30 Ways to calculate the Royal Cubit

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.

MethodValueDifference from π/6Abs differenceRound
π/60.5235987760.0000000000.0000000000.5236
Very close
cube roots0.5236003500.0000015750.0000015750.5236
(7π/5e)²/50.523596637-0.0000021380.0000021380.5236
28φπ/100e0.5236017170.0000029420.0000029420.5236
(φe)/8.40.5236038560.0000050800.0000050800.5236
ln(4)0.523591499-0.0000072770.0000072770.5236
((1 + π)/e) - 10.5236067910.0000080150.0000080150.5236
φ²/50.5236067980.0000080220.0000080220.5236
Close
π – (7π/5e)²0.5236094680.0000106920.0000106920.5236
(eφ³)/7π0.5236118780.0000131030.0000131030.5236
(10φ)/(11e + 1)0.5236169530.0000181770.0000181770.5236
tan(2φπe)0.523569002-0.0000297740.0000297740.5236
π – φ²0.523558665-0.0000401110.0000401110.5236
√(√5/(3e))0.5236421930.0000434170.0000434170.5236
e(2√2 - φ)/2π0.5236499490.0000511740.0000511740.5236
Less close
1/(√(ln365.25/φ))0.5236563730.0000575970.0000575970.524
2√5/πe0.5236856130.0000868380.0000868380.524
1/log(φ²π³)0.5237179010.0001191250.0001191250.524
(10/φπⅇ)²0.5237644410.0001656650.0001656650.524
(10√2)/270.5237828010.0001840250.0001840250.524
square roots0.523403737-0.0001950390.0001950390.523
cos(π(4φe+1))0.5238087940.0002100190.0002100190.524
e/(3√3)0.523133582-0.0004651940.0004651940.523
(∛3 ∛5 ∛7)/90.5241882200.0005894440.0005894440.524
(∛5φ²)/πⅇ0.5242288610.0006300860.0006300860.524
sin((φ²πe√2)0.5242537150.0006549400.0006549400.524
(φ)/(e + 1/e)0.5242869210.0006881450.0006881450.524
(√2+√3)/60.5243773950.0007786190.0007786190.524
√(7√2)/60.5243910470.0007922720.0007922720.524
π/(φ³√2)0.5244111940.0008124190.000812419.0524

Grand metre:

Grand Metre comparisons, 2018-12-04

MethodValueDifference from 1 + π/6Abs difference
1 + π/61.523598775598300.000000000000000.00000000000000
approx 1.5236
(1 + π)/e1.523606790962400.000008015364070.00000801536407
π - φ1.52355866483990-0.000040110758400.00004011075840
approx 1.524
(e√2/π) + 3φe/14π1.523659651828300.000060876230040.00006087623004
π^(1/e)1.523671054858900.000072279260630.00007227926063
1.2345²1.523990250000000.000391474401700.00039147440170
(1.1111²)²1.524096937357100.000498161758810.00049816175881
∛3∛5/φ1.524202885401600.000604109803320.00060410980332

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)

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