On the site of the Cheops pyramid, the filling stones in their cumulative journeys made 300 KM per day, round the earth every 6 months and each 5 years a journey to the moon!
An efficient mobility solution was therefore needed to meet this challenge because sleds sliding on lubricated tracks did not do the trick.
For the bipeds that we are, the solution found by nature to make us move on earth is to rotate the body on one leg by making it describe an arc of a circle, while moving it sideways so that the center of gravity remains in the support polygon, then the second leg takes over and so on. In fact our body MOVES IN THE AIR by taking support on the joints of our feet, however with each step, the pivoting on one leg raises our center of gravity, the work to give it this additional potential energy is given by our muscles, from this summit, the body then falls back into a circular path, transforming the energy acquired into kinetic energy, the horizontal component of this energy is recovered for the next step and maintains the speed of movement acquired, the vertical component is lost and is transformed into heat. A small stride is calorie efficient, a large stride consumes.
If a load is portable on a man’s back, it also travels in the air, but its mass increases the calorie consumption accordingly.
When the load is too heavy to be carried, it stays on the ground and the most immediate way to move it is to drag it.
There is a very efficient slip that was used very early on, it is that of a boat in water, a solution that is still current maritime or river, generally reserved for long journeys and very heavy loads, but on a plateau desert, you can forget that.
The other “obvious” solution is to slide the load, usually carried by a sled on a lubricated track, to reduce friction as much as possible. This apparently simple solution consumes too much manpower because the coefficient of friction of the sled on the track is of the order of 0.2 which is equivalent to raising the sled and its load on a slope at 20%.
More demanding in terms of technology, movement on a wheel, on circular rollers, or even on balls divides more than one hundred times the energy consumption compared to the sled.
For the very heavy stones of the pyramids, the solution of the wheel would have been possible at the time although they left no trace of it, except much later with that of the chariots, on the other hand the displacement on circular rollers appears to us in general as more easily accessible and “natural”, but these two solutions had 3 annoying drawbacks,
1- You needed a perfectly clean runway in an environment invaded by sand and rubble from the quarries, otherwise the roller gets bogged down, anyone who has done 4 x 4 in the desert or biking on a beach knows that !
2- It is impossible to change direction a load carried by wheels unless you have the technology of the steerable axle which under loads of up to a hundred tons would probably not have been accessible at the time , when with the circular rollers, they are impossible to orient under load except that they have the unfortunate natural propensity for a yes or a no to get in the way of the progression, blocking it.
3-Impossible to place such rollers and wheels on standby on a sloping track, without a locking device, for example to prevent the load hoisted on a slope from turning back alone in the event of an incident on the traction.
The ball bearing suffered from the same drawbacks, it would however have allowed changes of direction, the ancient Egyptians who accustomed us to the realization of perfect sculptures, would no doubt have been technically capable of making perfect balls, but to manufacture tens of thousands of them. identical would certainly have been too expensive.
It is possible to think that the ancient Egyptians of the Fourth Dynasty, careful observers of nature, chose to copy the solution that mother nature took millions of years to perfect.
MAKE THE STONES “WALK” IN THE AIR SO HEAVY THEY ARE.
Indeed all mammals progress in the air carried on their legs, in a kinematics of rotation / pivoting, passing their weight from one side to the other alternately to let the hind legs come back to the front.
These pivoting lower limbs are articulated on lubricated ball joints allowing rotations.
To copy nature would have been to make a sort of artificial tibia with a candle fitted with two “ball joints” on either side.
The principle is very simple, in a rotation / pivoting movement, this artificial tibia causes the load placed on the upper “plug” to move horizontally with a very slight vertical path, while the lower “plug” rolls on the moving pavement.
What made me think of that were these curious stones that are found by the thousands in the Petrie Museum in London , under the tag Egyptian weights and measures!
These stones of all sizes have in common that they have a top in the form of a segment of a sphere and a bottom in the form of a truncated cone, some bear indisputable signs of wear on the spherical part.
In fact these studs could have been the “kneecaps” of the artificial “tibia”.
To ensure that the rotation / pivoting is only done on one axis, we could have made a “double tibia” which I will now call “culbuto”:
Let’s take an example to understand why this move is efficient:
Taking into account the size of these studs, we can anticipate that the length of the tumbler is of the order of a “fist” in ancient Egyptian measure, or today 11.2 cm while the diameter of the sphere segment could have been of a “hand” or 9.35 cm.
Let us imagine that the tumbler receives a crutch to maintain it in a stable starting position, which gives it an inclination on the vertical of approximately 27 °, ie a cotangent = SKD of 2, an angle well known in the pyramids.
By correctly ballasting this stand, it will return the tumbler to its starting position when it is released from the load by having placed it on the next tumbler.
We thus obtain a tumbler which has a stable position.
In this configuration the angle formed with the vertical by the straight line which passes through the upper point of contact and the lower point of contact would be 4.3 °, i.e. a tangent = 0.075 which means that to make the tumbler take off at a standing start it is necessary apply to the top an horizontal force which makes 7.5% of the weight that loads it.
Between the starting point and the high point of the rotation the load rises by 2 mm, while it travels horizontally 4.33 cm of developed due to the rotation of 27 ° of the lower and upper spheres plus a displacement of 1.65 cm due to the 27 ° pivoting of the line which links the two centers of the spheres, i.e. 6 cm of total displacement, so when the tumbler has continued its downstream rotation of 27 ° the load will have returned to the same level horizontal, but will have moved horizontally 12 cm or almost the length of the tumbler.
In this displacement, it was necessary to spend energy to raise the load, for example by 1 t of 2 mm, or about 20 KJ, to obtain a displacement of 0.12 m, which would be the equivalent of an average force resistant to displacement of 20 / 0.12 = 167 KN or 1.7% of the weight of the load.
However, the same load sliding on a track would have presented a resistance of the order of 20% of its weight.
So moving through the air on a tumbler, without energy recovery is already 12 times more efficient than sliding a sled on a track.
The example above is there just for the understanding of the basic kinematics, because in reality one has interest to recover the kinetic energy acquired by the block during its fall.
There are two types of solutions to do this:
- The studded roller: the studs distributed over a circumference successively take the load in the rotation of the roller.
- A series of somersaults spread over the track: the next somersault picking up the load brought by the previous tumbler to continue the movement.
1 – Stud roller
For the implementation of this principle in the transport of the stones of the pyramid, I imagined that the builders could have used a roller with 9 studs on the circumference which gives excellent performance.
The problem of building the roller is to get precise geometry, one way to get it is to build the cylinder from segments of 1/9 straight in a triangular shape, for this example it is easy to make. machine with precision, then dig the housing of the studs.
The studs can be glued in their housings by resin, the housing of the studs only works in compression, this way of proceeding makes it possible to easily adjust the studs in a very precise manner.
Then, as for the manufacture of a barrel, the 9 segments are assembled and they are held together by a copper ring.
This process makes it possible for heavy loads to use the wood in axial compression which is more resistant than radial compression.
As many rollers as necessary can easily be produced in series with good reproducibility.
The roller does not need to be very big, a diameter of 11.2 cm (6 fingers or a fist) is sufficient, while its length can be of the order of 20 to 30 cm. In stone for the studs and wood for the body of the roller, in these dimensions its weight is around 1.5 to 3 KG
A roller thus formed has, compared to a perfectly circular roller, the double advantage of being easy to produce in all dimensions, for all loads, and of being less sensitive to the surface quality of the progression path because there is has pivoting in addition to rolling and holds still on slopes up to 8%, which the circular roller cannot.
The spherical shape of the contact allows the roller to circulate in a U- or V-shaped groove , so that it is guided and cannot get across like a circular roller.
The contact of the pad with the groove being at the origin of a point nature, the pressure generated is very high even for low loads.
It is therefore imperative that the material of the groove is less hard than that of the stud, so that the shape of the groove is modified by the pressure and not the stud.
With a diorite or granite stud, a fine limestone or even better copper groove would have been needed.
The copper being ductile, under the effect of the pressure of the stud would have been crushed giving a rolling track in cylindrical shape, with an increased contact surface until the contact pressure becomes lower than the elastic resistance to compression hardened copper.
Thus, by pushing to the extreme, the circulation track can only be made of two grooves, in fact two hollow copper rails, because it is the only place of contact between the rollers and the ground.
The icing on the cake, the contact of the stud on the track being of a punctual nature, the stud roller could follow a winding track which makes turns, not too tight however, which would be impossible with a cylindrical roller.
This property was very interesting because fairly long tracks, 400 m for the pyramid career path, 700 m for the Nile pyramid plain course, posed a problem of the difference in the coefficient of expansion between the copper rail and its limestone support, in making a slightly sinuous S-shaped course, this difference could be absorbed by a slight variation in the radius of the turns and at the same time eliminated the need for expansion joints.
Arbitrarily for the rest of the study, I will take rolls 10 cm in diameter and 20 cm in length, bearing 2 rings of 9 studs, each stud having a sphere radius of 3 cm.
However, if we wanted to take advantage of the very low resistance to forward movement to let the load on rollers move on a sloping track of the order of the % in full autonomy, that is to say to save the labor of work that would have accompanied it, it was necessary to solve the problem posed by the movement of the rollers following the load.
2 – Tumbler
Instead of distributing the pads on the periphery of a cylinder, they would have been directly distributed on the load progression track.
It can be seen in this example that the tumbler has two upper studs and two lower studs in granite or diorite, embedded in a hardwood body. In the rear part, a counterweight embedded in the wooden body allows the tumbler to return to the starting position once released from the load, a stop gives a precise starting position.
The load rests on the tumbler via a mini copper rolling track whose V-shaped ribs serve as a guide for the studs, a mini track following the same principle receives the lower studs.
The length of these mini tracks is slightly greater than the rolling development of the stud.
Let’s take a concrete case to develop the point:
The diameter of the sphere envelope of the stud would be of one hand (5 fingers) is 9.35 mm and the height of the tumbler which would make a fist (6 fingers) is 11.2cm of overall height, the consequence would be that the angle that would make with the vertical the line which joins the high point of contact with the low point of contact is about 4.3 ° while the tumbler would be inclined by 27 °.
Thus the “take off” force to rotate the tumbler would be 7.5% of the weight of the load it carries.
The tumbler would make a total pivoting of 54 °, the development of the lower part and upper part bearings added to the pivoting of the tumbler would give a displacement of the load of 12 cm per tumbler which is the distance at which we should find the next tumbler on standby to take over the load.
Thus at the end of its first movement, the load falls on the next tumbler and the kinematics of the movement are exactly the same as with the roller, so the resistance to advancement is identical for the same dimensional configuration.
Obviously to support the load-carrying plate it is necessary at least permanently to have 3 tumblers in charge.
The plate is alternately supported by 2 tumblers on the right and a tumbler on the left and vice versa at the next tilting. It looks like walking from one foot to the other.
For a better understanding the load-carrying plate has been removed from the animation.
After the passage of the plate, the released tumbler tilts only under the effect of the counterweight towards its waiting position, while the load continues its forward progression. As long as there are tumblers on the track, the load can progress by offering very low resistance to forward movement.
When the pairs of tumblers cross, the plate can thus remain in a stable position.
Thanks to the spherical shape of the contact between the stud and its track which is V-shaped, it is possible to rotate the load-bearing plate, the stud then putting itself slightly across the groove, the radius of the circle must be at least 30 times the width of the tumbler, ie approximately 3 to 4 m.
This can be achieved by gradually orienting the tumbler support on the track.
The load resistance of the block is of the order of 500 KG, which means that the plate always carried at least by 3 tumblers so 6 blocks can be loaded at 3 t. Out of 6 tumblers present on the track under the load carrier plate, only 3 are engaged due to the alternating right / left passage.
In the example illustrated, the minimum length of the platform would have been 0.72 m, the width of the tumbler being 0.1 m, it needed two widths permanently engaged, i.e. a floor area of 0.15 M² for 3 t with 6 supported studs i.e. a load density of 20 t per M² of platform
For heavy loads, we could have both increased the number of pins per tumbler, or put more tumblers in parallel in the same row and the number of rows of tumblers engaged by increasing the length of the load-carrying plate.
For example for a megalith of 65 t, it would have taken 3.25 M² of plate which was possible with its smaller face which measures 1.5 x 2.8 m.
For 130 loaded studs, we could have had 9 rows of loaded tumblers, ie 7 tumblers in parallel per row and a minimum width of the track of 1.4 m.
When you have so many studs supporting the load, a very important thing was the fair distribution of the load on the studs to avoid breakage, which required great dimensional accuracy of the tumblers. But not facilitating a homogeneous distribution of the load between studs, the fact that the body of the tumbler is made of wood gives a certain capacity for elastic deformation and on the other hand that the base of the tumblers, could rest on a wooden plank of a surface and of a thickness such that under the effect of the load the latter crashes more or less while remaining within its elastic deformation limit, which absorbed the slight dimensional deviations left by the manufacture of the tumblers.
Likewise, in the event of a change in slope of the runway, a very gradual connection would have been necessary so that the elasticity of the base of the tumblers could absorb the progressive inclination of the load.
This change of slope in dimension being carried out by bringing the following tumbler closer so that the plate can “climb” on it while its base is higher, and conversely in descent.
Another practical point, these very heavy loads had a journey of around 700 m to accomplish to pass from the landing dock to the base of the pyramid, but being very few in number, their progression (unlike the filling blocks ) could be very slow, thus leaving available the solution of using a fairly short and modular progression track, which operators could have unwound in front of the load by replacing the released modules at the rear.
On the other hand, for journeys on the base or in the access gallery to the freight elevator or in the path of the filling blocks from the quarries, given the traffic of around 500 blocks per day, tracks would have been necessary. completely filled with tumblers permanently.
With the dimensioning of the illustration, the density of tumblers on the track being 6 for 0.72 m, or 8 per linear meter, there were around 1 km of tracks to fill, i.e. 8,000 tumblers and 32,000 plots!
No wonder there are still by thousand of them in museums.
On heavily traveled routes such as those from the quarries to the pyramid, from the access gallery and the distribution of blocks on the course, it was advantageous to place the tracks on a slight slope so that the loads circulate without human intervention a almost like the mine wagons today.
It would then have been sufficient to raise the blocks at the start of the movement, then to make them accelerate on a launching track, to keep their acquired speed on a track whose slight slope compensates for the resistance to the advance of the tumblers, to the order of the %, to be at the end slowed down by a counter slope, then stopped on arrival.
Except for the end of the path of the blocks on the course which must be abrupt, stopped by a stop, so that the block tilts under the effect of its kinetic energy by falling on its own to its final place.
Conversely in the upward slope at 8% between the Nile plain and the pyramid, the rocking track had the considerable advantage of having a ‘sealskin’ behavior, ie very low resistance. when advancing forward, on the other hand in the opposite direction, the track presents a significant friction between the granite studs and their copper support with a coefficient probably of the order of 0.3 / 0.4, that is to say that it would have needed a slope of 30 to 40% for the load to slide back.
This advantage safely allowed non-continuous jerky traction.
Thus the blocks, even the megaliths outside the access ramp from the Nile plain to the pyramid, could move with a minimum of human handling.
In conclusion, the tumbling option was efficient and fairly easy to implement at the price, however, as always in large pyramids with great precision of execution.