Grand Gallery

Section 5.5.1 

The 10th of January 1988 when I and my wife came out from the ascending passageway and could straighten ourselves up, we saw the amazing Grand Gallery. 

A breathtaking view I will never forget. 

For this experience I owe King Khufu, his genius architects, his administrative and logistic staff  and also all his specialized workers many thanks for this fantastic result. The Grand Gallery looked like the path to heaven - (maybe this was exactly the intention).


Section 5.5.2

Here is a picture taken by Edgar in 1910 from the floor, the lowest part, of the Grand Gallery, looking South. Behind the helper you can see the passageway into the "Queens chamber".
You can hardly see the end of the Grand Gallery, it is ascending into the darkness like the sky in the night.

552 Grand Gallery gulv III DSC09475jpg

Source: The Great Pyramid Passages And Chambers" Volume 1 by John and Morton Edgar, 1910


Section 5.5.3

Flinders Petrie had these valuable comments to the angle and position of the Grand Gallery floor:
"For the angle of the passage, and its straightness, it will be well to consider it all in one with the gallery floor, as they were gauged together all in one length."
(S5orig-[S39]-P64-L27-29)

"This , when corrected for lower signal being .3 too high, gives 26° 12´ 50" for mean angle of both passage and gallery together."
(S5orig-[S39]-P65-L2-4)

My comments:
26° 12´ 50" = 26,2139°

"Hence the floor of the gallery intersects the S. wall at 1689.0 +/- .5 above the pavement; at 61.7 +/- .8 S. of the Pyramid centre;  "
(S5orig-[S45]-P72-L18-19)

Please observe the lengths informed by Flinders Petrie are always in British inches.
1689 inches = 81,91 cubits. ( = 819 pixels )
61,7 inches  = 2,99cubits   ( = 30 pixels )

From this we can calculate the coordinate where the floor of the Grand Gallery intersects the Southern wall, which of course is an imaginary point due to the top stone above the ramp.
For a better understanding please see my drawing below, the coordinate we are looking for is in the end of the dotted line:

The coordinate is: (4400/2-30, 2800-819 ) =  2170;1981
Now we have the direction and the point where the imaginary floor line meets the South wall of the Grand Gallery.


For 3D designers the line on the floor level from POI 3663;2717 to  the point where the imaginary floor line meets the South wall of the Grand Gallery 2170;1981 is:

Y= 0,49297X + 911,26122

Please observe that rounding of the numbers in the coordinates creates here and there very small deviations. If you for example calculate the angle of the ascending passageway inclusive the Grand Gallery based on the above mentioned coordinates you will get 26,2418°.  Compared to the accepted angle of  26,2139° there is a deviation of 0,0279°.  

Section 5.5.4

Let us start where we came into the Grand Gallery from the ascending passageway.  The old picture below is showing the end of the ascending passageway where it continues a bit into the Grand Gallery. On the right side you can see the lowest part of the eastern ramp-bench.


554 Plate CXLVIXPNG

Source: The Great Pyramid Passages And Chambers" Volume 1 by John and Morton Edgar, 1910

It seems there had been a mechanical abrasion in the middle part of the floor of the end of the ascending passageway. Please observe the picture was taken before 1910, which is before the massive entry of tourists as we see nowadays. There is a similar abrasion in the top stone in the high end of the Grand Gallery. Please see section 5.5.9

I assume the abrasions were caused during the building of the pyramid.



Section 5.5.5

If we turn around, then the left (eastern) side and the right (western) side at the floor of the Grand Gallery differs from each other. The workers removed a couple of stones on the western side of the ramp-bench near the northern wall to make a passage downwards to the descending passageway. The mouth of this passageway is shown here by some drawings and a photo by Mr. Edgar. 
555 Plate CXLVIIPNG
Source: The Great Pyramid Passages And Chambers" Volume 1 by John and Morton Edgar, 1910


Underneath is a drawing which shows the well from another angle.

555 Plate CXLVPNG
Source: The Great Pyramid Passages And Chambers" Volume 1 by John and Morton Edgar, 1910

The old photo underneath shows the same.  There is still a part of the original stone in the western corner of the Grand Gallery.
555 Plate CXLVIPNG
Source: The Great Pyramid Passages And Chambers" Volume 1 by John and Morton Edgar, 1910



Section 5.5.6

Let us have a closer look on the well showing the western wall and ramp-bench in the Grand Gallery to the left and the end of the ascending passageway to the right:


556 REV1 Entrance and ascending passage and Grand Gallery detail start - west view med blank closerPNG


(I have made a full line to show the supposed original intention of the corner where the upper part of the ramp-bench hit the Northern wall. Above picture in the previous section 5.5.5, you can see the corner is broken.)

Flinders Petrie measured the position of the well from the Northern wall:
"As, however, the position of its mouth has been supposed to have a meaning, it should be stated that the opening is from 21,8 to 49,0 horizontally from N. wall of gallery on floor."
 (S5orig-[S46]-P74-L26-28)

My comments:
21,8" = 1,06 cubits RM
49,0" = 2,38 cubits RM

Flinders Petrie measured also the offset on the eastern ramp to the floor from the North wall to the the position of the well from the North wall to the ramp end in the South:
"20,9"
 (S5orig-[S45]-P71-L24)

My comments:
20,9" = 1,0 cubit RM
Please observe that the offsets on the western and eastern ramp-benches varies from 20,7" (1,00 cubit) to 21,5" (1,04 cubits) and from 20,3" (0,98 cubits) to 21,6" (1,05 cubits), respectively.

On this point of the western ramp-bench the vertical height is :
1 / cos 26,2139° = 1,1 cubits

The mouth of the well in North South direction is then 2,38 cubits RM - 1,06 cubits RM = 1,32 cubits.


Please see additional calculations in section 5.5.7 regarding the floor level of the Queens chamber.
The height from the floor of the Queens chamber is 2384 - 2375 = 9 pixels = 0,9 cubits.

0,9 cubits - 0,52 cubits = 0,25 cubits


Concerning the depths of the Well, please see figure of Edgar in the section 5.5.7



With measures in cubits:


And coordinates:


Section 5.5.7

Calculations for the coordinates:

557 calcPNG

Special remarks:
Flinders Petrie has a measure from ground zero to the floor level 52" = 2,52 cubits from the door opening which is 858,4" = 41,6295 cubits, which corresponds to 416pixels.
In the calculation no. 5 I refer to this. My ground zero has the value 2800 minus 416 = 2384
The point of the mentioned measure is (2990 - 25, 2384) = 2965,2384
557 tablePNG
Source:  (S5orig-[S40]-P66-L4)


*In calculation no. 6 I have added 10 pixels = 1 cubit to find the coordinate. Please see the Plate XIII underneath made by Edgar. I have marked the height yellow.
This value of the height in the well is assumed, I cannot verify it in any sources. I have tried to compare drawings and photos, but in vain - so far.

557 Plate XIIIPNG
Source: The Great Pyramid Passages And Chambers" Volume 1 by John and Morton Edgar, 1910



Section 5.5.8

Internal notes for me only:
The drawings in section 5.5.6 are "close view" versions with another solution. Therefore the calculations and coordinates differs from the overview drawing.
The solution is 446 pixels per 2.6 cubits, which is 171.5 pixels per cubit.

1. 1043-182, 739-86 = 861,653
2. 861-226, 653-103 = 635,550
3. 635, 550-189 = 635,361
4. 635, 366+189+154 = 635,704
5. 635, 704+172 = 635,876
6. 635+226, 876 = 861,876

1,1 cubits = 189 pixels is the vertical height between ramp floor and ramp-bench.
0,9 cubits = 154 pixels is the vertical height between the floor to the "Queens chamber" and the ramp floor extended line.
0,6 cubits = 103 pixels is the vertical height between the ceiling of the passageway to the "Queens chamber" and the ramp floor.
1,0 cubits = 172 pixels is the vertical height between the floor of the well and the floor of the passageway to the "Queens chamber" .
0,33 cubits = 51 pixels is the vertical height of the step at the end of the ascending passageway.
1,3 cubits = 223 pixels is the vertical height between the floor of the well and the end of the ascending passageway .
1,32 cubits = 226 pixels is the horizontal length in the bottom of the well.
0,5 cubits = 86 pixels is the vertical height between the floor of the G.G. and the end point of the floor of the ascending passageway in G.G.
1,06 cubits = 182 pixels is the horizontal length from the well to the Northern wall in G.G.
1,5 cubits = 257 pixels is the vertical height between the ramp floor extended line and the end point of the ceiling of the ascending passageway.
1,1 cubits = 189 pixels is the vertical height between the floor of the ascending passageway and the ramp floor extended line at the Northern wall in G.G.
2,6 cubits = 739-293 = 446 pixels is the vertical height in the ascending passageway.
1043,293 is end point ceiling of ascending passageway
1043,550 is end point ramp-bench into the North wall of G.G.
1043,739 is end point floor of ascending passageway
 


Section 5.5.9

Let us take a detailed look into the Southern part of the Grand Gallery. As mentioned in section 5.5.3 we have the direction and the point where the imaginary floor line meets the South of the Grand Gallery which is (2170,1981).

First an overall view:

559 Grand Gallery overall without measues and coordinatesPNG

The complete length of the Grand Gallery:

559 Grand Gallery overview whole length without measues and coordinatesPNG

And closer to the top stone:

559 Grand Gallery top stone without measues and coordinatesPNG

Flinders Petrie:
"At the upper end of the gallery, we have already stated the S. wall to be 61,7 +/- ,8 S. of the Pyramid centre; and hence the face of the great step at the head of the gallery (which descends behind both floor and ramps) is (61,7 - 61,3) = ,4 +/- ,8 S. of the Pyramid centre."
Source: (S5orig-[S46]-P74-L33-36)

My comments:
The mentioned measures above are as usual i inches.
61,7" = 2,99 cubits RM
61,3" = 2,97 cubits
0,4"    = 0,02 cubits
My drawings is limited to an accuracy of 1 decimal in cubits, so the width is rounded up to 3 cubits RM. 
The 0,02 cubits cannot not be shown in my drawing as the measure is below the 1 pixel, but in this case I have chosen to show it on the drawing even 1 pixel is equal to 0,1 cubits, which is 5 times bigger than the real value. This decision does not have an impact on the position of the South wall.

Flinders Petrie:
"And the sloping floor seems to be also out of level by an equal amount in the opposite direction; since on the half width of the step (i.e. between the ramps) the height of the step face is 34,92 or 35,0 on E., and 35,80 or 35,85 on W."
Source: (S5orig-[S46]-P75-L9-11)

My comments: 
On East side:
34,92" = 1,69 cubits RM
35,0"   = 1,70 cubits RM
On West side:
35,80" = 1,74 cubits RM
35,85" = 1,74 cubits RM
It is clear the top stone or "step" is not perfectly horizontally. In my drawings it does not really matter as the accuracy is limited to +/- 0,1 cubits. The result is therefore the same on both sides : 1,7 cubits RM (which is 17 pixels).

We know from section 5.5.6 that the vertical height of the ramp-bench is 1,1 cubits. 
The vertical height from the ramp-bench to the edge of the top stone is therefore 1,7 cubits - 1,1 cubits = 0,6 cubits.

Flinders Petrie:
"Then the top of the step will (by above measures) be here 34,88 above actual floor end, and the step dips about ,64 to the S.wall at this part; so the top of the step at the S. wall is 34,88 + ,64 -30,08 = 4,16 (say+/-,2) above the virtual floor end at the line of taping."
Source: (S5orig-[S46]-P75-L18-21)

My comments:
34,88" = 1,69 cubits RM
0,64"    = 0,03 cubits RM
30,08" = 1,46 cubits RM
4,16"    = 0,2 cubits

With measures all in cubits:

559 REV1 Grand Gallery top stone with measures and without coordinatespng

It is impressive how close the Northern edge of the top stone is on the Apex - the middle of the pyramid. The distance is 0,02 cubits RM is equal to 1 cm !
I assume it was the intention to place the edge of the top stone in the middle of the pyramid. The idea might have been to mark a point between the ascending structure and the entrance of the passageway to the antechamber and the "Kings" chamber, which hereafter is horizontal.  

With coordinates:

559 REV 1 Grand Gallery top stone without measures and with coordinatespng

From section 5.5.3 we calculated the coordinate where the floor of the Grand Gallery intersects the Southern wall is (2170,1981).
The top stone intersects with the southern wall at (2170,1981-2) = (2170,1979).
The North-South length of the top stone is 3 cubits. the northern edge is therefore (2170 + 30,1979) = (2200,1979) which is shown in the drawing as ("2201",1979) to mark the small difference from Apex.
The height from the edge of the top stone to the ramp-bench is 0,6 cubits, the ramp intersects to the top stone at ("2201",1979 + 6) = ("2201",1985).
The vertical height of the ramp is 1,1 cubits, the floor of the ramp  intersects to the top stone at ("2201",1985 + 11) = ("2201",1996).

A much closer look on the intersection between the imaginary line and the upper face of the top stone is here:
559 Top Stone new drawing offset detail Ijpg

As you can see the imaginary line , which is prolonged from the floor of the ramp, hit the floor of the Antechamber only 0,4 cubits (=21 cm) from the edge of the Southern wall in the Grand Gallery.

Charles Rigano mentioned "The single block forming the Step does not stop at the Gallery wall but extends an additional 66" into the Ante Chamber".
Source: (S11-P53-L15-16)
My notes:
66" = 3,2 cubits

Flinders Petrie mentioned in a table:

559 Lngde af top stone i h t FlindersJPG
Source: (S5orig-P75-table)

My notes: 
61,32" = 3 cubits RM (Face of the Northern edge of the step to the Southern wall of Grand Gallery)
64,90" = 3,14 cubits RM (Length from the Southern wall to the joint where the granite begins)
As Flinders Petrie measured along the floor and the top stone is a lime stone then the length of the top stone in the North - South direction is 3 cubits + 3,14 cubits = 6,14 cubits RM in total.

I think it is an interesting structure as the Southern wall secure the top stone and prevents the stone from moving - especially downwards into the depths of the Grand Gallery. The top stone could withstand a huge downwards force during the construction of the pyramid. 

Are there any indications of forces which had been applied to the top stone? 

Edgar took a photo of the top stone, which gives you a better idea how it looks like. It was taken against the South - West corner of the Grand Gallery:

559 Topstone i Grand Gallery plate CVXIPNG
Source: The Great Pyramid Passages And Chambers" Volume 1 by John and Morton Edgar, 1910

Nowadays the top stone has been "repaired", maybe to avoid any accidents when tourists move around on the stone.
It is more or less the same kind of a mechanical abrasion in the middle part of the top stone as we saw in the floor of the Grand Gallery, see in section 5.5.4 . In 1910 it could still be seen. If you examine the picture carefully you can see a trace in the middle of the eroded part of the top stone. It looks like a rope has been running there. If this is so, then the top stone somehow has been a part of the building process of the pyramid.

Lowest in the picture you can see the ascending floor of the Grand Gallery and on both sides there are ramp-benches leading up the top stone. On the Western side a piece of the ramp-bench has been broken off. 

In the background you can see the entrance to the small passageway to the antechamber and hereafter the "Kings" chamber.


Section 5.5.10

Let us have a closer look on the old picture.

5510 Topstone i Grand Gallery plate CVXI closerPNG

Source: The Great Pyramid Passages And Chambers" Volume 1 by John and Morton Edgar, 1910

1. It is not possible to judge whether the missing part of the top stone is due to an abrasion or made for a purpose. If you ignore the angle from which the photo is taken, then it seems the top stone has a smooth surface in the rounded part from the vertical center of both the ramp-benches . There is a possibility the shape has been made by the workers for a purpose.
2. If you examine the line under "2", then it starts from the very top of the top stone and ends at the bottom of the rounded part.
3. Where the line "2" ends you see a clear round abrasion, which could have been made by a running rope. Under the rounded abrasion is a vertical abrasion, which could have been caused by a rough rope or by water to cool down the rope caused by the friction against the stone.
4. Just above the "4" point there is a line that could have been caused by a stone which the workers dragged up and hit the top stone hard because of an accident.

If any of my readers have another explanation, please let me know.

I have tried to make a drawing by hand from the North side:

5510 Top Stone new drawingjpg

It is hard to figure out how far the abrasion hit the top stone in the North - South direction, but a photo taken nowadays give us a good indication:

5510 Top stone after repairjpg
Source: https://commons.wikimedia.org/wiki/File:Herses-grande-pyramide.jpg

The line which forms a half circle behind the banister is the border between the original top stone and the repaired part. The repaired part seems to begin on the middle of the top stone.
 
The top stone seen from above:

With measures:

The width of the holes are 0,327 cubits RM or close to 1/3 cubits.

The East-West coordinates are:


Eastern hole :
2041;2200-30 = 2041 (E-W) ;2170 (N-S)
2041;2170+10 = 2041 (E-W) ;2180 (N-S)  
2041+3;2180 = 2044 (E-W) ;2180 (N-S)  
2044;2180-10 = 2044 (E-W) ;2170 (N-S)  

Western hole:
2081;2200-30 = 2081 (E-W) ;2170 (N-S)  
2081;2170+10 = 2081 (E-W) ;2180 (N-S)  
2081-3;2180 = 2078 (E-W) ;2180 (N-S)  
2078;2180-10 = 2078 (E-W) ;2170 (N-S)  


I have tried to make a 3D drawing of the top stone seen from North to South. I apologize for the low quality of my work - I am not used to make these drawings :
You can read the measures and the coordinates are mentioned below in a table.

5510 Top Stone 3d rev2jpg

The coordinate 1979  in cursive indicates the top stone is 0,05 cubits higher in the Western side than the Eastern side, but the measure is below the accuracy in my general drawings. 


Section 5.5.11

Let us focus on the top surfaces of the ramp-benches. On the picture in section 5.5.2 you can see rectangular holes cut down in the ramp-benches close to the walls. You can also see niches behind the holes cut into the walls.

Unfortunately Flinders Petrie did not register the number of holes, but he had following remarks:
"One remarkable point is that the holes are alternately long and short, on both sides of the gallery; the mean of the long holes is 23´32, with an average variation of ´73, and the mean of the short holes is 20´51, with average variation of ´40. Thus the horizontal length of a long hole is equal to the sloping length of a short hole, both being one cubit. This relation is true within less than half their average variations."
Source: (S5orig-[S46]-P72-L27-32)

My remarks:
As the sloping lengths for each hole and the spaces between them are not mentioned I must disregard from the average variations. Furthermore, it seems the variations are relatively small which confirms my decision.
Long hole: 23´32 inches = 1,13 cubits RM;   ´73 inches = 0,04 cubits 
Short hole: 20´51 inches = 1,0 cubit RM;      ´40 inches = 0,02 cubits

Another important measure Flinders Petrie mentioned:
"Ramp End ... 1815´5 Distance on Slope"
Source: (S5orig-[S45]-P71-L39)

My remarks:
1815,5" = 88,0456 cubits RM, which in my drawings I round down to 88 cubits.

Charles Rigano wrote following remarks:
"At the top of each platform 25 niches are cut into each wall (total 50) most of which have an angled trapezoidal feature chiseled across its face. There are 27 rectangular holes cut downwards into each side platform (total 54). The rectangular holes are next to the walls and generally centered on the niches cut into the side walls; the two rectangular holes against the north (lower) face of the Gallery do not have companion side niches.
All of the 50 side niches and trapezoidal features are completely filled with blocks and mortar of the type used elsewhere in the pyramid with the exception of the 7th and 11th side niches from the bottom on the west side, the first of which is empty and the second has a hole cut into the mortar. Because the depth of only two side niches is known, we cannot determine if opposing side niches were of different depths to facilitate insertion of a beam as is the case with the holes at the entrance to the Upper Horizontal Passage".
Source: (S11-P51-L16-26)

On the same page he wrote in a table :
"Depth (Into Platform)    7,0" - 11.5"  
Width (From Gallery Wall)  5,5" - 6,5"
Source: (S11-P51-L10 in the table)

On page 50 Charles Rigano mentioned following measures in the table:
"Width - Bottom Including Side Platforms  6´  10" (4 cubits)
Width - Between Side Platforms  3´  5"  (2 cubits)
Side Platforms   20,5" Wide  20,5" High (1 cubit square)"
Source: (S11-P50-L7 in the table)

My remarks:
7" = 0,34 cubits RM and 11,5" = 0,56 cubits RM. (depths)
5,5" = 0,27 cubits RM and 6,5" = 0,32 cubits RM. (widths)

Luca Miatello inform following in source 15:
"The holes are 14 cm wide, 18 cm deep, and their mean length alternates regularly between 52,1 cm (one cubit) and 59,2 cm (Petrie, 1883:72)."

My remarks about the holes in the ramp-benches:
14 cm = 0,27 cubits RM = 0,53 foots. (wide)
18 cm = 0,34 cubits RM = 0,49 remens. (depth)
52,1 cm = 1 cubit RM = 7 hands. (short length)
59,2 cm = 1,13 cubits RM = 8 hands. (long length)

There are 27 holes on each ramp-bench.
If you see the picture in section 5.5.9 it is obvious there is no hole adjacent to the top stone.
In section 5.5.5 you can see the picture showing where the ascending passageway end in the Grand Gallery. In the corner you can see the first hole. Taking in consideration that the hole is shorter than the horizontal length of the end of the floor of the ascending passageway, which is 1,06 cubits, I conclude the first hole is the short one of 1,0 cubit RM.



Section 5.5.12

The spaces between the holes are not measured, so I have to assume the sloping lengths of the spaces are the same.
As the ramps start with a short hole of a sloping length of 1 cubit RM and alternates with the long holes of the sloping length of 1,13 cubits RM and there are 27 holes in each ramp-bench and the series of holes and spaces end with a space and the total sloping length of each ramp-bench is 88,0456 cubits RM, we can calculate the sloping length of each space.
In fact we can see it as 13 sequences consisting of a short hole + a space + a long hole + space and in the end there is one short hole and a space before the top stone.
 
5512 calcPNG

The pixels are as usual for internal use only and are processed to calculate the coordinates and the drawings (10 pixels = 1 cubit).
The horizontal axis is calculated as the sloping length x COS 26,2139°.
The vertical axis is calculated as the sloping length x SIN 26,2139°.   



Section 5.5.13

The bottom of the holes have the same slope as the ramp-benches and the inner sides of the holes are perpendicular to the bottoms, see the drawings underneath.

It took some weeks before I realized the sides of the holes were not vertical, but perpendicular to the surface of the bench-ramp. This little misunderstanding cost me a lot of efforts to correct the calculations underneath. The incident also showed me with a cruel clarity how the small grey cells may mislead me. Afterwards I understood the holes may have been made before the stones were mounted in the Grand Gallery. 

There are two different sloping lengths L of the holes on these ramps: 1 cubits RM and 1,13 cubits RM.
Please see my drawing below with a horizontal line:


5513 Real hole in GG rev1JPG

Please see the drawing below of Mark Lehner showing an example of a hole (3) (here called notch) in the ramp (1), the niche (4) and the trapezoidal cutting (5) in the wall (2)  parallel to the ramp: 
5513 tegning af hul i rampe MarkPNG
Source S15,  "Niches, Slots, Grooves and Stains: Internal Framework in the Khufu Pyramid?" of Mark Lehner published in https://www.academia.edu/36645718/Niches_Slots_Grooves_and_Stains_Internal_Frameworks_in_the_Khufu_Pyramid 

Please see my detail drawing of the holes  of the Grand Gallery. It shows the holes in the eastern ramp-bench, the western ramp-bench is similar except the escape passageway near the northern entrance. Please see the coordinates underneath in section 5.5.14
s = Short hole
l = Long hole

Eastern ramp-bench:
5513 drawing of holes Eastern side ramp benchJPG


Western ramp-bench:
5513 rev 2 Western side Grand Gallery overview whole length without measues and coordinates with holes HIGH solutionpng

The corner at the Northern wall has been broken, please see the pictures in section 5.5.5, it is not possible to draw this section correctly. However in section 5.5.6. I have made a detail drawing to show the supposed original intention of the corner.

Section 5.5.14

As mentioned in section 5.5.11 the holes in the ramp-benches have these dimensions :
14 cm = 0,27 cubits RM = 0,53 foots. (wide)
18 cm = 0,34 cubits RM = 0,49 remens. (depth)
52,1 cm = 1 cubit RM = 7 hands. (short length)
59,2 cm = 1,13 cubits RM = 8 hands. (long length)

And according to section 5.5.12 the spaces between the holes are 2.198 cubits each.

As the holes alternate as short and long, all the holes have following lengths and coordinates on both ramp-benches (the western ramp-bench does not have the first hole "1s" at the Northern wall due to the escape passageway or well.):

According to section 5.5.6 the ramp-benches start at 2990,2375 and end at 2200,1985 according to section 5.5.9
(Please observe due to rounding the end point in the table is calculated to 2200,1986 which divert from the real coordinate.)

5514 Oversigt over coordinater West and East REV 3JPG

Calculations:
Sloping lengths at 26.2139° , cubits : The sloping lengths of the short and long holes are measured, the sloping length of the Spaces of 2.198 cubits is calculated in section 5.5.12
The total of cubits is a simple addition ends with 88.038 cubits.
The pixels used for the drawing in section 5.5.13 is calculated as cubits x 10

Lengths, cubits, X axis for the short hole: 1.0 cubit RM x cos 26.2139° = 0.8972 cubits (= 0.9 cubits rounded).
Lengths, cubits, Y axis for the short hole: 1.0 cubit RM x sin 26.2139° = 0.4417 cubits (= 0.4 cubits rounded).
Lengths, pixels, X axis for the short hole: 0.9 cubits x 10 = 9 pixels.
Lengths, pixels, Y axis for the short hole: 0.4 cubits x 10 = 4 pixels.

Lengths, cubits, X axis for the Space: 2.198 x cos 26.2139° = 1.9719 cubits (= 2.0 cubits rounded).
Lengths, cubits, Y axis for the Space : 2.198 x sin 26.2139° = 0.9709 cubits (= 1.0 cubits rounded).
Lengths, pixels, X axis for the Space : 2.0 cubits x 10 = 20 pixels.
Lengths, pixels, Y axis for the Space : 1.0 cubits x 10 = 10 pixels.

Lengths, cubits, X axis for the long hole: 1.13 cubit RM x cos 26.2139° = 1.013 cubits (= 1.0 cubits rounded).
Lengths, cubits, Y axis for the long hole: 1.13 cubit RM x sin 26.2139° = 0.4991 cubits (= 0.5 cubits rounded).
Lengths, pixels, X axis for the short hole: 1.0 cubits x 10 = 10 pixels.
Lengths, pixels, Y axis for the short hole: 0.5 cubits x 10 = 5 pixels.

Coordinate 2990,2375 is the start point at the Northern wall where the ramp-benches hit.
The upper Northern point of the first hole (on Eastern ramp-bench) is therefore 2990,2375.
All upper coordinates of the holes and Spaces are calculated this way:
For example is the Northern upper coordinate of the first hole "1 short": 2990, 2375 
The Southern upper coordinate of the first hole "1 short": (2990-9, 2375+4) = 2981,2371  , (the 9 and 4  are mentioned above).
The  Northern upper coordinate of the first Space starts at 2981,2371 and Southern upper coordinate is (2981-20, 2371+10) = 2961,2361
The next holes and Spaces are calculated in the same way.

The depths of the holes are measured to be 0.34 cubits RM.
The lower coordinates of the holes and Spaces are calculated from the upper set of coordinates.
Depth, X-axis: 0.34 cubits RM x sin 26.2139° = 0.1502 cubits (=0.2 cubits rounded).
Pixels, X-axes: 10 x 0,2 = 2 pixels.
Depth, X-axis: 0.34 cubits RM x cos 26.2139° = 0.3050 cubits (=0.3 cubits rounded).
Pixels, X-axes: 10 x 0,3 = 3 pixels.
For example is the Northern lower coordinate of the first hole "1 short": (2990-2, 2375+3) = 2988,2378  , (the 2 and 3 are mentioned above).
The Southern lower coordinate of the first hole "1 short": (2981-2, 2371+3) = 2979,2374 , (the 2 and 3 are mentioned above).
The other holes are calculated the same way.


Section 5.5.14

Top view of the ramp-benches.

Flinders Petrie noted:
"By plumb-line measure at the S. end, the roof on the E. side is inside the floor edge (or overhangs) 20.50, and on the W. side 20.40. On the S. end (eliminating the lean) the projections is 20.9, and on N. 20.4; mean of all, 20.55, for the sum of the seven projections of the laps, or one cubit, the laps being then one palm each in breadth. Thus the laps overhang the ramps along the gallery sides, and the space between the ramps (2 cubits), is equal to the space between the walls at the top."
Source: (S5orig-[S46]-P74-L15-21)

My notes:
Maybe it was easier just to measure the breadth of the ramp-benches. Anyway the note of Flinders Petrie gives us more information.
20.50" = 0,9942 cubits or 1 cubit NM.
20.40" = 0,9893 cubits or 1 cubit NM.
20.9"  = 1,0136 cubits or 1 cubit NM.
20.55" = 0,9966 cubits or 1 cubit NM.

With other words the breadth of the two ramp-benches is 1 cubit NM each and the space between the ramp-benches is 2 cubits RM, which is the same space between the walls in the ascending passageway. Please see the picture in section 5.5.4. The Western wall of the ascending passageway is aligned to the side of the ramp-bench. As the spaces between the walls are the same (2 cubits RM), then the Eastern wall is also aligned to the other ramp-bench.

According to the top view in section 5.4.15 the end coordinates in the East - West direction are 2041,2990 and 2061,2990 respectively.
As the breadth of the ramp-benches is 1 cubit NM each, then the Northern corners of the Eastern ramp-bench hits the Northern wall at the coordinates (2041-10,2990) = 2031,2990 and 2061,2990 respectively.
The Northern corners of the Western ramp-bench hits the Northern wall at the coordinates 2061,2990 and (2061+10,2990) = 2071,2990 respectively.

A similar calculation can be made where the ramp-benches hit the top stone at South.
The Eastern ramp-bench hits the top stone at 2031,2200 (at East) and 2041,2200 (at West).
The Western ramp-bench hits the top stone at 2061,2200 (at East) and 2071,2200 (at West).

Drawing for internal use only. (The Well or Pit is shown with coordinates):  


Section 5.5.15

The coordinates for the holes and spaces in the Eastern ramp-bench:

Holes and
spaces
EASTtopNE EASTtopSE EASTbottomNE EASTbottomSE EASTtopNW EASTtopSW EASTbottomNW EASTbottomSW
Top stone Z axis
E-W
(horiz.)
X axis
N-S
(vert.)
Z axis
E-W
(horiz.)
X axis
N-S
(vert.)
Z axis
E-W
(horiz.)
X axis
N-S
(vert.)
Z axis
E-W
(horiz.)
X axis
N-S
(vert.)
Z axis
E-W
(horiz.)
X axis
N-S
(vert.)
Z axis
E-W
(horiz.)
X axis
N-S
(vert.)
Z axis
E-W
(horiz.)
X axis
N-S
(vert.)
Z axis
E-W
(horiz.)
X axis
N-S
(vert.)
Space 2041 2220 2041 2200         2044 2220 2044 2200        
14 short 2041 2229 2041 2220 2041 2227 2041 2218 2044 2229 2044 2220 2044 2227 2044 2218
Space 2041 2249 2041 2229         2044 2249 2044 2229        
13 long 2041 2259 2041 2249 2041 2257 2041 2247 2044 2259 2044 2249 2044 2257 2044 2247
Space 2041 2278 2041 2259         2044 2278 2044 2259        
13 short 2041 2287 2041 2278 2041 2285 2041 2276 2044 2287 2044 2278 2044 2285 2044 2276
Space 2041 2307 2041 2287         2044 2307 2044 2287        
12 long 2041 2317 2041 2307 2041 2315 2041 2305 2044 2317 2044 2307 2044 2315 2044 2305
Space 2041 2337 2041 2317         2044 2337 2044 2317        
12 short 2041 2346 2041 2337 2041 2344 2041 2335 2044 2346 2044 2337 2044 2344 2044 2335
Space 2041 2366 2041 2346         2044 2366 2044 2346        
11 long 2041 2376 2041 2366 2041 2374 2041 2364 2044 2376 2044 2366 2044 2374 2044 2364
Space 2041 2396 2041 2376         2044 2396 2044 2376        
11 short 2041 2405 2041 2396 2041 2403 2041 2394 2044 2405 2044 2396 2044 2403 2044 2394
Space 2041 2424 2041 2405         2044 2424 2044 2405        
10 long 2041 2434 2041 2424 2041 2432 2041 2422 2044 2434 2044 2424 2044 2432 2044 2422
Space 2041 2454 2041 2434         2044 2454 2044 2434        
10 short 2041 2463 2041 2454 2041 2461 2041 2452 2044 2463 2044 2454 2044 2461 2044 2452
Space 2041 2483 2041 2463         2044 2483 2044 2463        
9 long 2041 2493 2041 2483 2041 2491 2041 2481 2044 2493 2044 2483 2044 2491 2044 2481
Space 2041 2513 2041 2493         2044 2513 2044 2493        
9 short 2041 2522 2041 2513 2041 2520 2041 2511 2044 2522 2044 2513 2044 2520 2044 2511
Space 2041 2541 2041 2522         2044 2541 2044 2522        
8 long 2041 2551 2041 2541 2041 2549 2041 2539 2044 2551 2044 2541 2044 2549 2044 2539
Space 2041 2571 2041 2551         2044 2571 2044 2551        
8 short 2041 2580 2041 2571 2041 2578 2041 2569 2044 2580 2044 2571 2044 2578 2044 2569
Space 2041 2600 2041 2580         2044 2600 2044 2580        
7 long 2041 2610 2041 2600 2041 2608 2041 2598 2044 2610 2044 2600 2044 2608 2044 2598
Space 2041 2630 2041 2610         2044 2630 2044 2610        
7 short 2041 2639 2041 2630 2041 2637 2041 2628 2044 2639 2044 2630 2044 2637 2044 2628
Space 2041 2658 2041 2639         2044 2658 2044 2639        
6 long 2041 2669 2041 2658 2041 2667 2041 2656 2044 2669 2044 2658 2044 2667 2044 2656
Space 2041 2688 2041 2669         2044 2688 2044 2669        
6 short 2041 2697 2041 2688 2041 2695 2041 2686 2044 2697 2044 2688 2044 2695 2044 2686
Space 2041 2717 2041 2697         2044 2717 2044 2697        
5 long 2041 2727 2041 2717 2041 2725 2041 2715 2044 2727 2044 2717 2044 2725 2044 2715
Space 2041 2747 2041 2727         2044 2747 2044 2727        
5 short 2041 2756 2041 2747 2041 2754 2041 2745 2044 2756 2044 2747 2044 2754 2044 2745
Space 2041 2776 2041 2756         2044 2776 2044 2756        
4 long 2041 2786 2041 2776 2041 2784 2041 2774 2044 2786 2044 2776 2044 2784 2044 2774
Space 2041 2805 2041 2786         2044 2805 2044 2786        
4 short 2041 2814 2041 2805 2041 2812 2041 2803 2044 2814 2044 2805 2044 2812 2044 2803
Space 2041 2834 2041 2814         2044 2834 2044 2814        
3 long 2041 2844 2041 2834 2041 2842 2041 2832 2044 2844 2044 2834 2044 2842 2044 2832
Space 2041 2864 2041 2844         2044 2864 2044 2844        
3 short 2041 2873 2041 2864 2041 2871 2041 2862 2044 2873 2044 2864 2044 2871 2044 2862
Space 2041 2893 2041 2873         2044 2893 2044 2873        
2 long 2041 2903 2041 2893 2041 2901 2041 2891 2044 2903 2044 2893 2044 2901 2044 2891
Space 2041 2922 2041 2903         2044 2922 2044 2903        
2 short 2041 2931 2041 2922 2041 2929 2041 2920 2044 2931 2044 2922 2044 2929 2044 2920
Space 2041 2951 2041 2931         2044 2951 2044 2931        
1 long 2041 2961 2041 2951 2041 2959 2041 2949 2044 2961 2044 2951 2044 2959 2044 2949
Space 2041 2981 2041 2961         2044 2981 2044 2961        
1 short 2041 2990 2041 2981 2041 2988 2041 2979 2044 2990 2044 2981 2044 2988 2044 2979
North wall 2041 2990               2990            


Section 5.5.16

The coordinates for the holes and spaces in the Western ramp-bench:

Holes and
spaces
WESTtopNE WESTtopSE WESTbottomNE WESTbottomSE WESTtopNW WESTtopSW WESTbottomNW WESTbottomSW
Top stone X axis
N-S
(horiz.)
Y axis
N-S
(vert.)
X axis
N-S
(horiz.)
Y axis
N-S
(vert.)
X axis
N-S
(horiz.)
Y axis
N-S
(vert.)
X axis
N-S
(horiz.)
Y axis
N-S
(vert.)
Z axis
E-W
(horiz.)
X axis
N-S
(vert.)
Z axis
E-W
(horiz.)
X axis
N-S
(vert.)
Z axis
E-W
(horiz.)
X axis
N-S
(vert.)
Z axis
E-W
(horiz.)
X axis
N-S
(vert.)
Space 2078 2220 2078 2200         2081 2220 2081 2200        
14 short 2078 2229 2078 2220 2078 2227 2078 2218 2081 2229 2081 2220 2081 2227 2081 2218
Space 2078 2249 2078 2229         2081 2249 2081 2229        
13 long 2078 2259 2078 2249 2078 2257 2078 2247 2081 2259 2081 2249 2081 2257 2081 2247
Space 2078 2278 2078 2259         2081 2278 2081 2259        
13 short 2078 2287 2078 2278 2078 2285 2078 2276 2081 2287 2081 2278 2081 2285 2081 2276
Space 2078 2307 2078 2287         2081 2307 2081 2287        
12 long 2078 2317 2078 2307 2078 2315 2078 2305 2081 2317 2081 2307 2081 2315 2081 2305
Space 2078 2337 2078 2317         2081 2337 2081 2317        
12 short 2078 2346 2078 2337 2078 2344 2078 2335 2081 2346 2081 2337 2081 2344 2081 2335
Space 2078 2366 2078 2346         2081 2366 2081 2346        
11 long 2078 2376 2078 2366 2078 2374 2078 2364 2081 2376 2081 2366 2081 2374 2081 2364
Space 2078 2396 2078 2376         2081 2396 2081 2376        
11 short 2078 2405 2078 2396 2078 2403 2078 2394 2081 2405 2081 2396 2081 2403 2081 2394
Space 2078 2424 2078 2405         2081 2424 2081 2405        
10 long 2078 2434 2078 2424 2078 2432 2078 2422 2081 2434 2081 2424 2081 2432 2081 2422
Space 2078 2454 2078 2434         2081 2454 2081 2434        
10 short 2078 2463 2078 2454 2078 2461 2078 2452 2081 2463 2081 2454 2081 2461 2081 2452
Space 2078 2483 2078 2463         2081 2483 2081 2463        
9 long 2078 2493 2078 2483 2078 2491 2078 2481 2081 2493 2081 2483 2081 2491 2081 2481
Space 2078 2513 2078 2493         2081 2513 2081 2493        
9 short 2078 2522 2078 2513 2078 2520 2078 2511 2081 2522 2081 2513 2081 2520 2081 2511
Space 2078 2541 2078 2522         2081 2541 2081 2522        
8 long 2078 2551 2078 2541 2078 2549 2078 2539 2081 2551 2081 2541 2081 2549 2081 2539
Space 2078 2571 2078 2551         2081 2571 2081 2551        
8 short 2078 2580 2078 2571 2078 2578 2078 2569 2081 2580 2081 2571 2081 2578 2081 2569
Space 2078 2600 2078 2580         2081 2600 2081 2580        
7 long 2078 2610 2078 2600 2078 2608 2078 2598 2081 2610 2081 2600 2081 2608 2081 2598
Space 2078 2630 2078 2610         2081 2630 2081 2610        
7 short 2078 2639 2078 2630 2078 2637 2078 2628 2081 2639 2081 2630 2081 2637 2081 2628
Space 2078 2658 2078 2639         2081 2658 2081 2639        
6 long 2078 2669 2078 2658 2078 2667 2078 2656 2081 2669 2081 2658 2081 2667 2081 2656
Space 2078 2688 2078 2669         2081 2688 2081 2669        
6 short 2078 2697 2078 2688 2078 2695 2078 2686 2081 2697 2081 2688 2081 2695 2081 2686
Space 2078 2717 2078 2697         2081 2717 2081 2697        
5 long 2078 2727 2078 2717 2078 2725 2078 2715 2081 2727 2081 2717 2081 2725 2081 2715
Space 2078 2747 2078 2727         2081 2747 2081 2727        
5 short 2078 2756 2078 2747 2078 2754 2078 2745 2081 2756 2081 2747 2081 2754 2081 2745
Space 2078 2776 2078 2756         2081 2776 2081 2756        
4 long 2078 2786 2078 2776 2078 2784 2078 2774 2081 2786 2081 2776 2081 2784 2081 2774
Space 2078 2805 2078 2786         2081 2805 2081 2786        
4 short 2078 2814 2078 2805 2078 2812 2078 2803 2081 2814 2081 2805 2081 2812 2081 2803
Space 2078 2834 2078 2814         2081 2834 2081 2814        
3 long 2078 2844 2078 2834 2078 2842 2078 2832 2081 2844 2081 2834 2081 2842 2081 2832
Space 2078 2864 2078 2844         2081 2864 2081 2844        
3 short 2078 2873 2078 2864 2078 2871 2078 2862 2081 2873 2081 2864 2081 2871 2081 2862
Space 2078 2893 2078 2873         2081 2893 2081 2873        
2 long 2078 2903 2078 2893 2078 2901 2078 2891 2081 2903 2081 2893 2081 2901 2081 2891
Space 2078 2922 2078 2903         2081 2922 2081 2903        
2 short 2078 2931 2078 2922 2078 2929 2078 2920 2081 2931 2081 2922 2081 2929 2081 2920
Space 2078 2951 2078 2931         2081 2951 2081 2931        
1 long 2078 2961 2078 2951 2078 2959 2078 2949 2081 2961 2081 2951 2081 2959 2081 2949
Space2078298120782961    2081298120812961    
1 short2078299020782981207829882078297920812990208129812081298820812979
North wall 2078 2990               2990            


Section 5.5.17

In section 5.5.13 you saw this excellent drawing of Mark Lehner:

5513 tegning af hul i rampe MarkPNG
Source S15, "Niches, Slots, Grooves and Stains: Internal Framework in the Khufu Pyramid?" of Mark Lehner published in https://www.academia.edu/36645718/Niches_Slots_Grooves_and_Stains_Internal_Frameworks_in_the_Khufu_Pyramid 

Compare the drawing to this photo:

niche ved hul i rampe IVJPG
Source: S17.  http://grahamhancock.com/phorum/read.php?1,269522,269576
(Hole 4s)

and to another photo:

niche ved hul i rampe VJPG
http://thepyramids.org/p-016-great-pyramid-khufu-cheops-233-039-cavity-grand-gallery.html

For those who has not visited the pyramid I can assure you both photos and the drawing are correct. Let me explain:
Mark Lehner call the rectangular holes for "slots" and above the "slots" are "niches" cut into the wall. Across the niches are chiseled a trapezoidal structure called "trapezoidal cutting".  In my opinion it is not a trapezoidal structure, but a parallelogram as both the opposite sides are parallel. 

Anyway, in "Niches, Slots, Grooves and Stains: Internal Framework in the Khufu Pyramid?" by Mark Lehner published in https://www.academia.edu/36645718/Niches_Slots_Grooves_and_Stains_Internal_Frameworks_in_the_Khufu_Pyramid  : 
"4. The niches, numbering 25 were cut, 18 to 20 cm deep and 67 cm high, into the side walls above all but the upper southermost and lower nothermost of the notches. The top and bottom edges are horizontal. the bottom north corners are even with the sloping top of the ramp benches while the south corner is 10 - 13 cm lower, consequently part way inside the accompanying notch. The outer top and north edge of at least some of the niches are bevelled.
5. The niches were closed with limestone patch (Flicken) or plugs. The beveled edges were filled with mortar. The bottoms of the patch stones slope even with the slope of the upper surface of the bench-ramp. Below this, the niches were filled with limestone fragments and mortar. L. BORCHARDT reported that, where the patches had broken or the mortar fill had fallen away, the well-trimmed inside surfaces of the niches were fresh, clean and white as though they were little used before being closed by the patches.
6. Trapezoidal cuttings, (Streifen, striures, incisions) about 2 to 3 cm deep, were chiseled across the patched niches and onto the adjacent wall surface. The cuttings are 10 cm above the tops of the ramp benches and roughly parallel to the slope of the ramp-benches. The lengths of the cuttings vary from 55 to 70 cm. The cuttings are missing from the northernmost patched niches at the low end of the Grand Gallery."
Source: (S15-P102-L17-32)
   
According to Charles Rigano there are 25 holes out of 27 in each ramp-bench which have these niche and trapezoidal features.  Charles Rigano wrote following in the book "Pyramids of the Giza Plateau":
"All of the 50 side niches and trapezoidal features are completely filled with blocks and mortar of the type used elsewhere in the pyramid with the exception of the 7th and 11th side niches from the bottom on the west side, the first of which is empty and the second has a hole cut into the mortar."
Source: (S11-P51-L21-23)

In the figure on the same page mention Charles Rigano some figures in a table:
"Side Niches (In Walls") :
Height: 26"
Width: 11"
Depth: 8"
Source: (S11-P51-table)
 
My comments: The first picture is hole marked 4s ( or the 7th hole ) showing the empty hole and the 11th hole is the hole marked "6s".

Section 5.5.18

With other words, the only niche we can describe is the one connected to hole marked 4s, which is a short one. All the other niches might be of the same sizes, but we cannot be sure. The consequence of this is, that I avoid to calculate the coordinates for all other niches.

As you can see on the picture, it has not been easy to measure the dimensions of the niche at the short hole 4s and different sources have different answers:

The hole 4s has following dimensions:
Length: 1 cubit RM
Width: 0,27 cubits RM
Depth: 0,34 cubits RM

The angle is: 26,2139°

The niche has following dimensions from sources in a certain order, where Mark Lehner is the most reliable :
Depth:
Mark Lehner: 18-20cm , Source: ( S15-P102-L17), equal to 0,34-0,38 cubits
Miatello: 20cm ,  Source: ( S16-P3-1C-L7) , equal to 0,38 cubits
Rigano: 8" ,  Source: (S11-P51-table) , equal to 0,388 cubits
The depth of the niche at 4s is 0,38 cubits.
  
Height:
Mark Lehner: 67cm , Source: ( S15-P102-L17) , equal to 1,28 cubits  
Miatello (source S16): 60cm (average) ,  Source: ( S16-P3-1C-L26) , equal to 1,146 cubits  
Rigano: 26" , Source: (S11-P51-table) , equal to 1,261 cubits
The height of the niche at 4s is 1,28 cubits.

Width:
Mark Lehner: ?
Miatello: 32 cm , Source: ( S16-P3-1C-L26) , equal to 0,61 cubits  
Rigano: 11" , Source: (S11-P51-table) , equal to 0,533 cubits
See below.

Southern bottom corner in the hole:
Mark Lehner: 10-13cm , Source: ( S15-P102-L20) , equal to 0,19-0,25 cubits
See below.

It is not the big secret the dimensions "Width" and "Southern bottom corner in the hole" puzzles me. Let me prove what is right and wrong.
Underneath is a drawing of the upper part of hole 4s which is aligned with the surface of the ramp-bench seen from the western side. 

5518 Projection af hul 4sjpg

The upper part of the hole 4s is illustrated by a 26,2139° sloping imaginary line with a sloping length of 1 cubit.
The dotted lines in the upper part of the drawing symbolizes the niche.
The northern lower corner of the niche hit the imaginary sloping line of the hole.
The southern lower corner of the niche descend 0,19 cubits to 0,25 cubits into the hole marked "c".
The sloping length of the hole where the niche cut the imaginary line is marked a´
The remaining parts of the sloping line are marked b´ so b´ + a´ + b´ = 1 cubit.

According to Miatello the width of the niche is 0,61 cubits  
Rigano has informed the width of the niche is 0,533 cubits
Let us make some calculations to judge who is right, we can make two calculations based on both c = 0,19 cubits and c = 0,25 cubits:
5518 calcJPG
It is obvious that Miatello is not right, it is not possible the width is 0,61 cubits.
Let us turn around and make the calculation of the length c based on the number of Rigano:
5518 calc IJPG
which is close to 0,25 cubits mentioned by Mark Lehner.

Rigano has informed the right figure of 0,533 cubits, if  the lower southern corner descent 0,25 cubits into the hole 4s.

Does it make sense ? Let us calculate the sloping lengths a´and b´ :
 5518 calc IIJPG

Result:
a´= 0,6 cubits
b´= 0,2 cubits

The so called  trapezoidal cuttings has these measures:
(The sides are parallel to the ramp-bench).

Depth:
Mark Lehner: 2-3cm, Source: ( S15-P102-L28), equal to 0,04-0,06 cubits
Miatello: 2-3cm , Source: ( S16-P3-1C-L23) , equal to 0,04-0,06 cubits
The depth of the trapezoidal cuttings at 4s is 0,04-0,06 cubits.
 
Height:
Mark Lehner: ?
Miatello (source S16): 20cm, Source: ( S16-P3-1C-L24) , equal to 0,38 cubits  
The height of the trapezoidal cuttings at 4s is 0,38 cubits.

Length:
Mark Lehner: 55-70cm,  Source: ( S15-P102-L17), equal to 1,05-1,34 cubits
Miatello: 55-70cm, Source: ( S16-P3-1C-L24) , equal to 1,05-1,34 cubits
The length of the trapezoidal cuttings at 4s is 1,05 cubits

Lower side above ramp-bench:
Mark Lehner: 10-13cm , Source: ( S15-P102-L29) , equal to 0,19-0,25 cubits


Section 5.5.19

The measures are approximate values, so in this case I have chosen not to add RM. As the figures are not precise it could be helpful to receive a second opinion from the readers.
All these figures helped me to issue a drawing, the measures are in cubits:


5519 Billede af tegning af niche rev 4jpg

The drawings show the same niche at the hole 4s but seen from two different angles.

As mentioned the measures are not precise. From that perspective I dare to point out that some of measures might have been intended to be the same.
The depth of the hole of 0,34 cubits could have been intended to the same measure as the height of the trapezoidal structure of 0,38 cubits and the depth of the niche of 0,38 cubits as well. Normally I would not play with figures, but in this case I cannot resist. If the intended measure was the average between 0,34 cubits and 0,38 cubits = 0,36 cubits then this value is equal to 0,5 remens or 10 fingers. Is it a coincidence? Maybe - maybe not.

Another interesting measure is the offset of the bottom of the trapezoidal structure above the ramp-bench of 0,19 cubits, which is very similar to the sloping distance of 0,2 cubits, which is the same as 1/5 cubits.

Furthermore the height of the niche is 1,28 cubits which is exactly the same as 9 hands.

We can conclude the measures of smaller structures were defined as a simple fraction of cubits or in natural numbers of remens, hands (palms) or fingers.

Section 5.5.20

Let us see a top view at the floors:

5520 Top view GG on DescendJPG

The black lines are the ascending and descending passageways. The green lines represent the Grand Gallery. 

A closer look on the Southern part at Apex (center of the pyramid):

5520 Top view GG on Descend closer viewJPGJPG

The passageways are East of the North-South center line.
The entrance from the Grand Gallery into the antechamber (southern part of the topstone) at the North-South coordinate 2170 is more or less in the same position as the entrance of the subterranean chamber at 2180, but of course placed higher up in the pyramid. The difference is 10 pixels or 1 cubit = 0,524 metres. 

Section 5.5.21

Let us focus on the walls and the roof in the Grand Gallery. For this purpose, we need an additional figure in the set of coordinates as the walls and the roof. Until now the coordinates consisted of two coordinates, either (N-S, Up-Down) or (E-W, N-S). Until now it has been sufficient, and we saved some space in the drawings.

In this section we need to define an expanded set of coordinates as (N-S, Up-Down, E-W) or (X, Y, Z) as in a "normal" coordinate system.    

The four corners where the walls hit the "floor" are:
The northern wall at the eastern corner hits the ramp-bench at: 2990, 2375, 2031
The northern wall at the western corner hits the ramp-bench at: 2990, 2375, 2071
The northern wall at the eastern corner hits the imaginary floor line at: 2990, 2386, 2031
The northern wall at the western corner hits the imaginary floor line at: 2990, 2386, 2071
The southern wall at the eastern corner hits the top stone at: 2170, 1979, 2031
The southern wall at the western corner hits the top stone at: 2170, 1979, 2071

Section 5.5.22

Edgar published a picture of the North side of the Grand Gallery, where you easily can see the corbelled walls on all sides.

5522 Grand Gallery Northjpg
Source: The Great Pyramid Passages And Chambers" Volume 1 by John and Morton Edgar, 1910

A recently taken picture of the walls:

5522 Grand Gallery pict of todayJPG
Source:  http://www.touregypt.net/featurestories/greatpyramid3.htm
Please ignore the staircase and the angle irons inserted in the walls to prevent fractions falling down.  I have chosen this "new" picture to give you a better idea how the walls look like. Even the roof was not made smooth, but the stones are angled. Flinders Petrie had a good explanation:

"The roof of the gallery and its walls are not well known, owing to the difficulty of reaching them. By means of ladders, that I made jointing together, I was able to thoroughly examine both ends and parts of the sides of the gallery. The roof stones are set each at a steeper slope than the passage, in order that the lower edge of each stone should hitch like a paul into a ratchetcut in the top of the walls; hence no stone can press on the one below it, so as to cause a cumulative pressure all down the roof ; and each stone is separately upheld by the side walls across which it lies."
(S5orig-[S46]-P72-L33-40)


Section 5.5.23

All the walls consist of a basis and 7 laps and in the 3rd lap there is a groove on the eastern and western walls, let us see which measures we have:

Flinders Petrie:
"The remarkable groove in the lower part of the third lap, along the whole length of the sides, was measured thus, perpendicularly :
(S5orig-[S46]-P73-L40-41)
5523 tableJPG
(S5orig-[S46]-P73-L42-45)

My comments:
N.W. 11,7" = 0,567 cubits RM close to 4 hands RM (palms)
N. E. 11,8" = 0,572 cubits RM = 4 hands RM (palms)
S. W. 11,2" = 0,543 cubits RM 
S. E. 11,0" =  0,533 cubits RM
Mean 11,4" = 0,553 cubits RM
N.W. 5,4" = 0,262 cubits RM
N.E. 5,7" = 0,276 cubits RM
S.W. 5,1" = 0,247 cubits RM close to 1/2 foot or 7 fingers
S.E. 5,1" =  0,247 cubits RM close to 1/2 foot or 7 fingers
Mean 5,3" = 0,257 cubits RM
Mean difference  -6,1" = 0,296 cubits RM or close to 2 hands (palms)

Flinders Petrie:
"At the S.W. it is cut to a depth of -8 inch, at the S.E. to -6 (?) ; the upper edge of it is often ill-defined and sloping."
(S5orig-[S46]-P73-L46-47)

My comments:
S.W 0,8" = 0,039 cubits RM close to 1 finger.
S.E 0,6" (?) = 0,029 cubits RM (?)


Section 5.5.24

The dimensions in the Grand Gallery is a challenge as my sources miss a lot of genuine measurements. Even my favorite source Flinders Petrie had difficulties to measure for example the vertical height of the Grand Gallery. I have different measures from different sources, and they might be correct all of them depending on where they measured. To understand the problem, I have copied the old drawing of Edgar. I have inserted 4 red lines to visualize the problem. The heights of each lap are missing, also the height of the basis of the walls under the laps.  

5524 Hvor er der mlt 2JPG
Source: The Great Pyramid Passages And Chambers" Volume 1 by John and Morton Edgar, 1910  (Red lines and blue numbers are added by me).

(You can also see the groove in the third lap mentioned in the previous section).

Line I: Even it has been extremely dangerous to measure this height, it should be possible.
Line II: I cannot imagine this measure has ever been made. But, this height is the "true" vertical height.
Line III: It could be possible to measure there, but it is not the "real" height".
Line IV: This measuring line is very possible and the most convenient, but is it the "true" vertical height?  


Section 5.5.25

Another tricky point is that the roof is not smooth, but the stones are angled as explained in section 5.5.22

If you see the yellow marked area of the roof of the drawing med by Edgar, then you might understand the measuring problem of the "true" vertical height:

5525 Roof Grand Gallery Side 64 EdgarJPG 
Source: The Great Pyramid Passages And Chambers" Volume 1 by John and Morton Edgar, 1910  (page 64)

Let us examine the different sources:


Section 5.5.26

Luca Miatello made following figure 3:

5526 Miatello fig 3JPG
Source:  S16. "Examining The Grand Gallery in The Pyramid of Khufu And its Features" by Luca Miatello in PalArch´s Journal of Achaeology of Egypt/Egyptology, 7(6) (2010)

My comments:
I'm not sure where Luca Miatello's measurement results come from, with all due respect I doubt he has measured them himself.  Let us for a moment accept the results and compare them with other sources. 

Luca Miatello:

"Projections of the seven corbels together: seven palms (one palm each);"
Source: (S16-P6-L22-23)

"Perpendicular height from the gallery floor to the top of the gallery: about 15 cubits;"
Source: (S16-P6-L32-33)
"Note 11: according to Maragioglio & Rinaldi (1965: pl. 6, fig. 1), the vertical height of the gallery varies from 8.48 to 8.74 m, mean 8.61 m, hence: x2 + x2/tan(63.8°)2 = 8.612; x = 7.73 m = 14.8 cubits."
Source: (S16-[C1]-P24-L50-53)

My comments:
The calculation of  Maragioglio & Rinald is very doubtful and it is not obviously clear from where the different factors are coming from. As I can calculate X=2,633 m in the above equation, which makes no sense.
8,48 m = 16,196 cubits = 23,904 remens
8,74 m = 16,692 cubits = 23,606 remens
8,61 m = 16,444 cubits = 23,255 remens
7,73m = 14,763 cubits = 20,878 remens
 
"Perpendicular height from the top of the ramp benches to the top of the gallery: about 14 cubits;" 
Source: (S16-P6-L34-36)
"Note 12: the perpendicular height of the ramp benches is 0.52 m, or one cubit; the perpendicular height from the top of the benches to the first corbel is about 1.65 m, or 3.1 cubits: see Maragioglio & Rinaldi (1965: pl. 6, fig. 1)."
Source: (S16-[C1]-P24-L54-56 and S16-[C2]-P24-L1)

"Perpendicular height of the seven corbels together: about 11 cubits."
Source: (S16-P6-L37-38)

My comments:
It is important to observe the measures are calculated and the little innocent word "about" is added in the text, but not in figure 3.

Please observe where the vertical measuring line is placed at the roof, is it placed there by a coincidence?

5526 Miatello fig 3aJPG
Source: S16. "Examining The Grand Gallery in The Pyramid of Khufu And its Features" by Luca Miatello in PalArch´s Journal of Achaeology of Egypt/Egyptology, 7(6) (2010)
(This is the upper part of fig. 3 - Yellow markings and blue circles are made by me.)

My comments:
8,6 m = 16,425 cubits = 23,228 remens


Section 5.5.27

Flinders Petrie had following comments:
"The depth of two of these ratchetcuts, at the S. end, I measured as 1,0 and 1,9 to 2,0) ; and the angles of the two slabs there 28° 0'  to 28° 18', and 27° 56' to 28° 30',
mean 28° 11' ; which on a mean slab 52,2 from N. to S., would differ 1,74 inches from the passage slope. The edge of the southernmost slab is 14,5 from the S. wall ; the next slab is 47,4 from N. to S."
Source: (S5orig-[S46]-P72-L40-41) followed by (S5orig-[S46]-P73-L1-4)  

My comments:
28° 0' to 28° 18' = 28,0° to 28,3° = 13,16 seked to 13,00 seked
27° 56' to 28° 30' = 27,9° to 28,5° = 13,22 seked to 12,89 seked
mean 28° 11' = 28,18° = 13,07 seked
52,2" = 2,532 cubits RM = 3,580 remens RM
1,74" = 0,084 cubits RM = 0,119 remens RM
14,5" = 0,703 cubits RM = 0,994 remens RM
47,4" = 2,299 cubits RM = 3,251 remens RM

As the measurements of the angle are so different from each other, it makes sense to choose the mean value as Petrie suggested, though the original intention might very well have been 13,0 seked NM equal to 28,3° NM.
See also section 5.5.32

Flinders Petrie:
"According to Prof. Smith the mean height of this lap above the gallery floor is 166,2 ± 0,8 vertically ; hence the groove is at 172,1 to 179,0 vertically over the floor, and its lower edge is therefore at half the height of the gallery, that varying from 167 to 172."
Source: (S5orig-[S46]-P73-L47) followed by (S5orig-[S46]-P74-L1-3)

My comments:
166,2" = 8,06 cubits RM = 11,399 remens RM
0,8" = 0,039 cubits RM = 0,055 remens RM = 1,086 fingers RM
172,1" to 179,0" = 8,346 cubits RM = 11,803 remens RM  to  8,681 cubits RM = 12,277 remens RM
167" to 172" = 8,099 cubits RM = 11,454 remens RM  to  8,341 cubits RM = 11,797 remens RM



Section 5.5.28

Charles Rigano mentioned:
"Incline at Ceiling: 28° 11'
Height - Vertical : 28' 2"   "
Source: S11-P50 in the table.

My comments:
28° 11' = 28,1833°
28' 2" = 16,397 cubits = 23,189 remens

Section 5.5.29

Mark Lehner and Zahi Hawass mentioned:
"Height 8,7 m (28 ft 6 in.)"
Source: S12-P152-[C1]-L3

My comments:
8,7 m = 16,616 cubits = 23,498 remens
28 ft 6 in = 16,591 cubits = 23,463 remens

Section 5.5.30

Mark Lehner mentioned:
"The Grand Gallery is a stupendous achievement: the roof soars to 8,74 m (26 ft) and is the glorious culmination a series of corbelled roofs seen at Meidum and Daschur."
Source: S13-P112-Lines under the picture of Grand Gallery

My comments:
 8,74 m = 16,692 cubits = 23,606 remens
26 ft = 15,14 cubits (I assume the 26 ft is a print error, maybe it should have been 28 ft) ?


Section 5.5.31

As you may understand, to measure the height of the Grand Gallery is not easy and to bring the results to the public is even worse because the roof has the ratchetcuts and the roof is angled 28,18°, which is more than the floor of the Grand Gallery. The floor and the floor of the internal ramp is smooth but the roof is not. Let us examine the roof in detail.

5527 GG Ratchetcutjpg
    
From section 5.5.27 we know following measures:

h : 1" and 1,9" to 2,0" = 0,048 cubits RM and 0,092 cubits RM to 0,097 cubits RM. (Petrie) (Basically two measures, we check which of them is correct). 
β : Mean angle 28,18° (Petrie) (There are 4 different measures  28,0° and 28,3° and 27,9° and 28,5°. With this diversity it is fair to accept the mean value).  
b2 or c2 : Mean slab 52,2" = 2,532 cubits RM (Petrie) (It is not crystal clear whether Petrie referred to the sloping c2 or horizontal b2, we will check this).
b1 : 14,5" = 0,703 cubits RM (Petrie) 

5531 kalk 1JPG

b2 or c2 ?

If we compare the calculations in step 4  and step 5 then the results in step 5 is much more aligned with  "Mean slab 52,2" = 2,532 cubits RM (Petrie)" than step 4.
We can hereby conclude that Petrie meant the sloping length c2. The mean sloping length of the slab is 2,532 cubits RM.

Section 5.5.32

Now we know c2 is 2,532 cubits RM, then we can calculate the real value of b2:

5532 calc 1JPG     Please observe b2 is revised.

h1 is:

5532 calc 2JPG

Let us check the mean angle 28,18° :
 
2,2319 cubits x 7 hands / 1,1957 cubits = 13,07 seked, which is very close to 13 seked. b2 and h1 is therefore verified.
If the angle should have been exact 13,0 seked, then h1 should have been 1,2018 cubits, a deviation of 0,0061 cubits, which is far under the accuracy in this work as  mentioned earlier.

5532 calc 3JPG   Depending on which of Petries measures can be verified.
"The depth of two of these ratchetcuts, at the S. end, I measured as 1,0" and 1,9" to 2,0"
Source: (S5orig-[S46]-P72-L40 ) se also section 5.5.27

Section 5.5.33

Back to the height of the Grand Gallery. In the sections 5.5.24 and 5.5.26 we saw different possible measuring lines and it is even more complicated as the slabs in the roof elevate with another angle than the floor of the Grand Gallery.  However it looks like the mean elevation of the roof is the same as the elevation of the floor. We can check this easily if we imagine a line from one upwards corner to the next downwards corner.

It is the same as in section 5.5.27, but in this I have inserted a triangle where the corners of the hypotenuse hit the downwards corners.

 The angle β = 28,18° as mentioned in section 5.5.27 and the angle alpha is the one we check whether it is the same angle as the elevation of the floor 26,2139°.

It is obvious the angle is not the same as the angle of the floor independent of the depth of the ratchet-cut are measured as  1,0" or 1,9" to 2,0" . So, this measure is still unclear, though the angle of the floor as 26,2139° is very close to 26,484°. which derived from the measure of the ratchet-cut as 2,0".

If the elevation of the ceiling was exactly the same as the elevation of the floor, them the depth of the ratchet-cut should have been: 

1,76" is in between of the set of measures, but 1,76" was not measured and cannot be accepted as a mean value.


Another interesting point concerning the measured and possible depths of the ratchet-cuts is the number of fingers:
1,0" = 1,4 fingers RM

(1,5" = 2,0 fingers) 

(1,76" = 2,4 fingers)

1,9" = 2,6 fingers RM)
2,0" = 2,7 fingers RM)

According to section 3.2.5 certain fractions was known, but they preferred integer values. 
Unless all depths of the ratchet-cuts are measured we cannot verify the exact depths and as the deviation is so small, we leave it as it is - not verified. 

Section 5.5.34

I have to introduce a new source S20 :  Life And Work at The Great Pyramid, during the months of January, February, March and April, A.D. 1865 by C. Piazzi Smyth, Vol. II, Edinburgh, Edmonston And Douglas 1867.  

The problem with this source is the way Piazzi Smyth measured. As I have learned the measurements were not accurate. On the other hand we have no other choice and the construction of the pyramid is not 100 % regular.

Piazzi Smyth measured following heights of the Grand Gallery:
 

I have tried to clean the results:

(Please observe there are 15 observations only - not 16 as Piazzi Smyth wrote)

I have calculated all measures in cubits RM :

And in remens RM as well :