Updated versions of the different ways of approximating the cubit. Also includes separate table for Grand Metre (1 plus royal cubit) == 1.5236…. == 1.524.
These are done with pi, phi, e, roots and powers (usually of basic primes), as well as ln, log, sin, cos and tan.
See square roots, cube roots and ln(4) for formulas not shown below.
Changelog at the bottom.See also The Magical Mystical Royal Cubit for the Pretty Picture version.
| Method | Value | Difference from π/6 | Abs difference | Round |
|---|---|---|---|---|
| π/6 | 0.523598776 | 0.000000000 | 0.000000000 | 0.5236 |
| Very close | ||||
| ((6√2/10)² + (6/100)²+(8√2/10000)²)² | 0.523598812 | 0.000000037 | 0.000000037 | 0.5236 |
| cube roots | 0.523600350 | 0.000001575 | 0.000001575 | 0.5236 |
| ((6√2/10)² + (6/100)²)² | 0.523596960 | -0.000001816 | 0.000001816 | 0.5236 |
| (7π/5e)²/5 | 0.523596637 | -0.000002138 | 0.000002138 | 0.5236 |
| 28φπ/100e | 0.523601717 | 0.000002942 | 0.000002942 | 0.5236 |
| (φe)/8.4 | 0.523603856 | 0.000005080 | 0.000005080 | 0.5236 |
| ln(4) | 0.523591499 | -0.000007277 | 0.000007277 | 0.5236 |
| ((1 + π)/e) - 1 | 0.523606791 | 0.000008015 | 0.000008015 | 0.5236 |
| φ²/5 | 0.523606798 | 0.000008022 | 0.000008022 | 0.5236 |
| Close | ||||
| π – (7π/5e)² | 0.523609468 | 0.000010692 | 0.000010692 | 0.5236 |
| (eφ³)/7π | 0.523611878 | 0.000013103 | 0.000013103 | 0.5236 |
| (10φ)/(11e + 1) | 0.523616953 | 0.000018177 | 0.000018177 | 0.5236 |
| tan(2φπe) | 0.523569002 | -0.000029774 | 0.000029774 | 0.5236 |
| π – φ² | 0.523558665 | -0.000040111 | 0.000040111 | 0.5236 |
| √(√5/(3e)) | 0.523642193 | 0.000043417 | 0.000043417 | 0.5236 |
| e(2√2 - φ)/2π | 0.523649949 | 0.000051174 | 0.000051174 | 0.5236 |
| Less close | ||||
| 10e/(36³√3) | 0.523542042 | -0.00005673 | 0.000056733 | 0.5235 |
| ln(10)/φe | 0.523520348 | -0.00007843 | 0.000078427 | 0.5235 |
| e/(3√3) | 0.523133582 | -0.000465194 | 0.000465194 | 0.523 |
| square roots | 0.523403737 | -0.000195039 | 0.000195039 | 0.523 |
| 1/(√(ln365.25/φ)) | 0.523656373 | 0.000057597 | 0.000057597 | 0.524 |
| 2√5/πe | 0.523685613 | 0.000086838 | 0.000086838 | 0.524 |
| 1/log(φ²π³) | 0.523717901 | 0.000119125 | 0.000119125 | 0.524 |
| (10/φπⅇ)² | 0.523764441 | 0.000165665 | 0.000165665 | 0.524 |
| (10√2)/27 | 0.523782801 | 0.000184025 | 0.000184025 | 0.524 |
| cos(π(4φe+1)) | 0.523808794 | 0.000210019 | 0.000210019 | 0.524 |
| (∛3 ∛5 ∛7)/9 | 0.524188220 | 0.000589444 | 0.000589444 | 0.524 |
| (∛5φ²)/πⅇ | 0.524228861 | 0.000630086 | 0.000630086 | 0.524 |
| sin((φ²πe√2) | 0.524253715 | 0.000654940 | 0.000654940 | 0.524 |
| (φ)/(e + 1/e) | 0.524286921 | 0.000688145 | 0.000688145 | 0.524 |
| (√2+√3)/6 | 0.524377395 | 0.000778619 | 0.000778619 | 0.524 |
| √(7√2)/6 | 0.524391047 | 0.000792272 | 0.000792272 | 0.524 |
| π/(φ³√2) | 0.524411194 | 0.000812419 | 0.000812419 | .0524 |
Grand metre:
Grand Metre comparisons, 2018-12-04
| Method | Value | Difference from 1 + π/6 | Abs difference |
|---|---|---|---|
| 1 + π/6 | 1.52359877559830 | 0.00000000000000 | 0.00000000000000 |
| approx 1.5236 | |||
| (1 + π)/e | 1.52360679096240 | 0.00000801536407 | 0.00000801536407 |
| π - φ | 1.52355866483990 | -0.00004011075840 | 0.00004011075840 |
| approx 1.524 | |||
| (e√2/π) + 3φe/14π | 1.52365965182830 | 0.00006087623004 | 0.00006087623004 |
| π^(1/e) | 1.52367105485890 | 0.00007227926063 | 0.00007227926063 |
| 1.2345² | 1.52399025000000 | 0.00039147440170 | 0.00039147440170 |
| (1.1111²)² | 1.52409693735710 | 0.00049816175881 | 0.00049816175881 |
| ∛3∛5/φ | 1.52420288540160 | 0.00060410980332 | 0.00060410980332 |
Putting things in perspective
If we for example look at the φe/8.4 formula, that differs from π/6 by 0.00000508028836 metres. That sort of number evokes three reactions… the mathematicians will say it’s not equal to ₢, engineers will say its “close enough for practical purposes” and the rest will have some variation of WTF or “what the heck does that mean?”
So to put things in perspective, I’ve converted the differences to microns and used some nice comparisons provided to WikiPedia to help us visualise just how small those differences are. Remember the royal cubit was typically used for building projects or land surveying, where such differences are not going to make a difference.
| Method | Value | Abs Delta | Microns | Less than / about |
|---|---|---|---|---|
| π/6 | 0.523598776 | 0.000000000 | ||
| Very close | ||||
| ((6√2/10)² + (6/100)²+(8√2/10000)²)² | 0.523598812 | 0.000000037 | 0.04 | length of a lysosome |
| cube roots | 0.523600350 | 0.000001575 | 1.57 | anthrax spore |
| ((6√2/10)² + (6/100)²)² | 0.523596960 | 0.000001816 | 1.82 | anthrax spore |
| (7π/5e)²/5 | 0.523596637 | 0.000002138 | 2.14 | length of an average E. coli bacteria |
| 28φπ/100e | 0.523601717 | 0.000002942 | 2.94 | size of a typical yeast cell |
| (φe)/8.4 | 0.523603856 | 0.000005080 | 5.08 | length of a typical human spermatozoon's head |
| ln(4) formula | 0.523591499 | 0.000007277 | 7.28 | diameter of human red blood cells |
| ((1 + π)/e) - 1 | 0.523606791 | 0.000008015 | 8.02 | size of a ground-level fog or mist droplet |
| φ²/5 | 0.523606798 | 0.000008022 | 8.02 | size of a ground-level fog or mist droplet |
| Close | ||||
| π – (7π/5e)² | 0.523609468 | 0.000010692 | 10.69 | width of cotton fibre |
| (eφ³)/7π | 0.523611878 | 0.000013103 | 13.10 | width of silk fibre |
| (10φ)/(11e + 1) | 0.523616953 | 0.000018177 | 18.18 | about minimum width of a strand of human hair |
| tan(2φπe) | 0.523569002 | 0.000029774 | 29.77 | length of a human skin cell |
| π – φ² | 0.523558665 | 0.000040111 | 40.11 | less than typical length of a human liver cell, an average-sized body cell |
| √(√5/(3e)) | 0.523642193 | 0.000043417 | 43.42 | less than typical length of a human liver cell, an average-sized body cell |
| e(2√2 - φ)/2π | 0.523649949 | 0.000051174 | 51.17 | typical length of a human liver cell, an average-sized body cell |
| Less close | ||||
| 10e/(36³√3) | 0.523542042 | 0.000056733 | 56.73 | less than length of a sperm cell |
| ln(10)/φe | 0.523520348 | 0.000078427 | 78.43 | thickness of sheet of paper |
| e/(3√3) | 0.523133582 | 0.000465194 | 465.19 | less than average length of a grain of sand |
| square roots | 0.523403737 | 0.000195039 | 195.04 | typical length of Paramecium caudatum, a ciliate protist |
| 1/(√(ln365.25/φ)) | 0.523656373 | 0.000057597 | 57.60 | less than length of a sperm cell |
| 2√5/πe | 0.523685613 | 0.000086838 | 86.84 | less than smallest distance that can be seen with the naked eye |
| 1/log(φ²π³) | 0.523717901 | 0.000119125 | 119.13 | diameter of a human ovum |
| (10/φπⅇ)² | 0.523764441 | 0.000165665 | 165.67 | length of the largest sperm cell in nature, belonging to the Drosophila bifurca fruit fly |
| (10√2)/27 | 0.523782801 | 0.000184025 | 184.03 | less than typical length of Paramecium caudatum, a ciliate protist |
| cos(π(4φe+1)) | 0.523808794 | 0.000210019 | 210.02 | typical length of Paramecium caudatum, a ciliate protist |
| (∛3 ∛5 ∛7)/9 | 0.524188220 | 0.000589444 | 589.44 | over 0.5 mm |
| (∛5φ²)/πⅇ | 0.524228861 | 0.000630086 | 630.09 | over 0.6 mm |
| sin((φ²πe√2) | 0.524253715 | 0.000654940 | 654.94 | over 0.6 mm |
| (φ)/(e + 1/e) | 0.524286921 | 0.000688145 | 688.15 | over 0.6 mm |
| (√2+√3)/6 | 0.524377395 | 0.000778619 | 778.62 | over 0.7 mm |
| √(7√2)/6 | 0.524391047 | 0.000792272 | 792.27 | over 0.7 mm |
| π/(φ³√2) | 0.524411195 | 0.000812419 | 812.42 | over 0.8 mm |
2018-11-26: added (e φ³)/(7π), as well as ((1+π)/e)-1, from Grand Metre table to cubit table, since it is so accurate.
2018-11-27: added (e√2/π) + 3φe/14π
2018-11-27: added e(2√2 – φ)/2π
2018-11-28: added (φe)/8.4, (10φ)/(11e + 1), and (φ)/(e + 1/e). Added 1/(√(ln365.25/φ)).
2018-12-02: Added 28φπ/100e. I’ve actually done the φπ/e sum many times, but tonight the lightbulb went on and I thought, hey, that 1.87 answer looks a lot like the value of 0.01875 for the length of a digit … I wonder what happens if I divide by 100 (to get decimal point in right place, else just skip that and work in cm) and multiply by 28, the number of digits in a royal cubit …. and voila … very nice answer. I suppose we could dub this the “royal digit” to differentiate it from the normal digit. The construction of the cubit is very complex, the sample we have in Turin has digits of different length on the last palm.
I also found another approximation using root(2π²)/2φ³ but this would be the worst of the lot so not adding it at the moment … comes out around 0.5244112.
2018-12-03: Added formulas based on 5φe/7π, split table into categories.
2018-12-04: Added pi^(1/e) to Grand Metre section, rearranged that table a bit.
2018-12-08: Added 2√5 / φe and 10√2 / 3³
2018-12-10: Added value from solving for x in ln(4) formula: ln(4) + x = 1/x
2018-12-12: Added 1/log(φ²π³), sin(φ²πe√2)
2018-12-13: Added tan(2φπe), cos(π(4φe+1))
2018-12-14: Added e/3√3
2018-12-15: Added √(√5/3e)
2018-12-18: Added π/(φ³√2)
2019-02-07: Added ln(10)/φe, assorted rearrangements
2019-02-10: Added two complicated formulas based on (6√2/10)²
2019-04-25: Added the “perspective” table.