Base Optimality

In the conclusion to the Factor Density topic at https://dozenal.forumotion.com/t51-factor-density, I concluded:

Phaethon, Tue 19th Jun 2022 at 19:44 wrote:This factor density as a computed measure of base efficiency does not penalise an increase in size of the base enough,

In order to furnish a measure of optimality of bases that moves large bases down the league, the factor density \(D_B\) can be multiplied by the measure of another property that decreases with the size of the base. One such property is the radix economy, which is optimal for very small bases near the base of the natural logarithm. I will use a modified form of radix economy generalised for alternating bases, as described previously elsewhere:

David Kennedy, 3:25 PM - May 29, 2017 wrote:a modified efficiency function, such as \(E_{B}=\displaystyle\frac{2(\log_{2}∏_{i}^{n}b_{i})^\frac{3}{2}}{\sqrt{n}\sum_{i}^{n}{(b_{i}-1)}}\)

I remove the factor of two from that expression which appeared to scale the efficiency of a binary base to two rather than one, but call it \(E_B\) instead of \( E_B \)/2 nonetheless. If this efficiency be multiplied by the factor density, the result tends to decrease as the base increases and does not provide much information about which bases are better than others. The influence of the factor density needs to be increased or weighted so that it is more important than the radix economy. Initially, I chose to make the factor density twice as important as the radix economy efficiency by raising the factor density to the power of two in the product. This resulted in twelve being the base with the highest score of all bases. Encouraged by this, I investigated variation of the exponent \(k\) to find the range of it for which twelve remains the top base, according to the formula:
\[ E_B * {D_B}^k \]
I found twelve to be the best base for \( 1.8686 \lesssim k \lesssim 3.206\). For \(k \lessapprox 1.8685\), the best base was the number six. For \(k \gtrapprox 3.207\), the best base was twelfty. When \(k\) is smaller, there is more emphasis on smaller bases being better, whereas when \(k\) is larger, bigger highly composite bases tend to be better. The exponent \(k\) should not be made too large or else the scores for larger highly composite numbers would keep on increasing, and there would be no final best base. This makes it highly probable to interpret that there is no better base than twelve if the factor density and radix economy are the only two contributions taken into account for the score.

Next, I chose a value for the exponent \(k\) such that the ratio between the scores for the top base twelve and the second best base would be maximised. By coincidence, this value of \(k \simeq 2.66770688\) happened to have the advantage of causing the scores for the second and third best bases to be nearly equal, so that one would not have to decide which was better than the other, and also the score for twelfty happened to be the same as binary. I wonder whether there is any insight implied by these coincidences. A table of results for a selection of bases with this exponent is shown below:

Table of Bases Ranked by Optimality Scores

Base	\(E_B *{D_B}^{2.66770688}\)
12	1.264630504
6	1.113341812
24	1.113330551
60	1.062100735
120	1.000000006
2	1
4	0.942809042
36	0.919522883
48	0.856010571
72	0.852782923
180	0.850009264
30	0.841843609
360	0.818695262
240	0.804982549
18	0.745171206
8	0.742307489
720	0.670640405
840	0.668366963
20	0.651182733
2.718281828	0.591569443
40	0.589910884
1260	0.577717466
42	0.555646281
1680	0.552970045
10	0.547218881
16	0.533333333
3	0.503212301
54	0.409054139
56	0.391419125
14	0.349933442
9	0.293077693
64	0.233284737
5	0.228484451
50	0.208483258

This measure of base optimality is interesting in that bases as apparently different in size such as binary and twelfty are could have the same score.

Conclusion:
If the best base is the number twelve and reasons are chosen so as to make twelve the best base, this has implications on the order or rank for optimality of other bases. This optimality score is not comprehensive in that it does not take into consideration the influence of factors of the base plus or minus one on its ability to accurately approximate non-terminating fractions.

References:
https://en.wikipedia.org/wiki/Radix_economy
https://www.tapatalk.com/groups/dozensonline/base-subdivisions-t1629-s24.html#p40009997
https://dozenal.forumotion.com/t24-dozenal-fifths-better-than-decimal-thirds

The previous rank of bases by a score involving a kind of radix economy and factor density had a tendency to promote large highly composite numbers too much. Using a variant of factor density produces a rank of bases by optimality in which larger highly composite numbers tend to be ranked not so close to the top. Let the variant of factor density of base \(B\) be called \(Ð_B\). Then the optimality score, say \(O_B\), may be represented similarly to before as
\[ O_B = {E_B} * {Ð_B}^k\]
As before, I investigated the values of the exponent \(k\) for which the number twelve is the optimal base according to this score and found this to occur for \( 2.97094 \lesssim k \lesssim 7.45851\). For \(k \lessapprox 2.97094\), the best base is the number six, but such a low value for the exponent promotes very small bases and is unlikely to be realistic. For \(k \gtrapprox 7.45852\), the best base was sexagesimal this time.  Maximising the ratio between the scores of the best base twelve and the next best base produces a value for the exponent of \(k \approx 4.6525\). This again has the effect of equalising the scores of the two next best bases. The following rank for selected top bases is tabulated:

Table of Bases Ranked by Optimality Scores

Base	\(O_B = {E_B} * {Ð_B}^{4.6525}\)
12	1.537085926
6	1.298585424
24	1.298584927
60	1.083056916
36	1.051025376
2	1
30	0.983095093
48	0.954698569
4	0.942809042
72	0.889480255
120	0.869012938
18	0.855567804
8	0.742307489
20	0.732312334

The top few bases are in the same order as before, but twelfty is further down the list now. One pertinent result is that the double dozen or four factorial, a base often neglected as too large by dozenists, is the second best base for \( 4.6526 \lesssim k \lesssim 6.871\). It might be possible for the double dozen to be the best base if twelve is too small, if the importance of the prime number two is higher, if greater accuracy of representing fractions by fewer significant figures is desired, and if data numbers and codes are to be compressed in number of characters so that they appear shorter.

To take into account further qualities contributing to suitability of bases, the aim would be to create scores for each of these qualities and form an optimality model \(O_B\) out of them:
\[ \displaystyle O_B = \prod_{i} X_{i} ^{k_i}\]
where the factor \(X_i\) as a function of the base \(B\) is the score for some quality weighted in some way by its exponent \(k_i\). The question to ask the audience, the end users of the base, is: "Ideally, what are the qualities that a base should have?" One quality yet to be scored here is the accuracy with which a base can represent fractions generally penalised by increasing number of significant figures and numerical symbols. This might be viewed as a generalisation of radix economy from integers to the rational numbers.

Reference:
https://dozenal.forumotion.com/t51-factor-density#162