The Beautiful Cubit System

The Beautiful Cubit System I Douglas 2019 The Beautiful Cubit System Ian Douglas, B.Sc ian@zti.co.za 16 September 2019 Version 1.0.6 DOI: https://doi.org/10.5281/zenodo.3263863 This work is licensed under the Creative Commons Attribution 4.0 International License. Abstract An analysis of the Egyptian Royal cubit, presenting some research and opinions flowing from that research, into what I believe was the original cubit, and how it was corrupted. I show various close arithmetic approximations and multiple ways of getting the divisions of the cubit, as well as some related measures. The cubit also encapsulates the basic components for the metric system. Keywords: Egyptology, metrology, royal cubit, cubit, metre, foot, metric system Contents 1. Introduction 2. Overview of current understanding 3. An alternative origin 4. Different ways of approximating the royal cubit 5. Different ways of getting the cubit divisions 6. Geometry, the Royal Cubit and the metric system 7. Bibliography 1. Introduction The cubit is a well-know ancient measure of length, used around various places in the Middle East and Mediterranean region in the distant past. 1 The Beautiful Cubit System I Douglas 2019 It is allegedly based on the length of a human (male) fore-arm. It is typically measured from the back of the elbow to some point between the wrist and the end of the outstretched middle finger, or in some variants, a point beyond that. The problem with this approach is that everyone’s arm is a different length. If the heights of the dynastic Egyptians is taken as representative, then their arms would have been too short to justify the accepted lengths. There is also the issue of a whole range of different cubit lengths, not only between different cultures, but even within the same culture. So I propose a different origin, based on mathematics, and dating back to a much earlier time. Changes from version 1.0.0: 1.0.1 to 1.0.5 : Added formulas with ρ 1.0.6 : Added Fractal dimension of the boundary of the dragon curve Cd,, and more exact approximation with ρ³. Added comments about ₢ being 0.5237. Added approximation for Grand Metre with ρ³. Added symbols table and assorted fix-ups. Symbols used in this and other papers: Name Archimedes’ constant Circle constant Euler’s number e-1 Golden ratio Plastic number Royal cubit Cubit Grand metre Foot, Imperial Foot, Egyptian “Megalithic yard” (₢ + F) Symbol π τ e é φ ρ ₢ Ͼ ℳ F Ⓕ Ɱ Approximate value 3.14159265... 6.283185... (2π) 2.71828... 1.7183... 1.618034... 1.324718... 0.5236m (π/6) 0.4488m (π/7) 1.5236m = 1m + 1₢ 0.3048m or 0.3047 (from ₢/é) 0.3000m or 0.2992m (from τ/21) 0.8284m I would have preferred a better symbol for the Egyptian foot but Unicode has a limited selection of F shapes 2 The Beautiful Cubit System I Douglas 2019 2. Summary of current understanding Mark Stone’s overview [1] covers the different cubit lengths in different cultures, as well as issues regarding human anatomy as the basis for the cubit and other measures. Quentin Leplat [2] analysed the Turin cubit, noting that it is 0.5236m long, and consists of 24 digits of 18.5mm, and 4 of 19.75mm. If I can summarize the current consensus regarding the cubit, it would be something like this: 1. The cubit was based on the length of a forearm, from the back of the elbow to some point from the wrist to the end of the extended middle finger, or possibly further. 2. Different cubits exist because different communities each made their own. 3. The standard may actually have been the arm of some king, at some point in time. 4. The divisions are similarly based on and named after various other body parts, like palm, span or digit. There are several problems with this consensus. 1. Measuring from the elbow: Depending on how hard the arm is pressed against some “zero point” backstop, you can change the measured length by a few millimetres. Given that cubit lengths are usually quoted down to fractions of a millimetre, this alone will give varying results. 2. If we take the height of the Egyptians as typical for populations in the area (or in any event, as a sample), then their heights do not support the standard short cubit of about 45cm. Consider: “The average height of the male population varied between 161 cm (5.28 feet) in the New Kingdom (about 1550–1070 BC) and 169.6 cm (5.56 feet) in the Early Dynastic period (about 2925–2575 BC), making an average of 165.7 cm (5.43 feet) for all time periods.” [3] 3. If we compare the short cubit of 45cm, or as is more usually stated, 18”, with the royal cubit of say 20.6”, then we have a different problem. The difference is 2.6 inches or 66mm. However, the royal cubit is a short cubit plus a palm, with a palm normally given as 75mm. So this does not work either. 3 The Beautiful Cubit System I Douglas 2019 3. An alternative origin As discussed in my other two papers [4] and [5], the cubit may be very much older than we think. So I propose that instead of saying it was based on the length of a forearm, we look at the numbers more closely, and at how a highly logical population would derive it. We start with the royal cubit rather than the regular cubit. I am convinced that those who say that the royal cubit was based on a circle with diameter one metre, are correct. The royal cubit would then be π/6 metres, or 0.5236m (to 4 places) long. One particular issue that pops up here, is the actual value of π. In his commentary [6] on a paper by Bauval & Bauval [7], as well as in other works, Sivertsen proposes that the ₢ is 0.5237 rather than 0.5236m, based on measurements by Petrie. He also suggests we should be using a π value of 22/7 rather than the correct value. In response, measuring blocks of stone is very difficult, and when we get down to ¹/₁₀th of a mm, we’re talking about a size that is less than the diameter of a grain of sand. Measuring that consistently correctly using a metal ruler can not be accurate, apart from thousands of years of wear-and-tear on the blocks affecting the measurements. As to the value of π, the dynastic Egyptians may have used 22/7, or 3.16 (based on 256/81), but it may be a moot point. In my opinion the ₢ originated long before the dynastic Egyptians, and judging by what else they appear to have known, I expect that they knew the proper value for π. The population that invented this disappeared a long time ago. Some time on this side of the last ice age, our forefathers in the middle east found one or more surviving cubit rods and adopted it, perhaps as “given by the gods.” This found cubit was copied and spread around. Bad copies led to varying lengths. At some point, people noticed it was “about” the length of their forearm plus hand, and back-named the length accordingly, as well as the subdivisions. I can’t answer the question of how they had the metre to start, but they clearly did. Perhaps the answer will surface in due course. 4. Different ways of approximating the royal cubit If we start with π/6, then there are two well-known approximations that produce values close to this, both based on π and/or φ, the golden ratio. These are φ²/5, and π - φ². However, there are other formulas that I either figured out or rediscovered, that give better approximations. These are listed in Tables 2, 3, 4 and 5 in decreasing order of closeness to π/6. 4 The Beautiful Cubit System I Douglas 2019 First we put the differences in perspective in Table 1, using values supplied by Wikipedia. [8] Microns Metres 0.04 1.5 2 3.5 Less than / about 0.00000004 Length of a lysosome 0.00000150 Anthrax spore 0.00000200 Length of an average E. coli bacteria 0.00000350 Size of a typical yeast cell 5 0.00000500 Length of a typical human spermatozoon's head 7 0.00000700 Diameter of human red blood cells 10 0.00001000 Transistor width of the Intel 4004 17 0.00001700 Minimum width of a strand of human hair 30 0.00003000 Length of a human skin cell 50 0.00005000 Typical length of a human liver cell 60 0.00006000 Length of a sperm cell 100 0.00010000 The smallest distance that can be seen with the naked eye 181 0.00018100 Maximum width of a strand of human hair 200 0.00020000 Typical length of Paramecium caudatum, a ciliate protist 500 0.00050000 Typical length of Amoeba proteus, an amoeboid protist Table 1: Putting small distances in perspective Table 2 has very close approximations for the Royal Cubit (henceforth ₢). Method Value Abs difference from π/6 Rounded π/6 0.523598776 0.000000000 0.5236 ((6√2/10)² + (6/100)²+(8√2/10000)²)² 0.523598812 0.000000037 0.5236 ρ³/⁶√7660 (see below) 0.523599000 0.000000225 0.5236 cube roots (see below) 0.523600350 0.000001575 0.5236 ((6√2/10)² + (6/100)²)² 0.523596960 0.000001816 0.5236 (7π/5e)²/5 0.523596637 0.000002138 0.5236 28φπ/100e 0.523601717 0.000002942 0.5236 φe/8.4 0.523603856 0.000005080 0.5236 5 The Beautiful Cubit System I Douglas 2019 Method Value Abs difference from π/6 Rounded ln(4) (see below) 0.523591499 0.000007277 0.5236 ((1 + π)/e) - 1 0.523606791 0.000008015 0.5236 φ²/5 0.523606798 0.000008022 0.5236 Table 2: Formulas giving approximations very close to π/6 The “cube roots” formula is 1 √ 3 ( √2⋅( √5−√ 3) 7− 3 3 3 10 ) The ln(4) formula is the solution to the equation ln (4)+ x= 1 x We should also point out that thanks to Euler and the Zeta function, we can also write π/6 ∞ as ζ (2) π or 1 ∑ n2 n =0 π as both equal to ₢ precisely. Also, thanks to the nature of the golden ratio φ and the plastic constant ρ, we can also write π π as π or 2 13 6 1 1 2∑ n 3∑ n n=1 ρ n =0 φ In the same way that we can approximate similarly use the plastic ratio ρ by 2 π with the golden ration φ by φ , we can 6 5 ρ3 ρ3 , or more exactly 6 4.4 √7660 The plastic constant ρ is a real root for the equation ρ + 1 = ρ³. φ and ρ are the only two morphic numbers greater than 1. I have yet to find the relevance of 7660 or the sixth root of anything, but I’ve included it for completeness. Perhaps the justification will surface in the future. The formulas are easier to follow when shown in conventional form: 6 The Beautiful Cubit System ∞ 1 ∑ n2 ζ (2) ₢ = π = π = n=0π 6 ≈ (( ) ( ) ( 6 √2 10 2 + 2 6 100 I Douglas 2019 + =π 2 = π 13 1 1 3∑ n 2∑ n n=0 φ n=1 ρ 8 √2 10000 )) 2 2 ≈ ρ3 ≈ 6 √ 7660 √ √2⋅( √5−√ 3) 7− ( ) ( 3 ( ) ( 3 (( ) ( ) ) 6 √2 6 + 10 100 2 1 3 3 10 ) ≈ 2 2 ) 2 7 φ π 28 φ π φe φ2 1 7π 1 1+ π ≈ ≈ ≈ ≈ ≈ ln(4)+x= ≈ −1 ≈ 5 5e 25 e 100 e 8.4 x e 5 Next are formulas giving close values in Table 3. Method Value Abs difference from π/6 Rounded π – (7π/5e)² 0.523609468 0.000010692 0.5236 eφ³/7π 0.523611878 0.000013103 0.5236 ρ³/4.44 = (ρ+1)/4.44 0.523585126 0.000013650 0.5236 (10φ)/(11e + 1) 0.523616953 0.000018177 0.5236 tan(2φπe) = tan(τφe) 0.523569002 0.000029774 0.5236 (φ²/(e-1))-1 0.523634799 0.000036024 0.5236 π – φ² 0.523558665 0.000040111 0.5236 √(√5/(3e)) 0.523642193 0.000043417 0.5236 e(2√2 - φ)/2π 0.523649949 0.000051174 0.5236 Table 3: Formulas giving close approximations of π/6 The conventional formulas are like this: ₢≈π− ≈ ( ( ) 7π 5e 2 ≈ ) e φ3 ρ3 10 φ ≈ ≈ ≈ tan(2 π φ e) ≈ tan( τ φ e) 7π 4.44 11 e+1 √ φ2 5 e (2 √ 2− φ) e (2 √ 2−φ ) −1 ≈ π −φ2 ≈ √ ≈ ≈ τ e−1 3e 2π Table 4 has less-close approximations of ₢, but still better than 0.5250. 7 The Beautiful Cubit System I Douglas 2019 Method Value Abs difference from π/6 Rounded (ρ³+√2)/(π+2²) 0.523543095 0.000055681 0.5235 10e/(36³√3) 0.523542042 0.000056733 0.5235 1/(√(ln365.25/φ)) 0.523656373 0.000057597 0.5237 ln(10)/φe 0.523520348 0.000078427 0.5235 2√5/πe 0.523685613 0.000086838 0.5237 1/log(φ²π³) 0.523717901 0.000119125 0.5237 ρ⁹/24 0.523479368 0.000119408 0.5235 (10/φπⅇ)² 0.523764441 0.000165665 0.5238 1/(ρ+2-√2) 0.523421984 0.000176792 0.5234 (10√2)/27 0.523782801 0.000184025 0.5238 square roots (see below) 0.523403737 0.000195039 0.5234 cos(π(4φe+1)) 0.523808794 0.000210019 0.5238 (ρφ+1)/6 0.523906447 0.000307671 0.5239 π.10^8/2c 0.523961255 0.000362479 0.5240 1/(2φ-ρ) = 1/(√5 +1-ρ) 0.523190410 0.000408366 0.5232 e/(3√3) 0.523133582 0.000465194 0.5231 (∛3 ∛5 ∛7)/9 0.524188220 0.000589444 0.5242 (∛5φ²)/πⅇ 0.524228861 0.000630086 0.5242 sin((φ²πe√2) 0.524253715 0.000654940 0.5243 (φ)/(e + 1/e) 0.524286921 0.000688145 0.5243 (√2+√3)/6 0.524377395 0.000778619 0.5244 √(7√2)/6 0.524391047 0.000792272 0.5244 π/(φ³√2) 0.524411195 0.000812419 0.5244 Table 4: Less-close approximations of π/6 The “square roots” formula is 1 √(√ 2+√ 5) Here are these conventionally: 8 The Beautiful Cubit System I Douglas 2019 ( ρ + √2) ln(365.25) 10 e 10 e ₢=π ≈ ≈ 3 ≈ 2 23 ≈ 2 φ 6 ( π +2 ) 36 √3 2 3 √3 ( 3 ≈ ) − 1 2 ≈ ln (10) 2 √ 5 ≈ φe πe 2 ρ9 1 10 √2 10 1 1 ≈ ≈ ≈ ≈ ≈ 2 3 3 24 π φ e ρ+2−√ 2 log10 ( φ π ) 3 √( √2+ √5) ( ) ≈ cos( π (4 φ e +1)) ≈ ρφ + 1 π 108 1 1 e ≈ ≈ ≈ ≈ 6 2c 2 φ −ρ (√ 5+1−ρ ) 3 √ 3 3 3 3 2 √3 √ 5 √ 7 ≈ √5 φ ≈ sin (√ 2 φ2 π e) ≈ 3 ≈ πe 2 3 ≈ φ e +e −1 √2+ √3 ≈ √ 7 √ 2 ≈ π 2×3 6 φ3 √ 2 5. Different ways of getting the cubit divisions 5.1 The different extant lengths It appears that apart from making bad copies, the ancients decided to “improve” the cubit subdivisions, in both directions. They did this by changing the length of the digit. One change was from 18.7mm to 18.75mm, leading to the 45cm short cubit and 52.5cm royal cubit. This digit size also produces the 30cm Egyptian Foot, as well as other measures based around 7.5cm intervals. The other change was a move to a digit of 18.5mm, which led to further complications, resulting in the curious Turin cubit with its two digit sizes. 18.7 x 24 = 448.8mm = short cubit (original). 18.7 x 28 = 523.6mm = royal cubit (original). 18.75 x 24 = 450mm = short cubit (variant 1). 18.75 x 28 = 525mm = royal cubit (variant 1). 18.75 x 16 = 300mm = Egyptian foot (variant 1). 9 The Beautiful Cubit System I Douglas 2019 18.5 x 24 = 444mm = short cubit (variant 2). 18.5 x 28 = 518mm = royal cubit (variant 2), which doesn’t work, hence they had to do 18.5 x 24 = 444mm, plus 4 x 19.75 = 79mm, giving 523mm. This is an explanation for the various cubit lengths ranging from 523 to 525mm. In truth, it is difficult for modern students with sharp pencils and accurate rulers, to differentiate between a line of 18.7 and 18.75mm. You need to use micrometer-style or slide-rule techniques as discussed by Monnier et al. [9] Figure 0 shows two lines, one 18.5mm and the other 18.7mm, to demonstrate how subtle the difference is. Obviously the difference between 18.7 and 18.75mm will be even harder to see. This is a screenshot of a drawing done with SVG and may print out slightly differently. Figure 0: 18.5mm vs. 18.7mm 5.2 The π method I first heard that the royal cubit was π/6 from Robert Bauval, but have seen references to someone back in the 1800’s who first proposed it, possibly Karl Richard Lepsius. The thinking is that you take a circle with diameter of 1 metre, which gives a circumference of πm. You then take one sixth of this (i.e. a 60° arc) and that is the royal cubit ₢. This division matches nicely with a six-spoked chariot wheel, and some Egyptian chariots had six spokes and a diameter of close to 1 metre. [10] If we accept that π/6 from a circle of diameter one metre was the origin of the ₢, then it is simple to generate the divisions of the cubit following the same pattern. These are compared to the “reference values” taken from Wikipedia [11], which we can use as “currently accepted” even though I disagree with them. They are similar to the figures from “The Cadastral Survey of Egypt” [12]. Table 5 has values for the divisions of the cubit, using π and τ, where τ is 2π. For the π values, we can use a divisor of 168, and a divisor of 336 for τ. We just need to multiply by the number of digits in each division to get the answer. 10 The Beautiful Cubit System I Douglas 2019 There appears to be conflicting opinions about the remen, one based on it being 20 digits, and the other setting it as half the diagonal of a square of 1₢ side, which is also the height measured from the diagonal. This method shows the beauty of the relationship between the short cubit (henceforth Ͼ) and ₢. The ₢ is π/6, and the Ͼ is π/7. That is the origin of this paper’s title. The Nby-rod, a measure used by builders, has its own special beauty in referencing π. Digits Length Reference Value Formula π Formula τ Value 1 Digit 0.01875m 1π = π 168 168 τ 336 0.0187m 4 Palm 0.0750m 4π = π 168 42 4τ = τ 336 84 0.0748m 5 Hand 0.0938m 5π 168 0.0935m 2π 67 5τ 336 τ 67 0.0938m 6 Fist 0.1125m 6π = π 168 28 6τ = τ 336 56 0.1122m 8 Double Handbreadth 0.1500m 8π = π 168 21 8τ = τ 336 42 0.1496m 12 Small span 0.2250m 12 π = π 168 14 0.2244m 14 Great span 0.2600m 14 π = π 168 12 12 τ = τ 336 28 τ 24 16 Foot 0.3000m 16 π 2π = 168 21 τ 21 0.2992m Remen 0.3702m π = ₢ 6 √2 √ 2 τ 12 √ 2 0.3702m 20 Remen 0.3750m 20 π 168 24 Cubit (standard) 0.4500m 24 π =π 168 7 5τ 84 τ 14 28 Cubit (royal) ₢ 0.523m or 0.525m 28 π =π 168 6 τ 12 0.5236m 32 Pole 0.6000m 32 π 4π = 168 21 4τ 42 0.5984m 11 0.2618m 0.3740m 0.4488m The Beautiful Cubit System I Douglas 2019 Digits Length Reference Value 36 Nby-rod (not on Wikipedia) 0.67 – 0.68m 64 Double pole (not 1.2000m on Wikipedia) Formula π Formula τ Value 36 π 3π = 168 14 3τ 28 0.6732m 64 π 8π = 168 21 8τ 42 1.1968m Table 5: Divisions of the cubit based on π or τ In their book The Lost Science of Measuring the Earth [13], Heath and Michell refer to a ‘sacred’ cubit of 2.057142857 feet, which converts to 0.627017m. This value slots into the above table nicely at π/5 = 0.62832m. The term ‘sacred cubit’ may be confusing as others use it as a synonym for the royal cubit. There is also Isaac Newton’s version at 25.025 British inches, which is supposed to give a 25 “pyramid inch” sacred cubit. 5.3 The √5/πe method I’m going to show alternative ways of dividing the cubit using famous mathematical constants, mostly π, φ, e, √2 and √5. First up is a version that produces values very close to Table 5, just a fraction larger as we get to the bigger lengths because the digit is fractionally larger. It is based on √5/πe. Digits Length Reference Value Formula Value 1 Digit 0.01875m 1 √5 14 π e 0.0187m 4 Palm 0.0750m 4 √5 14 π e 0.0748m 5 Hand 0.0938m 5 √5 14 π e 0.0935m 6 Fist 0.1125m 6 √5 14 π e 0.1122m 8 Double 0.1500m Handbreadth 8 √5 14 π e 0.1496m 12 Small span 0.2250m 12 √ 5 14 π e 0.2244m 14 Great span 0.2600m 14 √ 5 = 14 π e 16 Foot 0.3000m 16 √ 5 14 π e 12 √5 πe 0.2618m 0.2992m The Beautiful Cubit System Digits Length I Douglas 2019 Reference Value Formula Value Remen 0.3702m √2√ 5 πe 0.3703m 20 Remen 0.3750m 20 √ 5 14 π e 0.3741m 24 Cubit (standard) 0.4500m 24 √ 5 14 π e 0.4489m 28 Cubit (royal) ₢ 0.523m or 0.525m 28 √ 5 2 √ 5 = 14 π e πe 0.5237m 32 Pole 0.6000m 32 √ 5 14 π e 0.5985m 36 Nby-rod (not 0.67 – 0.68m on Wikipedia) 36 √ 5 14 π e 0.6733m 64 Double pole (not on Wikipedia) 64 √ 5 14 π e 1.1970m 1.2000m Table 6: Divisions of the cubit based on √5/πe. 5.4 The π/√2 method We then look at the problematic version where the digit is 18.5mm. This is based on π , √2 or by using a divisor of 120. Of necessity, the ₢, its half-value the great span, and one of the remen do not fit the digit-multiplier pattern. Digits Length Value Formula π 120 √ 2 Value 1 Digit 0.01875m 4 Palm 0.0750m 4π = π 120 √2 30 √2 0.0741m 5 Hand 0.0938m 5π = π 120 √2 24 √ 2 0.0926m 6 Fist 0.1125m 6π = π 120 √2 20 √ 2 0.1111m 8 Double Handbreadth 0.1500m 8π = π 120 √2 15 √2 0.1481m 12 Small span 0.2250m 12 π = π 120 √2 10 √2 0.2221m 13 0. 0185m The Beautiful Cubit System Digits Length I Douglas 2019 Value Great span 0.2618m Foot 0.3000m Remen 0.3702m 20 Remen 0.3750m 24 Cubit (standard) 0.4500m Cubit (royal) ₢ 0.5236m 32 Pole 36 64 16 Formula Value π π π 0.2618m = = 2 √ 18 √ 2 2 √ 36 12 16 π 2π 0.2962m = 120 √2 15 √2 20 π 0.3702m = π 120 √2 6 √ 2 24 π = π 120 √2 5 √2 π = π =π 18 √ √ 2 √36 6 0.4443m 0.6000m 32 π 4π = 120 √2 15 √2 0.5924m Nby-rod 0.67 – 0.68m 36 π 3π = 120 √2 10 √2 0.6664m Double pole 1.2000m 64 π 8π = 120 √2 15 √2 1.1848m 0.5236m Table 7: Poor divisions of the cubit based on π/√2 5.5 The π/φ² method We can now look at the various ways of getting the divisions of the other slightly larger cubit, of 0.525m, based on a digit of 18.75mm. The first version uses π and φ². These formulas handle both versions of the remen, great span and ₢ rather elegantly. Digits Length Value Formula Value 1 Digit 0.01875m 1π 64 φ2 0.01875m 4 Palm 0.0750m 4π 64 φ2 0.0750m 5 Hand 0.0938m 5π 64 φ2 0.0938m 6 Fist 0.1125m 6π 64 φ2 0.1125m 8 Double Handbreadth 0.1500m 8π 64 φ2 0.1500m 14 The Beautiful Cubit System Digits Length I Douglas 2019 Value Formula Value Small span 0.2250m 12 π 64 φ2 0.2250m Great span 0.2618m 0.2618m 14 Great span 0.2625m π− φ2 2 14 π 64 φ2 16 Foot 0.3000m 16 π 64 φ2 0.3000m Remen 0.3702m π− φ2 √2 0.3702m 20 Remen 0.3750m 20 π 64 φ2 0.3750m 24 Cubit (standard) 0.4500m 24 π 64 φ2 0.4500m Cubit (royal) ₢ 0.5236m π −φ 2 0.5236m 28 Cubit (royal) ₢ 0.5250m 28 π 64 φ2 0.5250m 32 Pole 0.6000m 32 π 64 φ2 0.6000m 36 Nby-rod 0.67 – 0.6 m 36 π 64 φ2 0.6750m 64 Double pole 1.2000m 64 π 64 φ2 1.2000m 12 0.2625m Table 8: Formulas for the large royal cubit using π and φ² 5.6 The φe/π method The next set of formulas are based on π, φ and e. The general form uses multiples of 3/224 of φe/π, except for the remen, great span and ₢, which flip the irrationals slightly and use πφ/e instead. Digits Length Value Formula Value 1 Digit 0.01875m 3φe 3 φ e 0.01875m =1 224 π 224 π 4 Palm 0.0750m 3φe 3φe =4 56 π 224 π 15 0.0750m The Beautiful Cubit System Digits Length I Douglas 2019 Value Formula Value 5 Hand 0.0938m 15 φ e 3 φ e 0.0938m =5 224 π 224 π 6 Fist 0.1125m 18 φ e 3 φ e 0.1125m =6 224 π 224 π 8 Double Handbreadth 0.1500m 3φe 3φe =8 28 π 224 π 12 Small span 0.2250m 9φe 3 φ e 0.2250m = 12 56 π 224 π Great span 0.2618m 7φπ 50 e 14 Great span 0.2625m 3 φe 3 φ e 0.2625m = 14 16 π 224 π 16 Foot 0.3000m 3φe 3 φ e 0.3000m = 16 14 π 224 π Remen 0.3702m 7 φπ 25 e √ 2 20 Remen 0.3750m 15 φ e 3 φ e 0.3750m = 20 56 π 224 π 24 Cubit (standard) 0.4500m 9φe 3 φ e 0.4500m = 24 28 π 224 π Cubit (royal) ₢ 0.5236m 7φπ 25 e 28 Cubit (royal) 0.5250m 3φe 3 φ e 0.5250m = 28 8π 224 π 32 Pole 0.6000m 3φe 3 φ e 0.6000m = 32 7π 224 π 36 Nby-rod 0.67 – 0.68m 27 φ e 3 φ e 0.6750m = 36 56 π 224 π 64 Double pole 1.2000m 6φe 3 φ e 1.2000m = 64 7π 224 π 0.1500m 0.2618m 0.3702m 0.5236m Table 9: Formulas for the large cubit divisions using π, e and φ. 5.7 The e/π∛3 method We now show formulas based on π, e and ∛3. These formulas are also starting to drift from the “accepted” values as per Wikipedia. The classic values for ₢, great span and remen can not be handled. 16 The Beautiful Cubit System Digits Length I Douglas 2019 Value Formula Value 1 Digit 0.01875m 1e 32 π √3 3 0. 01875m 4 Palm 0.0750m 4e 32 π √3 3 0.0750m 5 Hand 0.0938m 5e 32 π √3 3 0.0937m 6 Fist 0.1125m 6e 32 π √3 3 0.1125m 8 Double Handbreadth 0.1500m 8e 32 π √3 3 0.1500m 12 Small span 0.2250m 12 e 32 π √3 3 0.2250m Great span 0.2618m 0.2625m 14 e 32 π √3 3 0.2625m Foot 0.3000m 16 e 32 π √3 3 0.3000m Remen 0.3702m 20 Remen 0.3750m 20 e 32 π √3 3 0.3750m 24 Cubit (standard) 0.4500m 24 e 32 π √3 3 0.4500m 0.5250m 28 e 32 π √3 3 0.5249m 14 16 Cubit (royal) ₢ 28 0.5236m 32 Pole 0.6000m 32 e 32 π √3 3 0.5999m 36 Nby-rod 0.67 – 0.68m 36 e 32 π √3 3 0.6749m 64 Double pole 1.2000m 64 e 32 π √3 3 1.1999m Table 10: Formulas for the large cubit divisions using π, e and ∛3. 17 The Beautiful Cubit System I Douglas 2019 5.8 The √(π²+φ²)/e method The next formulas are more complicated, using π², φ² and e. They are also slightly more inaccurate. Digits 1 Length Digit Value 0.01875m Formula ( ( √π φ ( √π φ ( √π φ ( √π φ ( √π φ φ √π φ ( ( √π φ e Palm 0.0750m 4 16 2 + e 2 5 Hand 0.0938m 5 16 2 + e 2 6 Fist 0.1125m 6 16 2 + e 2 8 Double Handbreadth 0.1500m 8 16 2 + e 2 12 Small span 0.2250m 12 16 2 2 14 Great span 0.2618m 1 + e 2 2 + e 3 20 24 Foot 0.3000m Remen 0.3702m Remen 0.3750m + e Cubit (royal) ₢ 2 0.09375m −1 0.1125m −1 0.1500m −1 0.2250m −1 2 −1 ) 2 + e 2 2 2 + e 28 Cubit (royal) ₢ 0.5250m 28 16 2 + e 2 32 Pole 0.6000m 32 16 2 + e 2 36 Nby-Rod 0.67 – 0.68m 36 16 2 2 + e −1 ) ) ) 0.3750m 2 −1 ) ) ) −1 −1 −1 0.2618m 0.3000m −1 e 3 18 0.0750m −1 2 ( ( √π φ φ √π φ ( ( √π φ ( √π φ ( √π φ 0.01875m √ π 2+ φ 2 − 1 20 16 24 16 0.5236m 2 16 16 Cubit (standard) 0.4500m ) ) ) ) ) ) √ π 2+ φ 2 − 1 1 16 4 16 Value 0.4500m ) 0.5236m 0.5250m 0.6000m 0.3750m The Beautiful Cubit System Digits 64 Length Double pole I Douglas 2019 Value 1.2000m Formula 64 16 ( √ π 2+ φ 2 − 1 e Value ) 1.2000m Table 11: Formulas for the large cubit divisions using π², φ² and e. 5.9 The √2/π method The last set of formulas are the most inaccurate, and based on √2/π. Digits Length Value Formula Value 1 Digit 0.01875m 1√2 24 π 0.01876m 4 Palm 0.0750m 4 √2 24 π 0.0750m 5 Hand 0.0938m 5 √2 24 π 0.0938m 6 Fist 0.1125m 6 √2 24 π 0.1125m 8 Double Handbreadth 0.1500m 8 √2 24 π 0.1501m 12 Small span 0.2250m 12 √ 2 24 π 0.2251m Great span 0.2618m 2 √ 2 φ2 9π 0.2619m 0.2625m 14 √ 2 24 π 0.2626m Foot 0.3000m 16 √ 2 24 π 0.3001m Remen 0.3702m 20 Remen 0.3750m 20 √ 2 24 π 0.3751m 24 Cubit (standard) 0.4500m 24 √2 24 π 0.4502m Cubit (royal) ₢ 0.5236m 4 √ 2 φ2 9π 0.5238m 0.5250m 28 √ 2 24 π 0.5252m 14 16 28 19 The Beautiful Cubit System Digits I Douglas 2019 Length Value Formula Value 32 Pole 0.6000m 32 √ 2 24 π 0.6002m 36 Nby-Rod 0.67 – 0.68m 36 √2 24 π 0.6752m 64 Double pole 1.2000m 64 √ 2 24 π 1.2004m Table 12: Formulas for the large cubit divisions using √2/π This demonstrates that the divisions of the cubit can be calculated arithmetically in multiple different ways, with varying degrees of accuracy. The divisions do not need to have been based on actual measurements of some random, average or specific person. Table 13 has a few formulas that don’t slot in anywhere else. Foot and cubit are the “long” versions at 30cm and 45cm respectively. Note that the length of the British foot was “decreed.” Length Nby-rod Value 0.67 – 0.68m Formula Foot x Cubit x √π φ 2 √π φ 3 Value 0.6764m 0.6764m Table 13: Other assorted interesting formulas 5.10 The Grand Metre ℳ method The last set of formulas I want to demonstrate is based on what I call the “Grand Metre” (symbol ℳ) for lack of a better name. It is 1 metre plus ₢, totalling 1.5236m to 4 digits. I have no evidence that this was ever used, but it has popped up in various places, including the design of Menkaure, and the formulas are interesting. The curious thing is that we can approximate it rather well and easily, using the favourite π, φ and e, as follows: ℳ = 1+₢ ≈ 1+ π φ2 ≈ ≈ π −φ ≈ 1.5236m e é 20 The Beautiful Cubit System I Douglas 2019 Remember that é is also effectively the foot:₢ ratio. The plastic ratio also tries, but is closer to 1.525 than 1.5236: √ρ3 = 1.52470258 Curiously, a close approximation also pops up in the fractal dimension of the boundary of the dragon curve, designated as Cd, which only uses logs and roots: ( ) 1 + √73 − 6 √ 87 + √ 73 + 6 √ 87 log 3 = 1.52362708620249210627.. . [14] [15] log(2) 3 3 The value is also very close to 5 English feet (1.524m), or correct to 3 digits. Digits Length Value Formula Value 1 Digit 0.01875m ℳ 16 π φ 0.01873m 4 Palm 0.0750m ℳ 4 πφ 0.0749m ℳ 9 √π φ 0.0751m 0.0938m 5 Hand 0.0938m √ 6 Fist 0.1125m ℳ 6 √π φ 0.1126m 8 Double Handbreadth 0.1500m ℳ 2π φ 0.1499m 2ℳ 9 √π φ 0.1502m ℳ 100 √ 3 12 Small span 0.2250m ℳ 3 √π φ 0.2253m 14 Great span 0.2618m ℳφ 3π 0.2616m 16 Foot 0.3000m ℳ πφ 0.2997m 4ℳ 9 √π φ 0.3003m 5ℳ 4 πφ 0.3747m 20 Remen 0.3702m Remen 0.3750m 21 The Beautiful Cubit System Digits 24 I Douglas 2019 Length Value Formula Cubit (standard) 0.4500m 32 Value 3ℳ 2πφ 0.4496m 2ℳ 3 √π φ 0.4505m Cubit (royal) ₢ 0.5236m 2ℳ φ 3π 0.5231m Pole 0.6000m 2ℳ πφ 0.5995m ℳ 2 √φ 0.5989m 36 Nby-Rod 0.67 – 0.68m 4ℳ 9 0.6772m 64 Double pole 1.2000m 4ℳ πφ 1.1989m Table 14: Formulas for the large cubit divisions using ℳ 5.11 Other formulas Then there are a few formulas that produce interesting values, they have no name but round well to four decimal places. Length Value 1 metre 1.0000m 4 “Egyptian Feet” 1.2000m ? 1.3000m Formula Value 5 φ e 10 φ e = 7π 7τ π φ2 1.000m √ π2+ φ 2 1.3000m 1.2000m e ? φe π 1.4000m 1.4000m Table 15: Interesting lengths using famous irrationals. Table 16 has some assorted formulas, either related to the ₢, digit, ℳ, or other ancient units. At some point there were either “bad copies” or people actually using their shoes, or feet of a statue, as the basis for some official unit of length, which we can’t easily approximate mathematically. Official standards vary over time and complicate the problem, especially when standards get set by decree based on opinion rather than science. 22 The Beautiful Cubit System I Douglas 2019 Nevertheless, some relationships are interesting. Length Value Formula English inch 0.0254m English foot 0.3048m π ×e = πe (6 x 28) 2 336 Digit x e/2 Value 0.0254m ℳ 5 03047m 1.524 5 1 + π =ℳ 6 0.3048m Five English feet 60” = 1.5240m 1.5236m Six English feet 1.8288m 3 “Egyptian Feet” 0.9000m ₢é = Persian foot 0.32004m ℳ (π + φ ) 0.32011m Doric order foot ±0.324m π =₢ 6φ φ π √ 2 φ4 0.3236m π é 1.8283m πé 6 0.8997m 0.3241m Luwian foot ±0.323m π =₢ ϕ 6ϕ 0.3236m Attic foot 0.3084m √ 0.3086m ? ℳ 16 1 2φ 0.3090 m? ℳ 5 π 10 0.3047m Minoan foot ±0.304m Athenian foot ±0.315m Phoenician foot 0.3000m Megalithic yard 0.8275m 0.8297m Remen x √5 0.8279m 0.8275m 0.8297m ₢ + foot 0.8284m Nautical mile 1852m (currently) 0.3142m 3φe 14 π π = 4 φ2 100 π φ ( ) 1 ₢ 23 2 0.3000m 1854.1m The Beautiful Cubit System Length I Douglas 2019 Value Formula 100 π φ ( 1 0.524 ) 2 3600 φ π 5040 7 ! = e e Value 1851.3m 1854.1m 1854.1 m Table 16: Assorted interesting formulas 6. Geometry, the ₢ and the metric system One thing that has bothered me for a long time is the answer to the sceptic’s question, “If they had the metre, why didn’t they use it instead of the ₢?” I’m purposefully vague about who “they” were. I still don’t have an answer for that, but trying to find it led to something else. I received guidance that it was connected to the radian. About the same time, YouTube was constantly suggesting that I watch videos about the unit circle. I don’t think much of the traditional unit circle done with π, because the τ version is much better and more logical. In the end I gave in and watched part of one, mainly because it was by the very talented NancyPi. Little did I know that these were strong hints to the answer, which eventually came when I saw a website that pointed out that 30° in radians is π/6. Then the pennies started to fall into place. The usual way of describing the ₢ is as one-sixth of the circumference of a circle with diameter one metre, as in Figure 1. 24 The Beautiful Cubit System I Douglas 2019 Figure 1: The usual way of showing the ₢ Drawing one radian on that diagram does not help, because 1 radian is 57.295°, which is almost 60° and it’s hard to see any relationship. However, if we switch to using a unit circle, with a radius (instead of diameter) of one metre as in Figure 2, then suddenly things work much better, and I rediscovered the elegance. 25 The Beautiful Cubit System I Douglas 2019 Figure 2: The ₢ based on a 1 metre radius circle. As an aside, this divides the circle in 12, which may connect to things like the zodiac. So we have a radius of 1 metre, and an arc length of 1 ₢. The angle of the arc is 30°, which we can convert to radians: 30 ° = 30 π radians=0.5235987756 radians 180 We can restate that as: π radians. 6 The arc length is π metres. 6 The angle is The radius is 1 metre. The ₢ segment can be viewed as defining a pendulum, with a length of 1 metre, and a swing of 30°. This is (extremely close to) the seconds pendulum [16], where each swing takes 1 second for a period of 2 seconds. The arc of swing should not exceed 30°. I note the official length at 26 The Beautiful Cubit System I Douglas 2019 45° is actually slightly under 1 metre, this may imply that the force of gravity at Giza, or wherever the cubit originated, was slightly different a long time ago. [To be fair, I rechecked some videos I had watched previously about the seconds pendulum, and the presenter did mention that 30° in radians was numerically the same as the ₢, but didn’t join the rest of the dots. Nor did it trigger things for me at that time.] So Figure 2 has the metre and the second. From the metre and some water, we can get the kilogram. This is the basis of the metric system, all encapsulated in a circle showing the royal cubit. We can thus relabel Figure 2 as Figure 3: Figure 3: The metric system, summarised. Welcome to the beautiful cubit system. 7. Bibliography [1] M. H. Stone, ‘The Cubit: A History and Measurement Commentary’, Journal of Anthropology, 2014. [Online]. Available: https://www.hindawi.com/journals/janthro/2014/489757/. [Accessed: 24-Jun-2019]. [2] Q. Leplat, ‘Analyse métrologique de la coudée royale égyptienne’. 27 The Beautiful Cubit System I Douglas 2019 [3] R. Lorenzi, ‘Mummies’ Height Reveals Incest’, Seeker, 11-May-2015. [Online]. Available: https://www.seeker.com/mummies-height-reveals-incest-1769829336.html. [Accessed: 25-Jun2019]. [4] Douglas, Ian, ‘Diskerfery and the Alignment of the Four Main Giza Pyramids’. . [5] Douglas, Ian, ‘55,550 BCE and the 23 Stars of Giza’. . [6] H. Sivertsen, ‘THE SIZE OF THE GREAT PYRAMID’, The Size of the Great Pyramid A commentary on Robert Bauval's paper. [7] J.-P. Bauval and R. Bauval, ‘THE SIZE OF THE GREAT PYRAMID’. [8] ‘Orders of magnitude (length)’, Wikipedia. 19-Jun-2019. [9] F. Monnier, J.-P. Petit, and C. Tardy, ‘The use of the “ceremonial” cubit rod as a measuring tool. An explanation’, The Journal of Ancient Egyptian Architecture 2472-999X, vol. 1, pp. 1–9, Jan. 2016. [10] B. I. Sandor, ‘Tutankhamun’s chariots: secret treasures of engineering mechanics’, Fatigue & Fracture of Engineering Materials & Structures, vol. 27, no. 7, pp. 637–646, 2004. [11] ‘Ancient Egyptian units of measurement’, Wikipedia. 14-Jun-2019. [12] Egypt. Maṣlaḥat al-Misāḥah and H. G. (Henry G. Lyons, The cadastral survey of Egypt 1892-1907. Cairo : National Print. Dept., 1908. [13] R. Heath and J. Michel, The Lost Science of Measuring the Earth: Discovering the Sacred Geometry of the Ancients, 1st Ed. edition. Kempton, IL: Adventures Unlimited Press, 2006. [14] ‘List of mathematical constants’, Wikipedia. 19-Jul-2019. [15] ‘The Boundary of Dragon Curve’. [Online]. Available: http://poignance.coiraweb.com/math/ Fractals/Dragon/Bound.html. [Accessed: 20-Aug-2019]. [16] ‘Definition of SECONDS PENDULUM’. [Online]. Available: https://www.merriamwebster.com/dictionary/seconds+pendulum. [Accessed: 27-Jun-2019]. 28