pyramid's entrance passage nor as to the elevation of this entrance passage's original opening on the pyramid's northern face. While it is true that the differences between the Perring and Dorner findings on these points are not extremely large, the discrepancies are significant enough to justify further investigation. Such an investigation, I believe, can not only clarify some puzzling aspects found in the data of each of these surveys, but also possibly provide some new insight into the design of the Red Pyramid.

John S. Perring's Survey

Perring, trained as a civil engineer, conducted his survey of the Red Pyramid during the latter half of the 1830's. Although Perring included some roughly scaled drawings of the pyramid and it's interior chambers in his published findings, he unfortunately only gives the specifics of his measurements within his text.²

Perring determined that the pyramid's original base length had been 719 feet 5 inches, and he clarifies this fact by also stating that he measured the "present" base length to have been a much shorter 700 feet.³ He then states that the pyramid's entrance passage descends at an angle of 27°56', and that "it's original length was 205 feet 6 inches, of which 4 feet 6 inches have been destroyed by the removal of the external casing".⁴ We see, then, that in regard to both the pyramid's base length and the length of the descending entrance passage Perring was careful to distinguish between the length of each of these features as he then found them and the length which he computed that each must have originally been when the pyramid was new.

However, when listing the parameters of the exterior opening to the entrance passage, Perring merely states that the "centre of the entrance is 12 feet 6 inches to the eastward of the centre of the northern front, and the bottom of it is 94 feet perpendicularly higher than the base of the building".⁵ In other words, he makes no mention here of an original entrance elevation. Furthermore, as is quoted above, Perring says that bottom of the entrance opening "is 94 feet perpendicularly higher than the base of the building" - the operative word here being "is", not "was". I believe, then, that contrary to previous understandings the correct interpretation of his statement should be that "94 feet" is the height at which he found the existing entrance opening.

In addition, given the other pyramid parameters listed by Perring, the geometry of the situation essentially dictates that the 94 foot elevation must refer to the elevation of the extant, and not original, opening location. I have somewhat belabored this point because, as we shall see, it is crucial to a more tenable understanding of Perring's data. Our next step, therefore, is to take a cross-sectional look at the 'geometry of the situation'.

In Figure 1, Point Ex represents the extant upper end of the entrance passage floor - sitting where I am proposing that Perring found it - at about 94 feet above the pyramid's base. Point N is on the pyramid's original northern base edge in the same plane as Ex, and Point Ep is the point at which I am proposing that Perring's data predicts the entrance passage floor to have originally ended, on what Perring believed to be the original location of the pyramid's northern face. As quoted earlier, Perring concluded that the exterior end of the entrance passage was missing 4.5 feet of its original length, and so the reconstruction here has the distance from Point Ex to Point Ep identified as being 4.5 feet.

Perring determined the slope of the pyramid's original exterior side to be 43°36', and he measured the slope of the descending entrance passage to be 27°56' - with, he evidently believed, the bottom of this passage having its endpoint at the pyramid's base level.⁶ With these few parameters in hand, it is then a fairly simple task to calculate all of the remaining particulars seen in Figure 1. When this is done, one finds that Point Ep will be 2.1 feet higher than Point Ex. This means that Ep, the presumed original entrance location of Perring's data, will sit 96.11 feet above the pyramid's base level.

Computation will then show that in this proposed scenario the full original length of the entrance passage, from Point G to Point Ep, will be 205.17 feet - very nearly the 205.5 feet listed by Perring in his data. Note that if Perring had actually measured the elevation of Ex (the existing entrance location) to be 94.15 feet, and had he used this number in his calculations rather than the published value of 94 feet, then the full length of the diagram's hypotenuse (G to Ep) works out to be exactly 205.5 feet - the length which he gives.⁷ It may well be that Perring decided to round off the 94.15 foot amount to an even 94 feet when he was writing up his results for publication due to the uncertainties caused by the damage he found at the entrance passage opening site. Also note that using the 94.15 foot figure will result in raising the computed elevation of the original entrance opening (Point Ep) from 96.11 to 96.26 feet.

For these, and for the reasons about to be developed, we will see that the situation as shown in Figure 1 is likely to be very close to Perring's understandings.

The Maragioglio and Rinaldi Survey

In the early 1960's, Vito Maragioglio and Celeste Rinaldi conducted a comprehensive review of the survey literature pertaining to many of the Egyptian pyramids, and in the process undertook some field survey work of their own. In the case of the Red Pyramid, they performed their own detailed measurements of the pyramid's entrance passage and of the interior passageways and chambers. They apparently did not attempt to determine the elevation of the existing opening to the entrance passage relative to the pyramid's north base edge, but instead relied on data from the Perring survey for this factor.⁸

Their measurements for the interior compartments and horizontal passageways are all but exactly the same as those listed by both Perring and Josef Dorner. Of particular interest, however, are their measurements of - and their detailed scaled drawing of - the entrance passage itself.⁹ I propose that their data for the entrance passage can be used to provide a point of beginning from which to compare the related findings of the Perring and Dorner surveys.

 

Figure 2 is a representative sketch of Maragioglio and Rinaldi's drawing, showing an east to west view of the entrance passage. Maragioglio and Rinaldi's measurement for what they term the "conserved part" of the ceiling is given as 58.82 meters (= 193 feet).¹⁰ The corresponding distance on the passage floor is shown above as the distance between the points I have labeled "R" and "x". As their scaled drawing makes clear,

Maragioglio and Rinaldi considered that the floor continued upward for at least a foot beyond the ceiling endpoint, here labeled as the distance between Points "x" and "Ex".

In addition, a line from the ceiling's lower endpoint (Point "w" above) dropped straight down will intersect the line of the floor at a point here labeled as "m", and Maragioglio and Rinaldi's drawing shows that this Point "m" lies about 2 feet down slope from the spot labeled above as Point "R". And finally, their drawing shows that the entrance passage floor appeared to have extended about another .5 feet further down slope from Point "m", to an end point here labeled Point "g".¹¹ Adding these segments together, we have that Maragioglio and Rinaldi apparently concluded that the floor length of the entrance passage, from its bottom endpoint (Point "g") to the lip of what could be construed as its upper existing section (Point Ex), was very close to being 196.5 feet.

Comparing the Perring and Maragioglio & Rinaldi Data

The question next to be considered is how the above Maragioglio and Rinaldi result compares with what Perring appears to have found. From the analysis as presented in Figure 1, we can see that Perring implied in his published data that the surviving length of the entrance passage was essentially 201 feet (see Point Ex to Point G in Figure 1 and in Figure 3 below). This is 4.5 feet greater than the 196.5 foot length supplied by the Maragioglio and Rinaldi data.

Three explanations for this fairly large discrepancy present themselves. First, it is possible that the entrance opening had suffered substantial damage in the years since Perring's 1830's investigation, and that this damage had removed another 4.5 feet of material from the outer portions of the entrance area by the time of the Maragioglio and Rinaldi investigation. If this is the case, then going along with a 4.5 foot discrepancy one might expect the floor of the entrance opening to today sit at least 2 feet lower than the 94 foot elevation that has here been proposed for it from Perring's data, and that it would also now be about 4 feet further south of the pyramid's north base edge than the point observed by Perring..¹² Although this is a possibility, it is argued against by other evidence to be presented.

Second, it is possible that Perring made serious errors in his survey of the pyramid's exterior, and that his data should not here be trusted. However, as later corroborated by Dorner's survey, Perring evidently made an accurate determination of the pyramid's original base length, a feat of no mean accomplishment under the circumstances.¹³ We will for the moment, therefore, withhold from dismissing Perring's data out of hand. There is yet a third possibility, this being that a simple error was made either by Perring or an assistant as his field data was being compiled.

As proposed earlier, if Perring had determined that the then existing entrance opening sat at an elevation of 94.15 feet, and given that he had found the slope of entrance passage to be 27°56', then simple trigonometry would have led him to the finding that the hypotenuse of the resulting right triangle would be 201 feet - as is shown by the length G to Ex in Figure 3.¹⁴ Following this, Perring would then have been able to derive the elevation of the original entrance (Ep) as having been 96.26 feet, and also that the hypotenuse (G to Ep) of this new right triangle would be 205.5 feet (this being the distance that he gives for the original passage length). In addition, and as Perring states, according to his data the distance from the then existing entrance to the proposed original entrance location (Ex to Ep) computes to be 4.5 feet.

Since the difference between the 196.5 foot Maragioglio and Rinaldi finding for the existing length of the entrance passage and the 201 foot amount is also exactly 4.5 feet (this being G to Tp), it is possible that Perring or his assistant became confused by the coincidental duplicate appearance of the 4.5 foot quantity in the diagrammatic sketch of the situation (i.e., G to Tp equals Ex to Ep in Fig. 3 above). While correctly subtracting the 4.5 foot length of the now missing portion of the entrance passage (Ex to Ep), there may have been a failure to subtract this serendipitously same 4.5 foot amount again (Tp to G) to get what actual measurement would have shown to be the length of the surviving passage (that is, Tp to Ex = 196.5').¹⁵ In other words, the full length of the hypotenuse of the diagram was mistakenly taken to be the original descending passage length - which it is not. This scenario carries with it the implication that Perring should have found the lower end of the entrance passage to be located about 2 feet above the pyramid's base level (i.e., Tp in Figure 3).

In an effort to clarify this admittedly tricky situation, allow me to restate that it is possible that Perring should have calculated the original as-constructed length of the entrance passage to have been 201 feet (Tp to Ep = 196.5'+ 4.5') - and not the 205.5 foot amount he lists. From the 201 feet he should then have subtracted the 4.5 feet (= Ex to Ep) of the missing exterior casing stones - thus leaving behind the 196.5 foot amount of the extant descending passage length. Instead, the missing 4.5 foot exterior was erroneously subtracted from the 205.5 foot amount (this being the full diagrammatic hypotenuse of the right triangle whose height is 96.26 feet) - thus leaving the mistaken result of an extant passage length of 201 feet. It is not difficult to see how the double appearance of the 4.5 foot quantity in the diagram can be confusing and lead to error. Given all of this, Perring should have found that the entrance passage ends 2.11 feet above the pyramid's base level (and therefore, that it also ends 3.98 feet north of Point G).¹⁶

Before going on to see how all of this squares with Dorner's survey, a brief mention of W.M.F. Petrie's survey of 1887 will be in order.

W.M.F. Petrie's Survey

During his field work at Dahshur in 1887 Petrie was unable to obtain timely permission to clear away rubble and debris from the area, and was therefore unable to perform a thorough survey of the Red Pyramid. He did manage, however, to take careful measurements of the pyramid's "present slope of the rough surfaces of the core masonry".¹⁷ He arrived at slightly differing results for each of the structure's four sides, but for the purposes of the present analysis we will merely note that he measured the slope of the pyramid's northern face to be 44°42', and that he found the average slope of the four sides to be 44°36' - or about one minute of degree greater than Perring's finding of 43°36'.¹⁸ This is a small difference but, as shall soon be seen, a significant one.

Josef Dorner's Survey

Josef Dorner's published survey findings include measurements of the Red Pyramid that were taken on two separate occasions, the first in 1980 and the second in 1997.¹⁹ Dorner's finding for the pyramid's base length was 718.76 feet, or about 8 inches less than the 719.42 feet found by Perring (see footnote 13). His measurement of the Red Pyramid's exterior slope was based upon sightings made along the surfaces of a few of the remaining casing stones on the east face, and his efforts indicated an exterior angle of 44°44'.²⁰ This nearly exactly matches Petrie's result for the pyramid's north face, and thus it is also about one degree greater than the 43°36' claimed by Perring.

Dorner reported that the original, as-built, opening to the entrance passage was located 100.95 feet above the pyramid's base level, and that it was likewise 100.95 feet south of the pyramid's north base edge.²¹ In addition, he implied that the original length of the entrance passage was 204.2 feet, and that it descended into the pyramid at an angle of 26°34'.²² All of these various factors differ from those given by Perring, and perhaps the most straightforward way to proceed next is, oddly enough, to return again to the Perring diagram.

The only difference between Figures 3 and 4 above is that in Figure 4 we are answering the question of what Perring's diagram might have looked like had he, too, measured the pyramid to have had an exterior slope of 44°44'. The extant entrance location in Figure 4 remains at Ex, and so it is located at an elevation (J to Ex) of 94.15 feet, and it is also at its previous distance of 105 feet south of the pyramid's north base edge (J to N). The presumed extant passage length (Tp to Ex) also remains at 196.5 feet.

With the entrance passage extended past Ep to a Point E at the 44°44' side (in Figure 4), it will meet that side at an elevation of 97.61 feet (Z to E), and the proposed original passage length (Tp to E) will now be 203.89 feet. This is only 3.7 inches shy of the 204.2 foot original passage length given by Dorner.

In the scenario of Figure 4 above, the original entrance location (Point E) turns out to be 3.34 feet lower in elevation (that is, 100.95' - 97.61'), and 2.42 feet further north (that is, 100.95' - 98.53'), than where Dorner's data places it (see Figure 5 below). However, as just shown, the length of the entrance passage curiously works out to be very nearly the same as that listed by Dorner. Our next step will be to more fully examine the results of a work-up of Dorner's findings in relation to the proposed Perring analysis.

 

From the details which Dorner presents, it is apparent that he found the extant length along the slope of the entrance passage to be 191.53 feet (Point 1 to Point T in Figure 5).²³ This is nearly 5 feet shorter than the 196.5 foot length delineated for the remnants of the extant passage by Maragioglio and Rinaldi. The implication would therefore appear to be that either Dorner interpreted the entrance passage situation very differently, or between the time of the Maragioglio and Rinaldi investigation in the early 1960's and the 1980/1997 Dorner field surveys, nearly 5 feet of the area around the entrance opening had, for whatever reason, become removed.

It is interesting to note that Dorner's Point S is located 195.59 feet up slope from the bottom end of the passage (Point T), thus making Point S appear to be very close to the location of the upper passage endpoint that had been detailed by Maragioglio and Rinaldi. Dorner emphasizes that Point S is beyond the existing passage floor or ceiling upper endpoint, and he computes its original elevation above the pyramid's base to have been 97.08 feet.²⁴ Furthermore, his data indicates that Point S is located 108.65 feet horizontally south of the pyramid's north base edge (N to Sb in Figure 5).

Perhaps a more useful way in which to compare Dorner's results with the Perring survey data is to visualize a hypothetical Point X positioned 9/10ths of a foot up the passage slope from Dorner's Point S. This imaginary Point X will then be 195.59' + .9' Å 196.5 feet from the bottom end of the entrance passage (as presented by Dorner's survey), and it can therefore be seen as simultaneously coinciding with the upper passage endpoint as had been determined by the Maragioglio and Rinaldi survey and the upper endpoint that has been here postulated to have been that found by Perring in his survey.

The resulting distance of Point X down-slope from Dorner's proposed original opening of the entrance passage (Point Ed above) will be 7.7 feet, (i.e., 9/10ths of a foot less than the 8.6 foot down-slope distance from Point S to Ed).²⁵ As postulated in Figure 4, if Perring had measured the pyramid's exterior angle to have been 44°44', he would have calculated the slope distance from the then surviving passage opening (Point Ex) to the pyramid's 44°44' original exterior (Point E) to have been 7.39 feet. This is only 4 inches less than the 7.7 foot distance from our Point X to the pyramid's original exterior in Figure 5, and as such would appear to logically follow from the fact that the 203.89 foot entrance passage length of the Perring analysis (see Figure 4) so closely matches that registered by Dorner's survey.

However, Point X is located 107.84 feet south of the pyramid's north base edge (as computed from Dorner's findings in Figure 5), this being about 2.8 feet further south than the 105.05 feet that is here proposed for the location of Perring's finding for the extant passage opening (Point Ex in Figures 3 and 4). Furthermore, the elevation above the pyramid's base of Point X in Figure 5 works out to be 97.48 feet, and this is about 3.33 feet higher than the 94.15 feet here proposed for Perring.

We are then left to understand that Dorner's overall findings for the upper end of the pyramid's entrance passage are about 3 feet higher above the pyramid's base than the corresponding Perring results, and that they are also about 3 feet further south of the pyramid's northern base edge than that of Perring. As illustrated in Figure 6 above, the discrepancies for the entire passage do not remain in a constant ratio point to point, but vary somewhat north to south due to the circumstance that the two surveys have different values for the slope of the entrance passage - Dorner having measured the passage angle to be 26°34', while Perring found it to be 27°56'. ²⁶ This fact becomes especially evident at the passage's lower end.

A Final Consideration

Although it ignores the inherent inconsistency in Perring's data, the usual interpretation of his work holds that the 94 foot elevation that he listed for the entrance opening was in reference to an original entrance location, and not to the extant location as he then found it.²⁷ For the sake of completeness, then, Figure 7 explores a treatment of this usual interpretation to see to what extent it correlates with the other Red Pyramid surveys we have considered.

In Figure 7, Point Ex will again represent the location of the existing entrance opening, and Point Ep is again the location where the original entrance meets a presumed 43°36' exterior side. The premise of this scenario is to place Ep at an elevation of 94 feet above the pyramid's base level (note that in Figure 1 it was Point Ex that was considered as being 94 feet above the pyramid's base). Ep is then also at a distance of 4.5 feet upslope from Point Ex (as per Perring's data).

From all of these particulars we can learn that the entrance passage (Ep2) would meet a 44°44' pyramid side at a distance of 96.2 feet south of the pyramid's north base edge. This is 2.33 feet closer to the north base edge than the corresponding original entrance location as given in the proposed Perring interpretation of Figure 4, and it is 4.75 feet closer to the north base edge than that given by Dorner (see Figure 5).

In addition, by adding up the horizontal distances in Figure 7 one arrives at a total distance from the pyramid's north base edge (Point N) to the bottom end of the descending passage (Point G) of 275.99 feet.²⁸ Perring indicates (as does Dorner) that the center of the pyramid's second chamber coincides with the center of the pyramid, and his figures indicate that the center of the second chamber is 75.77 feet southward from the bottom endpoint of the descending passage (Point G).²⁹ Therefore, 275.99 feet plus 75.77 feet yields 351.76 feet. This means that in the scenario of Figure 7, Perring's data would place the center of the second chamber 7.95 feet north of the center of the pyramid, a finding that is in stark contradiction to his other information.³⁰ It is difficult to see a mechanism by which Perring could have overlooked so blatantly extreme an error, and so one must seriously call into question the initial assumption of this section.

The foregoing scenario does, however, offer the interesting consistency with the Maragioglio/Rinaldi data in that the existing length of the entrance passage (G to Ex) here computes to be 196.16 feet (as seen earlier, the Maragioglio/Rinaldi diagram delineates it to be 196.5 feet). However, Figure 7 implies that the full length of the passage extended to a 43°36' exterior side will be 200.67 feet, and so one must then ask how and why Perring mistakenly added a 9.34 foot amount to the extant passage length (i.e., 196.16 feet) in order to arrive at the 205.5 foot original passage length that he lists.

Taken all together, I find the factors which argue against a 'Figure 7 scenario' interpretation of Perring's data to be persuasive.

Concluding Thoughts

In the absence of new survey data for the Red Pyramid, this analysis by necessity leaves a number of questions unanswered. With Dorner's 44°44' finding for the pyramid's exterior slope essentially corroborating Petrie's 1887 survey, we can perhaps assume this particular factor to be largely settled. However, Maragioglio and Rinaldi's finding of 27°56' for the slope of the entrance passage corroborates Perring's measurement of this feature , and these two findings run counter to Dorner's 26°34' determination..³¹ Are we to then accept Dorner's finding for the passage slope simply because it is the most recent?

The relevance of these issues is that we may still not yet know with clear certainty the location of the original entrance to the Red Pyramid's descending passage in relation to the pyramid's exterior north base edge, and thus we may similarly have reason to still be in doubt as to where the central axis of the pyramid lies relative to the pyramid's interior chambers. If the proposed analysis of Perring's findings (as seen in Figure 4) does in fact present the more accurate picture of the Red Pyramid's entrance passage location, then this would imply that the bottom end of this passage (Point Tp in Figure 4) lies nearly 4 feet further to the north than is presently realized. If so, this would then indicate that the center of the pyramid's second chamber is not - as is presently believed - coincident with the center of the pyramid, but is rather about 4 feet to the north of it.

Such a conclusion is in direct contradiction with Dorner's findings. It is also in contradiction with the conclusion that Perring himself presents, although this essay has endeavored to show the likely manner in which Perring erred in compiling his own survey data. Given the differences between the Perring and Dorner findings, and given the issues raised in this essay from an analyses of these two surveys, it is to be hoped that a resurvey of the Red Pyramid's entrance passage situation will in the near future be undertaken..³² With the Red Pyramid being of such studied importance, it would be well to resolve any possible doubts that there might be regarding the precise location of the internal passages and chambers relative to this pyramid's central axis.

Notes

1. H. Vyse (with J.S. Perring), Operations Carried On at The Pyramids of Gizeh in 1837, Vol. 3, (London, 1842); W.M.F. Petrie, Pyramids and Temples of Giza, (London, 1883) and also A Season in Egypt, 1887, (London, 1888); V. Maragioglio and C. Rinaldi, L'Architettura Delle Piramidi Menfite, Parte 3, (Rapallo, 1964); J. Dorner, 'Neue Messungen an der Roten Pyramide', in Heike Guks (ed.), Stationen: Beiträge zur Kultur- geschichte Ägyptens. R. Stadelmann gewidmet,(Mainz, 1998).

2. This is true in both Vyse's Operations Carried On cited above, and in Perring's own The Pyramids of Gizeh From Actual Survey and Admeasurement (London, 1839). As regards the Red Pyramid, the text is the same in both works. However, there appear to be minor differences between the sets of accompanying drawings.

3. Vyse, op. cit., 65.

4. Ibid., 64.

5. Ibid., 64.

6. Perring implies that the lower end of the entrance passage meets a horizontal passage, and that the floor of the First Chamber is simultaneously "on a level with the base of the Pyramid" and at the same level with this horizontal passage. This is indeed what is depicted in his drawings. See Vyse, op. cit., 64, and also Figure 1 page 64a, and Figure 1 page 64b.

7. The sine of 27°56' is .46844, and so "h" in Figure 1 is found from h/4.5' = .46844. Ep to K is then 94.15'+2.108' = 96.258', and Ep to G = 96.258'/.46844 = 205.5 feet. Note that with Ex at 94.15 feet, the length Ex to G then becomes 201 feet.

8. Maragioglio and Rinaldi, op. cit., 128 and Plate 18 fig. 3.

9. Maragioglio and Rinaldi, op. cit., Plate 19.

10. Ibid. On page 128 of the text the ceiling length is listed as 58.8m, although on Plate 19 it is labeled as 58.82m. Note that I have added the 43°36' pyramid exterior side to Figure 2 for analysis purposes. In their original scaled drawing Maragioglio and Rinaldi included a 44°36' pyramid exterior side (which, as will be explained shortly, is based on W.M.F. Petrie's finding). They also apparently tried to fit Perring's data into this scheme by assuming that Perring had begun his measurement from the lower end of the entrance passage ceiling - and not it's floor. One reason for their doing this may have been that Perring makes it clear that the lower end of the passage was filled with much rubble, implying that he had difficulty making a direct measurement from the floor itself. However, since Perring states that the perpendicular height of the horizontal passage was about 6 inches greater than that of the entrance passage it seems likely that he would have added to his passage floor measurement at least the 2 foot amount listed in the present essay's Figure 2 as the distance between Points m and R. None of this, however,affects the Maragioglio and Rinaldi finding for the existing length of the floor as they found it. Note that from here on in this essay I will be converting into feet all measurements originally reported in meters.

11 The reason for the added .5 feet is that Maragioglio and Rinaldi's measurements show that the point at which the ceilings of the horizontal and entrance passages meet lies about this amount north of where the lines of the passage floors meet. The Maragioglio and Rinaldi survey lists a 7.5 inch difference in height between these two passages, Perring lists a difference of 6 inches. W.M.F. Petrie, in his Pyramids and Temples of Gizeh, op. cit., 144, agrees with Perring's finding that the height of the entrance passage is 47.5 inches. (He provides no measurement for the horizontal passage.) Dorner, op. cit., 27, on the other hand, gives an average height for the entrance passage of 52.5 inches - with this basically matching the height found by both Perring and Maragioglio/Rinaldi for the horizontal passage It is possible that Dorner may here be giving not the perpendicular height across the entrance passage (as for instance seen in Line w-r in Figure 2), but rather its vertical height (e.g., Line w-m in Figure 2).

12. Perring found the entrance passage to have a slope of 27°56'. The sine of this angle is .46844, and the cosine is .8835. Therefore, every foot of run along the passage slope results in about a 1/2 foot change in elevation and about an 8/9ths foot change in lateral distance. Dorner's figures, op. cit, 26, indicate that he found the existing opening to be at an elevation of 94.75 feet above the pyramid's base (as opposed to the 94.15 foot elevation here proposed for Perring).

13. Perring found the base length to be 719.42 feet. Dorner, op. cit., 25-26, lists it as 718.76 feet. Dorner mentions (p. 23) that due to the great amount of rubble covering the pyramid's base a direct measurement of the border lengths at the base level was not possible.

14. 94.15 ÷ .46844 (the sine of 27°56') yields a hypotenuse of 201.

15. It should be noted that the possibility exists that the G to Tp amount may not have been precisely 4.5 feet, but that nevertheless the inadvertent error was made to not subtract out this amount from the 201 feet.

16. As explained in footnote 6 of this essay, Perring all but specifically states that the entrance passage ends at the pyramid's base level. It is perhaps worthwhile to reemphasize that Perring's data, as he presented it, does not add up. An error was made somewhere along the way. See footnote 27 below.

17. Petrie, op. cit., A Season in Egypt, 27.

18. Ibid., 27.

19. Dorner, op. cit., 23.

20. Ibid., 25. Petrie, op. cit., A Season in Egypt, 27, gives a measure of 44°32' for the angle of the core masonry of the pyramid's east face.

21. Dorner, op. cit., 25-27, refers to the pyramid's exterior slope as expressing a "1:1 angle relationship", implying it slopes upwards at 45°. This runs counter to his earlier statement that he found the average for the exterior angle to be 44°44'. This difference of only 16 minutes translates to about a 1 foot difference either in the elevation of the opening, or in the lateral distance of the opening from the north base edge. Which of these is actually the case would appear to depend upon which of these two parameters (elevation or lateral distance) Dorner determined by direct measurement, and which he determined as a computational resultant of the other.

22. Ibid., 26 -27. Dorner does not directly give the original length along the slope of the entrance passage, but instead states that the slope of the passage is "1:2" (i.e. 26°34'), and that the passage's full, as originally-built, horizontal distance is182.64 feet. The length along the slope can then be calculated from these data.

23. Ibid., 26, Table 2 and Illustration 2. As per the previous footnote, it is assumed that Dorner computed these horizontal distances based on his field measurements of the entrance passage angle and of the length along its slope. Note that T, 1 and S are the designations used by Dorner in his Illustration 2.

24. Ibid., 26. Dorner's Illustration 2 clearly shows that Point S sits above and beyond the existing passage endpoint, and his Table 2 does not list an 'as now' elevation for it - as it does for all of the other points listed - it merely tabulates its 97.08 foot elevation 'as original'. Dorner adds on p. 27 that "the axis Point S" is located at the corridor's upper end "where the floor is ripped out".

25. The 8.6 foot distance is as computed from Dorner's data.

26. This difference in findings for the entrance passage slope is one aspect whichallows both published surveys to ostensibly locate the bottom end of the entrance passage at the same horizontal distance south of the north base edge. However, Dorner places this lower endpoint at 9.61 feet above the pyramid's base. Perring, in both his drawings and in his text, appears to have found the entrance passage to end at the pyramid's base level. As is argued here, I believe it likely that given a correct interpretation of his data Perring would have found the entrance passageto end about 2.1 feet above the pyramid's base level (see Figures 3 and 4).

27. If one assumes an original entrance location at a 94 foot elevation, and given an entrance passage slope of 27°56', then from the sine of this angle one will find the hypotenuse of the triangle (and hence an entrance passage length) to be 200.67 feet, and not the 205.5 feet given by Perring.

28. The tangent of 27°56' is .53022, and so G to P1 is found from 94' ÷ .53022 = 177.29'. Following this, 177.29' + 98.7' = 275.99'.

29. Vyse, op. cit., 64-65, 64a, and Figures 3 and 5. Dorner, op. cit., 28, Fig. 3 and Table 3. Dorner's measurements add up to 75.79 feet for this same extent.

30. Vyse, op. cit., 65. Perring measured the pyramid's full base length to be 719.42feet, and so his half-base length is 359.71 feet.

31. Maragioglio and Rinaldi, op. cit., 128.

32. I refer here to Dorner's assumption (op. cit., 25) that the pyramid was designed on the basis of a 45° exterior slope angle, even though the average of his own measurements yielded 44°44'; and to Dorner's finding of 26°34' for the slope of the entrance passage - a value whose tangent is 1/2. Dorner may have fallen victim to Petrie's observation (Petrie, op. cit., Pyramids and Temples, 163) that some of the pyramid slope angles that researchers report in their studies are "so extremely near to theoretical angles, that they seem to have been modified by the observer". As I have endeavored to show, small changes in these values can lead to significant differences when making a determination of the location of interior features relative to the pyramid's exterior.

References

Dorner, J. 1998. "Neue Messungen an der Roten Pyramide" in: Heike Guks (ed.), Stationen: Beiträge zur Kulturgeschichte Ägyptens. R. Stadelmann gewidmet. Mainz.

Lehner, M. 1997. The Complete Pyramids. London: Thames and Hudson.

Maragioglio, V. & Rinaldi, C.A. 1964. L'Architettura Delle Piramidi Menfite Parte III . Rapallo.

Perring, J. S. 1839. The Pyramids of Gizeh From Actual Survey and Admeasurement. London.

Petrie, W. M. F. 1888. A Season in Egypt, 1887. London. Petrie's text is currently available on-line at: http://digi.ub.uni-heidelberg.de/sammlung6/allg/buch.xml?docname=Petrie1887

----------- 1883. The Pyramids and Temples of Gizeh. London. The text of Petrie's original 1883 publication is currently available on line at http://www.ronaldbirdsall.com/gizeh/index.htm.

Rossi, Corinna. 2004. Architecture and Mathematics In Ancient Egypt. Cambridge.

Vyse, H. 1842. Operations Carried On at the Pyramids of Gizeh in 1837, V. 3. London