## Regular Figure Cycles

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
@Pinbacker, I meant to post here that I included the "log wheel" data in this post.

 Posts 114
Casual Member
Pinbacker
Casual Member
Joined: Feb 7 2014, 03:32 PM
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.
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.
But not below Ï = 3, nor Ï = 8. I think we should simply choose Ï = 1 instead (just taking the divisors). This is mathematically purer and more natural, and it also makes the calculations much more easy.

If you can't do it or don't have time to do it, then could you please give me the list of all the spacings between every divisors in each of the previously said bases?

wendy.krieger
wendy.krieger
The trouble is that you can't take p=1, because this will never fill the space between 1 and 2. Even in decimal, one uses p=2 to get 2.5 and 4, both common enough, and p=3 to get 8, 1.25.

For any appreciable base, there are no more than nine numbers in a logrithmic range of 16 (1/4, 1/3, 1/2, 1, 2, 3, 4). In the case of sixty, this is two-thirds of the circle, the remaining 5, 6, 10, 12, fill the other third. For twelfty, the above range is 4/7 of the circle, the divisors 5, 6, 8, 10, 12, 15, 20, 24 (ie 8) fill the other 3/7. (ie 1/18.3 per divisor against 1/31.6 for 30, 40, 60, 1, 2, 3, 4.

The real utility of three-prine bases only becomes apparent when you select p=2 and p=3 over p=1. For example, with p=2, there are 47 regular mantissa for twelfty, or 7 per octave, eg 1, 124, 130, 140, 160, 172, 180, 2 You don't get this in dozenal until you take p=7. With p=3 (which is pretty much unavoidable, since we live in 3d, and you end up with the cubes of the divisors), you get 97 regulars, of which only three are greater than 1.00.00.

The density of even the regulars to 240 is suffice for the sumerians to effectively avoid division by using interpolation of the recriprocals of the regulars. The sumerians didn't go for regulars by ripples, but just raw regulars.

 Posts 237
Regular
jrus
Regular
Joined: Oct 23 2015, 12:31 AM
This is great! It seems a little unfair to include equal number of "digits" for every base though. You probably would be better off showing all the "regulars" up to some fixed size as integers (say, 10^20 or something), and plotting numbers on the vertical axis on a log scale in a kind of barberpole style (consistent scale across bases) instead of in horizontal rows (which amounts to the rounded-down log, with a separate scale for each base).

Also, show those images inline instead of as attachments!

 Posts 237
Regular
jrus
Regular
Joined: Oct 23 2015, 12:31 AM
Did folks see the recent brouhaha about Plimpton 322?

The popular press has been a bit ridiculous but the paper is actually pretty interesting in its speculations
http://www.sciencedirect.com/science/ar ... 6017300691

And shows a concrete example of where having numerous well-spaced base-regular numbers has (or might have had) a real practical use.

 Posts 237
Regular
jrus
Regular
Joined: Oct 23 2015, 12:31 AM
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.
I’m not sure if Pinbacker still frequents this forum, but I’ve been thinking about how to quantify this.

I think the thing to do is look at all the log-scaled gaps between base-regular positional fractions in the range [1/base, 1) out to some certain number of digits, and then take the sum of squares of the log of each gap, divided by the log of the base. This metric can be then plotted for each additional digit, with the independent axis scaled like the log of the base for comparing multiple bases against each-other, and the vertical axis also log-scaled. The results should be lines that slope down to the right for every base, with the steepness and offset telling us how well the base does.

This gets at the idea that we want to measure evenness of spacing as well as efficiency vs. total number of representable values.

That might be hard to comprehend from a prose description. I’ll try to make some graphical pictures in the not too distant future.

wendy.krieger
wendy.krieger
The 'regular mantissas' are the string of digits that form regular numbers. Thus, for example, '2' represents 2, 20, 2000000, 0.0002, &c.

The spread is more governed by the prime divisors, so 10 and 20 have a simular spread, while 12 and 18 have a different spread. While the smallest 13 regular mantissa of base 20 are very evenly spread (in steps of 1.25 or 1.28 dec), this works against this when one wants to go past this. 12, 18, 24 &c are largely governed by the semitone sequence, where lg 2 = 12, lg 3 = 19. This produces an equality of 531441=524288.

Bases with two prime divisors produce a finite number of regulars of any given number of digits. This is ln(B )^2 / ln(p1)ln(p2), where p are the separate number of primes. There are other useful measures here, but this is the main one.

Bases with three primes produce a variable R = Prod(ln(B )/ln(p_i)), over the divisor primes. For 120, R=89.532, for 60, R = 56.002. The number of regular mantissa of n places is given by hR-A, where h is the number of places, and A is a constant.

One can measure the 'tightness' of a base. This is the ratio between a mantissa and its reciprocal. In essence, the reciprocal table of a tight base would have similar-sized numbers on both sides. eg dozenal has 346 vs 368 as reciprocals.

The measure here is to find ln(p^a × q^B )/ln(p^a/q^B ). Bases like 12, 56, 72, do quite well here. Where the larger power is the smaller prime (as 12, 56), the largest examples of similar-sized reciprocal pairs are quite large. For 16000, this runs to hundreds of places. But they can be quite impressive in any base, eg

72: 8 30 58 48 vs 8 39 6 54.

56: 7.20 28 7,47 35 51,35 21 24,28 vs 7.33 41 3,13 3 10,55 31 10,16

The sad part of 2-prime bases is that these regulars occur all too early in the list of regulars. The examples for 56 fall before #126 rows. It is made more palatable in that this same table contains the remarkable 3,7 51 53,26 20 7, where pi() is 3.7 52 1,52 21 32,

Bases with 3 prime factors

These provide a fairly uniform spread of regulars, but there are some devices that one can use, apart from the value R above.

You draw three vertical lines, for each prime, and then plot the points (base 120 as example), 3, 5, 8, and 1/3, 1/5, 1/8. Draw the hexagon 3, 15, 5, 40, 8, 24, 3. If this crosses the middle line (ie logb = 1/2), either twice or six times. (42 crosses it only twice). Bases that cross it twice, have much fewer cases of similar-sized regular to recriprocals than the six-crossers.

Another measure is suppose a triangle whose vertices are 3x/ln 2,0,0 ; 0,x/ln 5,0; 0, 0, x/ln 3 on the axies. This represents for example, the regulars of this particular size. This would intersect the cuboid formed by 6x, 2x, 2x to 0, 0, 0. The relative portion of the triangle that falls inside the prism is then some measure of efficiency of the base.

This is maximal when the three power-divisors are the same size, eg 7,8,9 for 504, or 3,4,5 for 60. Its maximum is 8/9. As this gets more skew, this tends to go down somewhat. I've measured it for 120, it's something like 2/3 or something.

One looks for large holes, and the last integer to 'fill in'. Bases like 105 the several primes fall very close to multiples of ln(3)/13 = ln(5)/19 = ln(7)/23. This means that the logrithms of 3-5-7 numbers fall close to some x ln(105)/55.

2-3-5 numbers like 120, take longer to fill every lead digit than bases 2-3-7 like 126. For example, 95 requires something like a 7-place mantissa to appear. For 126, there are four-place mantissa for every number from 1 to 126.

7-smooth numbers become granulated at eight decimal digits, and it can take something like 100 decimals to the right before you find instances that fill this.

The smallest 17-smooth number, produces something like 1,000,000 three-place mantissa, but i did not grovel through the tape before deleting it.

To fill the 83 semitones from 1 to 120 (lg 2 = 12, lg 3 = 19, lg 5 = 28), you have to go as far as the fourth ring. But it takes something like 11 rings of dozenal to fill the 43 semitones in that base.

I did a scale where log 2, 5, 7 to base 70, were 31, 72, and 87, and log 70 was 190. In this scale, you quickly find the close pair 3125=3136, but the regulars appear in two fan-out points.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Polar plots of base n logarithms of digits k.

This morning I finally broke through, and am able to automate polar plots of base n logarithms of digits k. (Labels are uncial).

These charts demonstrate some of the multiplicative properties of base n. We start by drawing a unit circle, placing 1 at the top, and proceed clockwise by pointing off each digit in red at 2π logn k radians, joining these with a red line. Then we mark divisors d | n in bold black; since we can write n = × d', 1 = logn d + logn d', and the pair of multiplicatively complementary divisors occur on the same horizontal line. We strike a vertical dashed blue line to indicate the square root, the line of multiplicative symmetry among the divisors d of n. This is a chart of the dozen.

The above chart of the number three dozen shows that it is a perfect power, which we indicate by a bold light blue line.

This is a chart of ten dozen, showing its many divisor pairs, a testament to its flexibility. The above is a chart of hexadecimal, the light blue cross showing that it is a fourth power.
This program is key in automating the Tour des Bases, it can chart bases as large as uncial 15466, though in these cases I would likely lose the labeling and eliminate the pointing except for divisors.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Regular Figure Maps.
I have managed to produce a polar map of regular figures that perhaps makes better sense than the old linear maps posted here long ago. These can now be fully automated without post-processing.

These first maps show figures for decimal (in decimal). The leftmost map shows divisors, the central shows decimal regular figures with richness 2, finally, the rightmost shows decimal regular figures with richness 3.
"Richness" is the smallest positive integer power of base n that a regular number divides. The number 1 is regular to all numbers and divides n^0. The divisor d divides n^1. Richness-2 regular numbers divide the square of the base and enjoy unit fraction termination in two base-n digits. Example, 25 divides 100 and 1/25 = .04. Likewise, richness-3 regular numbers divide the cube of the base and enjoy 3-digit unit fraction expansions in base n.

Bear in mind that any regular number that yields an integer if divided by n^e with e < richness is not a regular figure. If a base n regular number ends in a zero expressed in base n, it is not a regular figure. The notion of the regular figure is that it is a "primitive" regular number in that all regular numbers in base n that start with all the digits in the figure can be produced by adding zeros to the end of the number. Example; 20 has richness 2 in decimal, since 20 | 100. But we can divide 20 by 10 to get 2. Because the maps only chart the mantissae of the logarithms, the numbers 2, 20, 200, 2000, etc. would be coincident on the chart, and all we need chart is 2, the regular figure of all those numbers.

Perhaps a better term for "regular figure" is "proper regular", as the proper divisors of a number n are all divisors except n itself; n in base n is written "10", i.e., it ends in a zero, and no other divisor can end in zero. The proper divisors {1, p, d, ...} (p prime, d composite) have richnesses of 0 (for 1) and 1 (for p and d). All other proper regulars have richnesses greater than 1.

We like to consider regular figures as they abbreviate the list of regular numbers of a given richness in base n. Decimally, for nonzero richnesses there are 2 regular figures. Uncially we have 4 regular figures per nonzero richness level: {{1}, {2, 3, 4, 6}, {8, 9, 14, 16}, {23, 28, 46, 54}, {69, a8, 116, 194}, ...}. Compare this to the list of uncial regular numbers at given richnesses: {{1}, {2, 3, 4, 6, 10}, {8, 9, 14, 16, 20, 30, 40, 60, 100}, {23, 28, 46, 54, 80, 90, 140, 160, 200, 300, 400, 600, 1000}, {69, a8, 116, 194, 230, 280, 460, 540, 800, 900, 1400, 1600, 2000, 3000, 4000, 6000, 10000}, ...}. Prime powers p^e have a finite number of regular figures: namely e. Hexadecimal has the regular figures {{1}, {2, 4, 8}}.

Many of the smallest regular figures, seen in base n by a longtime user, will seem "round" in that base. The list of decimal proper regulars contain many often-used "surrogates" or "stand-ins" for the negative powers of 2: {{1}, {2, 5}, {4, 25}, {8, 125}, {16, 625}, {32, 3125}, ...}. This is the reason regular figures are important in base n.
The next set of maps examine uncial (figures in uncial), with the leftmost showing divisors, the middle richness 2, the rightmost richness 3. We observe that there are more "horizontal lines", i.e., multiplicatively complementary pairs that produce some positive integer power of the base in uncial than in decimal. This can be read as greater flexibility of the dozen. The following two charts show the divisors of base sixty (figures expressed decimally). We need only go to 2 places to observe more multiplicatively complementary pairs for this large base; it is more flexible than the dozen, but is of course much larger a base than uncial. These charts show hexadecimal at left, and base three dozen (divisors center, richness-2 regulars right). Any number n with ω(n) = 1, i.e., that has one distinct prime divisor, only has its proper divisors as regular figures. This illustrates a sore lack of flexibility for hexadecimal and any prime power base (octal, nonary, etc.) since we can't "cover more ground" by going to multiple digits to get additional complementary pairs. Example: 2 × 8, 20 × 80, and 200 × 800 all boil down to a power of sixteen in hexadecimal. We cannot employ surrogates outside of {1, 2, 4, 8} in hexadecimal, whereas in decimal we see 25 → 1/4 (of 100), 125 → 1/8 (of 1000), etc. In sexagesimal, the multiplicative complementary pairs were the basis of ancient Mesopotamian arithmetic (the "reciprocal divisor method").
For hexatrigesimal (base three dozen) we observe that, despite the fact three dozen is the square of six, we do in fact have access to additional complementary pairs by turning to multiple digits. 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. Below is base one hundred five as she'd discussed above, taken to richness 2:
The regular figures of ten dozen and those of other bases were discussed in issue ten dozen of the Duodecimal Bulletin.

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, and some aspects of the US customary system (though it is not coherent and its hodgepodge of factors throughout the system leaves a lot to be desired.)
Unless anyone has any input this would seem to be my last post on regular figures.

Obsessive poster
Obsessive poster
Joined: Sep 10 2011, 11:27 PM
icarus wrote: Polar plots of base n logarithms of digits k.
This is an extremely nifty tool for taking what would otherwise be abstruse properties of numbers and making them more intuitive!
... at 2π logn k radians,...
FWIW, this could be typeset as: $$\displaystyle\tau\ \log_n k$$ radians, ...  yeah I had to sneak tau in there!
... 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', ...
$$\displaystyle d | n$$  ...  $$\displaystyle n = d \times d'$$  ...  $$\displaystyle 1 = \log_n d + \log_n d'$$
... and the pair of multiplicatively complementary divisors occur on the same horizontal line.
Now, that is really quite elegant.  A little reflection and the reasoning becomes obvious.
We strike a vertical dashed blue line to indicate the square root, the line of multiplicative symmetry among the divisors d of n.
And that is equally elegant!
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)

Obsessive poster
Obsessive poster
Joined: Sep 10 2011, 11:27 PM
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.
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.
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, ...
Indeed, it's by such complementary factors that Primel (⚀) units and quasi-TGM (⊖) units can dovetail as mutually friendly auxiliaries:

 ⚀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)

But note that where we'd use richness-1 factors for length auxiliaries, we'd need richness-2 factors for area auxiliaries:

 ⚀morsel·area (⚀areanel) × 9 = ⚀thumb·area (⊖bicia·areanel ≈ bicia·Surf) × 14z = ⚀hand·area (⚀biqua·areanel) ⚀hand·area (⚀biqua·areanel × 9 = ⚀foot·area (⊖areanel ≈ Surf) × 14z = ⚀ell·area (⚀quadqua·areanel)

And we'd need richness-3 factors for volume auxiliaries (and likewise for mass, force, weight):

 ⚀morsel·volume (⚀volumel) × 23z = ⚀thumb·volume (⊖tricia·volumel ≈ tricia·Volm) × 54z = ⚀hand·volume (⚀triqua·volumel) ⚀hand·volume (⚀triqua·volumel) × 23z = ⚀foot·volume (⊖volumel ≈ Volm) × 54z = ⚀ell·volume (⚀quadqua·volumel)

We might even need richness-4 factors for energy (work, heat, potential) auxiliaries, since e.g. ⚀workel = ⚀forcel × ⚀lengthel

 ⚀morsel·work (⚀workel) × 69z = ⚀thumb·work (⊖quadcia·workel ≈ quadcia·Werg) × 194z = ⚀hand·work (⚀quadqua·workel) ⚀hand·work (⚀quadqua·workel) × 69z = ⚀foot·work (⊖workel ≈ Werg) × 194z = ⚀ell·work (⚀octqua·workel)

So for applications to metrologies, taking your diagrams to richness-3 or even richness-4 might wind up being useful.

Unless anyone has any input this would seem to be my last post on regular figures.
Wouldn't it be an enhancement to generate such diagrams for each of your entries in your Tour des Bases?
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Thanks Kode! Indeed, your last statement is the idea. I needed to automate the diagrams so as to make a ton of them in one fell swoop and not worry about labeling, etc. I did make these graphs in 2009, but they were done in AutoCad, ported to Illustrator and Photoshop where they stack and can be toggled in InDesign. Quite an involved workflow!

Indeed one of the benefits of regular figures (at least that's what I call them, for lack of a better term) are "surrogates" in bases like decimal, where they are sometimes used as "modules" in coherent decimal systems of weights, measures, packing, and currency. The application of "binary fractions" (i.e. 1/2^k for 2 ≤ k ≤ 4) is one example of these; the cup-pint-quart-gallon units and as fractions of the inch (these are available to any even base). The powers of 5, i.e. 25, 125, are used as "one quarter (of a hundred)", etc.
This diagram "turns off" the digits and the polygon, etc. to simply render uncial regular figures to richness 5. Surely 3 and 4 would seem to be the most useful figures uncially. Indeed, approximation of the roots of the smallest numbers (2, 3) would be useful scaling methods. If we consider the richness 2 regular figures (i.e., 8×16, 9×14, 23×54, 28×46), we cover the full circle adequately. (I am fond of 9×14 personally to cover the extreme area that is hard to hit with a divisor).

One of the things we observe is the "stacking" of the decimal regular figures. This diagram above shows decimal regular figures to richness 12. Once we get to richness 5 we seem to have occupied the ten nodes that merely get increasingly populated by subsequent figures. The diagram for richness 20 merely shows four figures clustered around the tenth-roots. Vigesimal and base 15 also suffer this "stacking" effect.
Senary (left) and octodecimal (right) also has a nice, even spacing. Base 6 is taken to richness 6, 18 to 5. Tetradecimal has good spacing once we reach richness 9 (i.e., products of either or both 2^e and 7^e with e ≤ 9).
In an above reply, Wendy talked about the 83 semitones in base-120, requiring richness 4 to fill. This is shown above at left.

This is not to criticize this observation because it is sound, but we note that once we have so many "snap points" that we essentially have painted in the entire circle, we seem to lose the function of figures as "handles" or "surrogates" that human beings can distinguish. This said, this base does have great coverage except for the extremes merely through its divisors (right). Trigesimal (base thirty, above left) fills nicely at richness 2. Looking only at divisors, we see that it has pretty even coverage that has 5 and 6 pretty tight around the square root at "6 o'clock" on the diagram. (Thirty is an oblong number like the dozen).
Sexagesimal (above right) has a notoriously wide separation between divisors 6 and 10 that is mitigated somewhat by richness 2. Finally, base two-dozen (above left) and three-dozen (above right) have quite nice spacings taken to richness 4.

The inclusion of these diagrams in the tour is fairly simple now that they are automated.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Regular Figure Symmetry

Second post of the day!

Here we examine some symmetry of the proper regulars of various bases. These charts mark roots in light yellow, and I am showing digits as red points. The red dot in the left image immediately below is the senary digit 5.
 Senary (base 6): 5th root Senary (base 6): 6th root
We observe that richness 5 is sufficient to occupy the fifth roots (sixth roots for "12" and "43") in base 6.
 Octal (base 8) Hexadecimal (base sixteen)
For prime and prime power bases, we have a finite set of regular figures and can map them all! (Prime numbers have a single regular figure: 1). From these it is easy to extrapolate a chart to any prime or prime power.
 Decimal (base ten) Uncial (Dozenal, base twelve)
Decimal and uncial regular figures neatly populate the tenth and the ninth roots, respectively. Ten populates in richness 5 and uncial in just 3.
 Tetradecimal (base fourteen) Octodecimal (base eighteen)
Tetradecimal and octodecimal neatly fit the ninteenth and twenty-first roots. In base fourteen we need to go to richness 9 to occupy the roots, while octodecimal only requires 5.
 Vigesimal (base twenty) Tetravigesimal (base twenty four)
Base twenty fits the thirteenth in richness 5, while base twenty four fits the twenty third root in richness 4.
 Base thirty Base thirty-six
Base 30 populates root 19 in only 2 while base 36 populates root 26 in richness 5.

Dozens Demigod
Double sharp
Dozens Demigod
Joined: Sep 19 2015, 11:02 AM
This is wonderful! Is there a way to see these charts for any base we'd like (within reason)?

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.

Obsessive poster
Obsessive poster
Joined: Sep 10 2011, 11:27 PM
Double sharp wrote: This is wonderful! Is there a way to see these charts for any base we'd like (within reason)?
Hey, good to see you here again, DS! ☺Yeah, this is some pretty awesome stuff.

Icarus, are those nth roots synonymous with nth roots of unity? They look like they're rays going out to the corners of regular n-gons, just like complex nth roots of unity, but it's a little hard to tell because those yellow lines are a bit faint, especially on my mobile.
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.
The only caveat I have to this is if you want even something like 9 or 14z or 21z as a regular in a base like ᘔ0z 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 30z and 84z and 100z are exceptions, but they are exceptions that prove the rule. They really are exactly equivalent to two digits of their root base.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Hi Double sharp. If you have a base you want to see I'll cook up a chart. I can control the appearance entirely. The app looks like this (Mathematica):
Bezel is the thin circle, Poly the red line that joins digits, Root the display of roots (square, cube, fourth - indeed this is the yellow in the above diagrams; it shows yellow when I don't have "Pwr" checked). Pwr shows the powers of primes when base n is a prime power; Digits plots the mantissa of digits k in base n; Sym shows either radial or complementary symmetry; and Anno identifies the regulars in the output base. I don't use the app for pictures here: instead I generate it with non-dynamic code and that is even more flexible.

So if you have a base I can write a chart.

Dozens Demigod
Double sharp
Dozens Demigod
Joined: Sep 19 2015, 11:02 AM
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, 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).
Kodegadulo @ 18 Jun 2018, 02:19 wrote:The only caveat I have to this is if you want even something like 9 or 14z or 21z as a regular in a base like ᘔ0z 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 30z and 84z and 100z are exceptions, but they are exceptions that prove the rule. They really are exactly equivalent to two digits of their root base.
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. ^_^

EDIT: I notice the current selection accords really quite well with what I would want to see (though maybe I'd include base seven dozen), so I'll leave further suggestions to others. ^_^
Last edited by Double sharp on Jun 19 2018, 03:28 PM, edited 1 time in total.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Those interested can PM me. I have some notes that go along with it. I'd like to share it. If you're interested in it, talk to me.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
I've come to find that CDF can be web-embedded. I'd tried some experiments but it as of yet does not work. If this is ultimately feasible and client sided, it opens up a tremendous opportunity for a variety of dynamic content. Nearly any of the diagrams recently posted, for instance, could be posted and manipulated by anyone with the CDF player!