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What I would like is exactly that: to classify/sort all the previous bases according to the spacing of proper regulars with richness below a certain threshold Ï.Icarus wrote:In order to fully consider the effect of proper regulars on the usefulness of a number base toward general arithmetic, we might consider a factor that weighs the spacing of proper regulars. This might be a sort of statistical measure that examines the standard deviation of the spaces for proper regulars with richness below a threshold. Iâ€™ve used Ï = {3, 8} but there is nothing sacred about these thresholds.
I’m not sure if Pinbacker still frequents this forum, but I’ve been thinking about how to quantify this.Pinbacker @ Jun 9 2014, 09:33 AM wrote: Could anyone please try to approximately classify/sort every of the following bases, from best to worse, based on how they fare (do well) on the log wheel thingy: 2, 4, 6, 8, 12, 16, 24, 30, 36, 48, 60, 120, 210, 360, 2310, 2520, 5040.
This is an extremely nifty tool for taking what would otherwise be abstruse properties of numbers and making them more intuitive!icarus wrote: Polar plots of base n logarithms of digits k.
FWIW, this could be typeset as: \(\displaystyle\tau\ \log_n k\) radians, ... yeah I had to sneak tau in there!... at 2π logn k radians,...
\(\displaystyle d | n\) ... \(\displaystyle n = d \times d'\) ... \(\displaystyle 1 = \log_n d + \log_n d'\)... joining these with a red line. Then we mark divisors d | n in bold black; since we can write n = d × d', 1 = logn d + logn d', ...
Now, that is really quite elegant. A little reflection and the reasoning becomes obvious.... and the pair of multiplicatively complementary divisors occur on the same horizontal line.
And that is equally elegant!We strike a vertical dashed blue line to indicate the square root, the line of multiplicative symmetry among the divisors d of n.
But the "closer" two complements are to each other on the wheel, the closer they both approximate the square root of the base. When it comes to populating useful auxiliary units in between the metricized powers of a base, the closer the auxiliary factors can approximate the square root of the base, the better. At least, I would think so, on the grounds that it more evenly divides up the logarithmic scale.icarus wrote:One argument Wendy posed was that the complements joined by lines ought to be as evenly distributed around the "wheel" as possible for optimum coverage. We can see that 3 and 4 are "a bit close" concerning the dozen, and the richness-5 decimals 32 and 3125 are extremely close.
Indeed, it's by such complementary factors that Primel (⚀) units and quasi-TGM (⊖) units can dovetail as mutually friendly auxiliaries:The regular figures of a base contribute to a more flexible system of weights and measurements, putting units and derivatives closer to a desired quantity, the "jump" of the powers of the base more evenly divided. This is part of how the primel system works, ...
⚀morsel·length (⚀lengthel) |
× 3 = | ⚀thumb·length (⊖uncia·lengthel ≈ uncia·Grafut) |
× 4 = | ⚀hand·length (⚀unqua·lengthel) |
⚀hand·length (⚀unqua·lengthel) |
× 3 = | ⚀foot·length (⊖lengthel ≈ Grafut) |
× 4 = | ⚀ell·length (⚀biqua·lengthel) |
⚀morsel·area (⚀areanel) |
× 9 = | ⚀thumb·area (⊖bicia·areanel ≈ bicia·Surf) |
× 14_{z} = | ⚀hand·area (⚀biqua·areanel) |
⚀hand·area (⚀biqua·areanel |
× 9 = | ⚀foot·area (⊖areanel ≈ Surf) |
× 14_{z} = | ⚀ell·area (⚀quadqua·areanel) |
⚀morsel·volume (⚀volumel) |
× 23_{z} = | ⚀thumb·volume (⊖tricia·volumel ≈ tricia·Volm) |
× 54_{z} = | ⚀hand·volume (⚀triqua·volumel) |
⚀hand·volume (⚀triqua·volumel) |
× 23_{z} = | ⚀foot·volume (⊖volumel ≈ Volm) |
× 54_{z} = | ⚀ell·volume (⚀quadqua·volumel) |
⚀morsel·work (⚀workel) |
× 69_{z} = | ⚀thumb·work (⊖quadcia·workel ≈ quadcia·Werg) |
× 194_{z} = | ⚀hand·work (⚀quadqua·workel) |
⚀hand·work (⚀quadqua·workel) |
× 69_{z} = | ⚀foot·work (⊖workel ≈ Werg) |
× 194_{z} = | ⚀ell·work (⚀octqua·workel) |
Wouldn't it be an enhancement to generate such diagrams for each of your entries in your Tour des Bases?Unless anyone has any input this would seem to be my last post on regular figures.
Senary (base 6): 5th root | Senary (base 6): 6th root |
Octal (base 8) | Hexadecimal (base sixteen) |
Decimal (base ten) | Uncial (Dozenal, base twelve) |
Tetradecimal (base fourteen) | Octodecimal (base eighteen) |
Vigesimal (base twenty) | Tetravigesimal (base twenty four) |
Base thirty | Base thirty-six |
Hey, good to see you here again, DS! ☺Yeah, this is some pretty awesome stuff.Double sharp wrote: This is wonderful! Is there a way to see these charts for any base we'd like (within reason)?
The only caveat I have to this is if you want even something like 9 or 14_{z} or 21_{z} as a regular in a base like ᘔ0_{z} you still have to go to richness-2, which then means four digits. I think large bases just have to suck it up and pull their own weight just like the small bases. We shouldn't have to bail them out and give them a pass just because they're "too big to fail."I think that in the written analysis (as opposed to the automated diagrams), this ought to be weighted for the size of the base. After all, the richness-1 regulars of base gross are exactly the richness-1 and richness-2 regulars of base dozen combined. So if we are going to compare the surrogates of base 120 to those of a human-scale base, it seems to me that looking at the divisors in base 120 is enough, as its use as "parts per 120" analogous to percent would act like two digits of a normal human-scale base. Indeed in that form we might use it as an auxiliary just like we use sexagesimal, without really using difficult mixed-radix arithmetic.
Well, different people may want to see different bases, so perhaps the best option to cater to everyone's taste might be something like your digit map CDF that we can all slide through ourselves. ^_^ It would certainly spare you the need to post dozens and dozens of pictures for everybody, since I think at least the first two or three dozen bases should be of general interest (plus maybe a few HCNs; five dozen and ten dozen ought to be included for sure, so it's great that they're already there).icarus @ 18 Jun 2018, 05:13 wrote:Hi Double sharp. If you have a base you want to see I'll cook up a chart. I can control the appearance entirely
Well, the same is true in square bases. To get 8 as a regular in base 36, you have to go to richness-2 as well, which is four digits of senary. It's true that the last will be zero, but the same will be true in base 120 for 1/16, which is 0.07'60 (or 0.07'6) in 12-on-10 notation. It's interesting to generalise this a little and consider surrogates in a mixed base as possibly having rational richnesses. ^_^Kodegadulo @ 18 Jun 2018, 02:19 wrote:The only caveat I have to this is if you want even something like 9 or 14_{z} or 21_{z} as a regular in a base like ᘔ0_{z} you still have to go to richness-2, which then means four digits. I think large bases just have to suck it up and pull their own weight just like the small bases. We shouldn't have to bail them out and give them a pass just because they're "too big to fail."
Square bases like 30_{z} and 84_{z} and 100_{z} are exceptions, but they are exceptions that prove the rule. They really are exactly equivalent to two digits of their root base.