Immutable laws of friction: preparing and fitting stone blocks into the Great Pyramid of Giza.

Immutable laws of friction: preparing and fitting stone blocks into the Great Pyramid of Giza. Introduction The exact techniques employed by ancient Egyptian craftworkers inthe construction of the Great Pyramid Great Pyramid,the Cheops’ tomb, built 4,600 years ago, nearly 500 feet high, with bases 755 feet long. [Egypt. Arch.: Brewer Dictionary, 735]See : Wonders, Architectural of Khufu at Giza during the FourthDynasty (c.2649-2513 BC) are still uncertain. Two of the major problemsconcern the preparation and fitting of the large stone blocks, whichwere achieved to a high degree of accuracy. A key factor was thefriction developed between two surfaces, which controlled the degree ofsliding of one stone block over another. Here, data obtained fromexperiments in measuring the blocks shows how plane surfaces could beprepared which were nearly perfectly flat. Other experiments showed howthe blocks could be moved, with the use of lubrication lubrication,introduction of a substance between the contact surfaces of moving parts to reduce friction and to dissipate heat. A lubricant may be oil, grease, graphite, or any substance—gas, liquid, semisolid, or solid—that permits free action of , to lessen theeffects of the immutable IMMUTABLE. What cannot be removed, what is unchangeable. The laws of God being perfect, are immutable, but no human law can be so considered. laws of friction. Preparing surfaces The tasks of the mason consist of producing horizontal and verticalsurfaces which are precisely flat, and these would require cutting andshaping tools and measuring instruments. Replicated and reconstructedcopper, bronze and stone tools for shaping hard and soft stones havebeen manufactured and tested (Zuber 1956: 180, figures 18-20; Stocks1986; 1988: 1, 17-99, II, 246-73). The tests indicated that stones ofhardness Mohs 3, or below (including soft limestone), could effectivelybe cut with copper and bronze chisels and adzes. Stones harder than Mohs3, including even calcite calcite(kăl`sīt), very widely distributed mineral, commonly white or colorless, but appearing in a great variety of colors owing to impurities. (Egyptian alabaster alabaster,fine-grained, massive, translucent variety of gypsum, a hydrous calcium sulfate. It is pure white or streaked with reddish brown. Alabaster, like all other forms of gypsum, forms by the evaporation of bedded deposits that are precipitated mainly from ) had to be worked withdifferent combinations of stone tools--pounders, hammers, picks, axes,chisels, punches, scrapers and sandstone rubbers. In addition to coppertools, stone implements were sometimes employed for shaping andsmoothing soft limestone objects (Petrie 1938: 30) Preparing the surfaces of the Great Pyramid's limestone core-and casing-blocks was a two stage process. The average size of theblocks, according to according toprep.1. As stated or indicated by; on the authority of: according to historians.2. In keeping with: according to instructions.3. W.M.F. Petrie (1883: 210, note) is 50 x 50 x 28inches (1.27 x 1.27 x 0.71m). The bottom surfaces were already flattenedand smoothed before inserting them into the structure of the pyramid(Edwards 1986: 283). The blocks' top surfaces were only made trulyhorizontal, flat and smooth after being fitted into the pyramid (Clarke& Engelbach 1930: 100): this system ensured that any block'stop and bottom surfaces were parallel, essential for making each layerof blocks horizontal throughout the pyramid. The four vertical sides ofa core-black were only roughly finished (Clarke & Engelbach 1930:81), and not intended to fit closely to neighbouring blocks. However,abutting end-faces on casing-blocks formed tightly fitting Adj. 1. tightly fitting - fitting snugly; "a tightly-fitting cover"; "tight-fitting clothes"tight fitting, tight-fitting, tightfitting, skinnytight - closely constrained or constricted or constricting; "tight skirts"; "he hated tight starched collars"; rising-joints. Ancient masons needed reliable tools for checking that thehorizontal joint surfaces of all stone blocks were made accurately flatand truly horizontal, in addition to making flat and parallel therising-joint surfaces of adjacent casing-blocks. Known instruments fortesting horizontal and vertical surfaces all depend upon a hanging plumbline. Such instruments were the frame for testing horizontal planes,shaped like the letter "A", and the vertical testing frame,both made of wood. Models of the horizontal and vertical testing toolswere found in the Nineteenth Dynasty (c.1315-1201 BC) tomb of thearchitect Senedjem at Deir el-Medina, an Upper Egyptian workers'village (Petrie 1917: 42, plate XLVII, B57, 59). The earliest plumb bobs(Petrie 1917: 42, plate XLVIII, B64, 65) date to the Third Dynasty(c.2687-2649 BC). Calibrating a replica 'A' flame (Stocks 1988: II, 368)required the two bottom ends to touch the surface of still water, whilesimultaneously marking a vertical line on the horizontal bar horizontal barEvent in men's gymnastics competition in which a steel bar fixed about 8 ft (2.4 m) above the floor is used for swinging exercises. Competitors generally wear hand protectors and perform routines that last 15–30 seconds. exactlybehind the hanging plumb line. This tool proved to be as reliable as amodern spirit level (Figure 1). A replica vertical testing tool was alsoconstructed (Stocks 1988: II, 369). Provided the two horizontal piecesof wood were accurately made and fitted to the vertical piece, thetool's reliability also compared favourably with a spirit level(Figure 2). Although there is no direct evidence to prove that these twoframes were in use at the Great Pyramid, the evidence for plumb linespredating the Fourth Dynasty and the ability of the masons to createtruly horizontal and vertical surfaces at Giza do support theproposition. [FIGURES 1-2 OMITTED] W.M.F. Petrie (1883: 213) and M.Z. Goneim (1956: 42) noticed redmarks on stone masonry, and suggested that it had been rubbed with anaccurate facing-plate smeared with red ochre Red ochre and yellow ochre (pronounced /'əʊk.ə/, from the Greek ochros, yellow) are pigments made from naturally tinted clay. It has been used worldwide since prehistoric times. to test a surface'sflatness. Petrie (1909: 72) stated that a stone's surface wasconsidered flat enough if the red ochre touched the high points atintervals of not more than an inch (2.5cm). Besides these red ochremarks, there is no evidence to support the use of facing-plates fortesting surface flatness in ancient Egypt. However, there is very goodepigraphic ep��i��graph?n.1. An inscription, as on a statue or building.2. A motto or quotation, as at the beginning of a literary composition, setting forth a theme. and archaeological evidence for a simpler surface testingtool. A scene in the Eighteenth Dynasty (c.1569-1315 BC) tomb ofRekhmire at Thebes, Upper Egypt (Davies 1943: II, plate LXII) shows tworods held upright against a block's perpendicular surface. A tautstring connects the top of each rod. Two other rods are held against thestring to check for high points on the stone. On a perpendicularsurface, a taut string's slight sag, or catenary curve, actstowards the ground, whereas on a horizontal plane horizontal planen.A plane crossing the body at right angles to the coronal and sagittal planes. Also called transverse plane.horizontal planethe string curvestowards it, deceiving a mason into producing a concave ConcaveProperty that a curve is below a straight line connecting two end points. If the curve falls above the straight line, it is called convex. surface. Petrie (1890: 27, plate IX, 13) found a set of three rods atTwelfth Dynasty (c. 1991-1786 BC) Kahun, a workers' town near tothe Fayum: the hole drilled into each of the outer rods is just largeenough for a 2 mm-diameter string. Each rod (Petrie 1890: 27) measures4.96 inches (12.6cm) in length, equal within two or three thousandths ofan inch (0.005cm). How and why did a craftworker make the Kahun rods soaccurate to one another? In the early 1880s, Petrie (1883: 44) measuredthe rising-joints separating several of the remaining largecasing-blocks on the northern side of the Great Pyramid. He found thatthe mean variation of the cutting of the stone from a straight line andfrom a true square equalled 0.01 inch (0.25mm) up a joint 75 inches(1.90m) high. These joints, with an area of some 35 square feet (3.3m)each, were not only worked as finely as this, but also cementedthroughout. It is likely that both the core- and the casing-blocks' bottomand top surfaces were similarly prepared to this accuracy. Could therods and string tool, by itself, have enabled ancient masons to flattena stone block's surface to an accuracy of 0.25mm, and therefore toindicate the tool's use at Giza in the Fourth Dynasty? In trying toanswer this question, a set of three replica rods was manufactured froma seasoned tree branch for testing (Stocks 1987: 45-6, figure 24; hereFigure 3). Each rod was cut to the same length between two stones setfirmly into the ground. This crude, yet effective, calliper ensured thatthe three rods matched each other in length. The accuracy of the Kahunrods points to the use of such a calliper. It did not matter that theactual lengths of the rods in units of measurement Units of measurementValues, quantities, or magnitudes in terms of which other such are expressed. Units are grouped into systems, suitable for use in the measurement of physical quantities and in the convenient statement of laws relating physical quantities. were unknown: extantrod sets do not conform to a standard measurement. For example, each rodin a Twelfth Dynasty set from Beni Hasan, Upper Egypt, measures 8.6cm inlength (Petrie 1917: 42, plate XLIX, B49). [FIGURE 3 OMITTED] Each replica rod's length was checked with a modern instrumentand all were equal within plus or minus 0.005cm. Two rods were drilledfor the string. The experiments (Stocks 1987: 48-50) began by obtaininga horizontal surface, whose flatness was checked with a steelstraight-edge. The test rods were stood upon this surface, with thestring under considerable tension (Figure 3). Measurements indicatedthat test tensioned strings between 1.2 and 2m in length sagged byapproximately 0.25mm, similar to the variation measured by Petrie on thebase casing-blocks at Giza. Ancient masons may have used the rods andstring tool, completely stretched out in a straight line, as an insidecalliper for testing parallelism between the rising-joint surfaces ofadjacent stone blocks, before fitting them into a building (Stocks 1987:48, figure 25). Surface high spots could have been marked by a fingertip fin��ger��tipn.The extreme end or tip of a finger. coated in red ochre (Stocks 1987: 48)--not necessarily by afacing-plate--when a rod was removed from its testing position next tothe string. Subsequently, other masons dressed the high spots down, andas the work became closer to a flat surface, the spacing between the redochre marks would decrease. In ancient times, a surface would be deemedflat when the third rod just touched the underside of the taut stringalong its length. Of course, a block's surface prepared in ahorizontal position horizontal position,n a posture in which the body lies flat and the feet and head remain on the same level. Also calledsupine. would end up slightly concave. However, the surfaceof a block actually shaped and tested in a vertical position, asillustrated in the tomb of Rekhmire, would not suffer such concavity con��cav��i��tyn.A hollow or depression that is curved like the inner surface of a sphere.concavity,n 1. the condition of being concave.n 2. . Friction and force: moving the blocks into position Friction between sliding surfaces of large blocks of limestoneposed a serious problem to craftworkers moving them. The friction thatmust be overcome to move a block is proportional to the coefficient offriction coefficient of frictionn. pl. coefficients of frictionThe ratio of the force that maintains contact between an object and a surface and the frictional force that resists the motion of the object. [micro] and the normal force N (Timoshenko & Young 1956:50). The coefficient of friction is a function of the type of surfacesin contact and the Normal force is the vertical force of gravity actingon the block. The force F required to move a block is F = [micro]N. If Fis taken as the force necessary to start sliding, [micro] is called thecoefficient of static friction. (If F is taken as the somewhat smallerforce necessary to maintain sliding, [micro] is called the coefficientof kinetic friction). The coefficient of static friction is the tangentof the angle of a ramp on which a block just starts to slide down. Itcan therefore be measured experimentally. It can be seen that the forcerequired is independent of the areas in contact, and since the weight isfixed, the ease of moving a block can only be altered by altering thecoefficient of friction, that is the character of the surfaces incontact. This what the Egyptians did. In the Twelfth Dynasty tomb of Djehutihotep, at el-Bersheh, UpperEgypt (Newberry 1895: I, plate XV), there is an illustration of analabaster statue of him, thought to weigh about 60 tonnes; which isbeing hauled along a level surface on a sledge by 172 men. A man ispouring some liquid, probably water, in front of the sledge'srunners to maintain a muddy track to ease the friction. Once on abuilding, Egyptian masons' use of gypsum gypsum(jĭp`səm), mineral composed of calcium sulfate (calcium, sulfur, and oxygen) with two molecules of water, CaSO4·2H2O. It is the most common sulfate mineral, occurring in many places in a variety of forms. mortar as a slidinglubricant (Clarke & Engelbach 1930: 78-80) between blocks alsosignificantly reduced the friction between the horizontal surfaces ofone block and the one below. Automatically, and essentially, the fillingof the slight spaces between imperfectly fitting horizontal joints withmortar prevented blocks from cracking, the mortar setting hard andevenly transmitting the load over supporting blocks' top surfaces(Clarke & Engelbach 1930: 78-9). This suggests that sledgeslubricated with mud were used to transport the blocks to the pyramid andonce laid on the course of stones, they would be moved on a layer ofgypsum. Since the surface area involved is not contributory to the forcerequired, experiments were carried out to measure different coefficientsof friction by masoning two small blocks of limestone to a tolerance of0.25mm (one hundredth of an inch) (Figure 4). The prepared blocks'dry flat surfaces were placed in contact, one block above the other, thebottom block being slowly tilted until the top block just began to slideacross its surface. The angle of tilt was 36 degrees. The tangent ofthis angle gives a coefficient of static friction of 0.73. The test wasthen repeated with liquid mortar applied to the bottom block's topsurface. The upper block now commenced sliding at an angle of 8 degrees,giving a coefficient of static friction of 0.14. Another experimentrevealed that a wooden sledge runner on liquid mud produced a similarcoefficient of static friction. [FIGURE 4 OMITTED] Petrie (1883: 44) stated that a base casing-block positioned on theGreat Pyramid's northern side weighed 'some 16 tons' (16300 kg). To find the force, F, to start this block to slide dry on aflat and smoothed stone surface, its weight must first be converted tothe Normal force, N, in Newtons, i.e., 16 300 x 9.8 = 159 740 Newtons.The sliding force, F, can now be calculated by multiplying thecoefficient of static friction of 0.73 by the Normal force, N. F = 116610 Newtons. To find the force, F, needed to start the same blocksliding on a surface lubricated with liquid mortar, the lessercoefficient of static friction of 0.14 must be used, giving F = 22 363Newtons. These results show that just over five times less force isneeded to start a lubricated block moving than a similar dry block. Thisreduction factor applies to all blocks, no matter what their weight andarea of surface contact. The Djehutihotep illustration suggests that one worker was capableof initiating and maintaining a pulling force of about 500 Newtons(about 50 kg) in order to start the statue moving from rest. Therefore,about 45 workers could have started a lubricated 16 300 kg block movingon a horizontal surface. Once started, the force required to keep theblock moving would drop, allowing it to be pulled forwards at a constantrate. A smaller, lubricated Great Pyramid casing-block of about 2750 kgwould require an initial force of 3770 Newtons (about 385 kg). Eightworkers could easily start a block of this weight moving on a levelsurface. Hauling a block on a sledge up a slope, as would be required to fitit into a pyramid, required a balance between the force required and theangle at which slippage occurred. The force required to haul a block upa plane inclined at the angle of slippage is twice that required on theflat (Timoshenko & Young 1956: 162-7). This and the risk of losing ablock through slippage means that the ramp should be inclined at lessthan the angle of slippage. This might explain why the angle of slopefor some ancient ramps was less than eight degrees, the angle ofslippage for a mud-lubricated sledge (above). For example, the gradientof a ramp left in the unfinished Fourth Dynasty mortuary temple ofMenkaure at Giza is about one in eight, or just over seven degrees(Edwards 1986: 280). Also, two stone-built ramps excavated at thesouthern end of the Gebel el-Asr region, Lower Nubia (Shaw et al. 2001:34), where gneiss gneiss(nīs), coarse-grained, imperfectly foliated, or layered, metamorphic rock. Gneiss is characterized by alternating light and dark bands differing in mineral composition and having coarser grains than those of schist. was extracted from the quarries there, slope at sevendegrees. Ramps sloping upwards at eight degrees and higher, are likelyto have been used dry, it being both counterproductive and dangerous tolubricate lu��bri��cate?v. lu��bri��cat��ed, lu��bri��cat��ing, lu��bri��catesv.tr.1. To apply a lubricant to.2. To make slippery or smooth.v.intr.To act as a lubricant. such a ramp. Conclusions The experiments with the three replica surface-testing toolsindicate their presence at Giza in the Fourth Dynasty: they, alone,could have enabled craftworkers to prepare the limestone blocks fittedinto the Great Pyramid of Giza "Great Pyramid" redirects here. You may have been looking for the Great Pyramid of Cholula in Mexico.The Great Pyramid of Giza is the oldest and largest of the three pyramids in the Giza Necropolis bordering what is now Cairo, Egypt in Africa, and is the only remaining with the accuracy that has been observed.The sliding experiments revealed significant advantages in moving stoneblocks, and loaded sledges, along mortar- and mud-lubricated horizontalsurfaces, and suggest an optimum of around seven degrees for alubricated ramp. Acknowledgement I sincerely thank Jeffrey Stocks for patiently explaining the lawsof friction to me. However, the results obtained by applying these lawsin the context of moving Fourth Dynasty limestone blocks at Giza areentirely my responsibility. References CLARKE, S. & R. ENGELBACH. 1930. Ancient Egyptian masonry.London: Oxford University Press. DAVIES, N. DE G. 1943. The tomb of Rekh-mi-Ree at Thebes. New York New York, state, United StatesNew York,Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of :Metropolitan Museum of Art. EDWARDS, I.E.S. 1986. The pyramids of Egypt. Harmondsworth: Viking. GONEIM, M.Z. 1956. The buried pyramid. London: Longmans, Green andCo. NEWBERRY, P.E. 1895. El-Bersheh. London: Egypt Exploration Fund. PETRIE, W.M.F. 1883. The pyramids and temples of Gizeh. London:Field and Tuer. --1890. Kahun, Gurob and Hawara. London: Kegan Paul, Trench,Trubner and Co. --1909. The arts and crafts arts and crafts,term for that general field of applied design in which hand fabrication is dominant. The term was coined in England in the late 19th cent. as a label for the then-current movement directed toward the revivifying of the decorative arts. of ancient Egypt. Edinburgh &London: T.N. Foulis Ltd. --1917. Tools and weapons. London: British School of Archaeology inEgypt. --1938. Egyptian architecture. London: British School ofArchaeology in Egypt. SHAW, I., E. BLOXHAM, J. BUNDURY, R. LEE, A. GRAHAM & D.DARNELL. 2001. Survey and excavation at the Gebel el-Asr gneiss andquartz quarries in Lower Nubia (1997-2000). Antiquity 75: 33-4. STOCKS, D.A. 1986. Tools of the ancient craftsman. PopularArchaeology 7 (6): 25-9. --1987. Experimental stone block fitting techniques: proposed useof a replica ancient Egyptian tool. Manchester Archaeological Bulletin2: 42-50. --1988. Industrial technology at Kahun and Gurob: experimentalmanufacture and test of replica and reconstructed tools with indicateduses and effects upon artefact See artifact. production. MPhil thesis, University ofManchester The University of Manchester is a university located in Manchester, England. With over 40,000 students studying 500 academic programmes, more than 10,000 staff and an annual income of nearly ��600 million it is the largest single-site University in the United Kingdom and receives . TIMOSHENKO, S. & D.H. YOUNG. 1956. Engineering mechanics.Tokyo: McGraw-Hill Kogakusha Ltd. ZUBER, A. 1956. Techniques du travail TRAVAIL. The act of child-bearing. 2. A woman is said to be in her travail from the time the pains of child-bearing commence until her delivery. 5 Pick. 63; 6 Greenl. R. 460. 3. des pierres dures dansl'ancienne Egypte. Techniques et civilisations 29: 161-80. Denys A. Stocks, 6 Greenacres Drive, Burnage, Manchester M19 1SB(Email: dstocks@ukonline.co.uk)