## Le Tour Des Bases

This forum examines bases other than twelve and less than sixty.
Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Added bases 60, 70, 72, and 84 this afternoon, in excitement over Wendy's comparison of 70 and 72. These will set the pace for the mid-scale bases. May still add some of the frills of the human scale bases, and will update the larger (grand) bases.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Bases 90, 96, 108, and 120 have been added, along with the Midscale Mashup. These bases round out the upper midscale bases. These also incorporate SDN and a more canonical "common lexicon" name based on greek units and latin decades.

Have a fine weekend!

 Posts 775
Dozens Disciple
Oschkar
Dozens Disciple
Joined: Nov 19 2011, 01:07 AM
Why are 18, 20 and 21 mid-scale bases? The multiplication tables have 171, 210 and 231 products, still within the reach of memorization, and divisibility tests still cover the majority of their digits. The primes 17 and 19 still have their uses, however rare they might be (19, for example, is used sometimes to pack pencils in a hexagon). The abbreviated multiplication tables of 22, 25 and 26 products are still in the senary-septimal range, and the bases only have 6 or 4 factors, small bases for themselves.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Oschkar, brilliant question and excellent thoughts! Let's debate it, see this thread. I really don't know the answer but the thread outlines my thoughts. I invite anyone to add their thoughts, because perhaps these classes don't make sense. They don't come from anywhere but here, so we can decide how to classify them. My concept of classes has to do with educating children to wield a number base, i.e., might this base actually stand as a civilizational number base.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
I've added bases 48, 80, and 112, as well as a midscale mash-up of {32, 48, 80, 112}. These might be the last tour bases for a while as a large project is upon the office >yay<. Like ostmark loves, time to make some wampum! ( in increments of twelfty).

Dozens Disciple
Treisaran
Dozens Disciple
Joined: Feb 14 2012, 01:00 PM
icarus wrote:These might be the last tour bases for a while as a large project is upon the office >yay<.
Pity, I was hoping bases 24 (double-dozen) and 36 (triple-dozen) would make it into the tour. They and the long hundred are, IMO, the only alternatives I'd consider to dozenal for general-purpose use.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Oh, those bases will certainly make it in the tour. I have 32 partially done and just need to set it aside for maybe a week or so. There might be windows, but I also have family, etc.

Dozens Disciple
Treisaran
Dozens Disciple
Joined: Feb 14 2012, 01:00 PM
icarus wrote:Oh, those bases will certainly make it in the tour.
Once they're in, the only dozenal-multiple base outside the tour will be 132 (levanunqual), admittedly a useless base (Ï‰ is prime, Î± is 7Â·19).

I'd like to make, myself, a single-thread mini-tour of all dozenal-multiple bases up to *140 (unquadranunqual, unquadnilimal, 192) inclusive. But I'd better wait with that post till you've had your say on bases 24 and 36, because I don't do coloured tables so well.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Treisaran,

Shortly before the big crunch I was preparing two mashup threads. The first compared the first twelve multiples of the dozen. This was a lengthier mashup than the Midscale Mashup, comparing all twelve multiples in groups of similarity in this or that aspect. The thread has all the tables including binunqual, trinunqual, and levanunqual. We'll see if I can't finish it because the tough part was finished last week.

Basically the thread considers bases that are integer multiples 12k with 0 <= k <= 12,
In terms of intrinsic properties there are the groups k = {1, 2, 3, 4, 6, 8, 9, 12} (2 distinct primes), {5, 7, 10, 11} (3 distinct primes). The extrinsic properties (neighbor factors) are also considered as well.

Dozens Disciple
Treisaran
Dozens Disciple
Joined: Feb 14 2012, 01:00 PM
I find the dozenal-multiple bases interesting because not just because they're related to the dozen, but also because they have some interesting groupings:
• The 3-smooth community: 12, 24, 36, 48, 72, 96, 108, 144, 192.
• Prime factor exogamists: 60, 84, 120, 132, 156, 168, 180.
• Quinary-haters: 48, 72, 108, 132, 168, 192.
• Primeflanked Flacks: 12, 60, 72, 108, 180, 192.
• Omega Warriors: 36, 96, 156.
• The Alpha Team: 24, 48, 84, 132, 168.
• Extroverts: 120, 144.
• Squares: 36, 144.
I wouldn't go beyond 192, *140, 0xC0 in a survey, but there are some interesting multiples still, some of them already covered in your tour: 216 (cube, flanked by composites), 240 (primeflank), 300 (flanked by composites, first multiple of the dozen divisibly by the decimal hundred) and 360.

 Posts 775
Dozens Disciple
Oschkar
Dozens Disciple
Joined: Nov 19 2011, 01:07 AM
On a side note, I'd like to suggest base 64. Being a power of two, it might seem that it is lacking in diversity, but its neighbors are 3Â²Ã—7 and 5Ã—13, giving it an advantage over the first six primes, excluding 11 (but then again, only the multiples of 11 and their neighbors are fair to it).

Code: Select all

`````` &nbsp; &nbsp; 7 &nbsp; 2^3 &nbsp; &nbsp; 3^2
&nbsp; 3Ã—5 &nbsp; 2^4 &nbsp; &nbsp; &nbsp;17
&nbsp; &nbsp;31 &nbsp; 2^5 &nbsp; &nbsp;3Ã—11
3^2Ã—7 &nbsp; 2^6 &nbsp; &nbsp;5Ã—13
&nbsp; 127 &nbsp; 2^7 &nbsp; &nbsp;3Ã—43
3Ã—5Ã—17 &nbsp; 2^8 &nbsp; &nbsp; 257
&nbsp;7Ã—73 &nbsp; 2^9 &nbsp;3^3Ã—19``````
All other power-of-two bases, except eight, have large primes in their factorizations, which makes 64 a rather convenient base. The product of 63, 64 and 65, 262080, is the quadruple of the already suggested radix 65520, very close to a binary power and with 168 factors (an important base in itself, 2^3Ã—3Ã—7).

Dozens Disciple
Treisaran
Dozens Disciple
Joined: Feb 14 2012, 01:00 PM
Oschkar wrote:On a side note, I'd like to suggest base 64.
Yes, probably the best base to choose if you're confined to a binary world to such a degree that you need to do all your calculations in it. (Hexadecimal isn't the product of so dire straits, it's just an auxiliary.)
Being a power of two, it might seem that it is lacking in diversity, but its neighbors are 3Â²Ã—7 and 5Ã—13, giving it an advantage over the first six primes, excluding 11 (but then again, only the multiples of 11 and their neighbors are fair to it).
Base 64 (*54, SDN pentquadral) enjoys the relationships that any sixth power does. There is a regularity for powers that can be obversed, and I assume also grounded in modular arithmetic:
• Every number has the prime 3 as its divisor or neighbour.
• Every square has the prime 5 as its divisor or neighbour.
• Every cube has the prime 7 as its divisor or neighbour.
• Every fifth power has the prime E as its divisor or neighbour.
• Every sixth power has the prime *11 as its divisor or neighbour; and as every sixth power is also a square and a cube, it also has a relationship with the primes 5 and 7.
Doubtless the list could go on for higher primes. For a number to have a relationship with all first six primes (2, 3, 5, 7, E, *11), you need to pick either a pentanunquate* (=sixtieth) power of a number (like 2â†‘*50, already astronomical) or a sixth power of a multiple of E (like 1Xâ†‘6).

â”€â”€â”€

* I assume 'pentanunquate' is the correct form on the basis of Kodegadulo's coinages 'binate' and 'trinate' to replace the confusing 'quadratic' and 'cubic' (for equations with a second-power term and a third-power term, respectively).

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Oschkar…the other mashup is…the power of 2 mashup! This goes to 256. Nearly have the dozens mashup done, have to little pieces here and there. Next week I think Ill be double loaded, so won't be any progress there.

 Posts 775
Dozens Disciple
Oschkar
Dozens Disciple
Joined: Nov 19 2011, 01:07 AM
icarus @ Sep 18 2012, 11:34 PM wrote: There might be windows
There's always Windows, even when you use another operating system in your daily life, just like there's always decimal although you use a different number base. (Or, in icarus's, wendy's, or my case, all of them.)

 Posts 775
Dozens Disciple
Oschkar
Dozens Disciple
Joined: Nov 19 2011, 01:07 AM
Small lower-mid scale bases 17, 18, 19, 22, 24, those which Icarus has ignored up to this point.

17: A just-beyond-manageable prime with no uses that I know of (a heptadecagon can be constructed with a compass and straightedge, but is virtually never used in practice.) It has maximal expansions of 1/5, 1/7 and 1/11 (0.36da36da36..., 0.274e9c274e..., 0.194adf7c63...), but it is intimately familiar with the dozen, having 2âµÃ—3Â² as the omega-2. There are 7 opaque totatives {5, 7, 10, 11, 13, 14, 15}, and, although SPD can alleviate the 5-situation, there's not much to do in heptadecimal.
18: There is something intriguing about octodecimal. Eighteen is an Î±Â²Î² type of base, like twelve or twenty, but the square factor is the larger prime in this case, something that does not happen again until 50. Because of this, base 18 acts more like 10 and 14 in some cases. but like 12 and 20 in others. 1/5 and 1/11 are maximally recurrent (0.3ae73ae73a..., 0.1b834g69ed...), and the neighbors are the useless primes 17 and 19. Base 18 has 6 divisors {1, 2, 3, 6, 9, 18} and 4 opaque totatives {5, 7, 11, 13} (once more, SPD helps with 5 and 13). The non-divisor regular digits {4, 8, 12, 16} are treated to short two or three digit expansions (somewhat like {4, 8} in bases 10 and 14), and the remaining digits {10, 14, 15} are semitotatives.
19: Another just-beyond-manageable prime, but a little bit friendlier on the neighbors. The omega-2 is 360, a highly composite number. On the other hand, the alpha-2 offers no help. 1/7 and 1/13 have maximal expansions. There are 5 opaque totatives {7, 11, 13, 14, 17}, and SPD is no help, since 362 is only a semiprime.
22: Duovigesimal is the divisibility-test paradise, the first six primes have relationships with the square and the cube. It is a semiprime, like 10 and 14, but the eleven-factor causes all sort of strangeness. The divisors {1, 2, 11, 22} are few and far between, and all of the remaining regular digits are powers of two {4, 8, 16}. There are 3 alpha-inheritors {3, 7, 21}, but the 6 opaque totatives {5, 9, 11, 13, 15, 17, 19} really hinder computation. The rest of the digits {6, 10, 12, 14, 18, 20} are semitotatives.
24: The diametrical opposite of 22 is 24, the double dozen. With eight divisors and a very useful relationship with the alpha, we are able to render {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 23} as Â«transparentÂ», a full 62.5% of all digits. The remainder can be classified in two types, the Â«negativesÂ» {14, 19, 21, 22} where the multiplication rows are the inverse of those of the transparent digits, and the Â«irrelevantÂ» digits {7, 11, 13, 17}, two prime pairs.

 Posts 775
Dozens Disciple
Oschkar
Dozens Disciple
Joined: Nov 19 2011, 01:07 AM
When are 17, 19, 22, 28, 32, 40 and 56 coming? And as weird bases go, I'd like to see 351, simple divisibility tests for 5 powers of 2, 3 powers of 3, 2 powers of 5, and 1 power of 7, 11, 13.

Dozens Disciple
Treisaran
Dozens Disciple
Joined: Feb 14 2012, 01:00 PM
Oschkar wrote:And as weird bases go, I'd like to see 351, simple divisibility tests for 5 powers of 2, 3 powers of 3, 2 powers of 5, and 1 power of 7, 11, 13.
Among what are called 'grand bases', I've recently thought about 2640 as offering a bit more bang for the buck than 2520:

2640 = 2â†‘4 Â· 3 Â· 5 Â· 11, with its Ï‰ = 7Â· 13 Â· 29, providing for all the primes needed for calendric use (I'm talking about a lunisolar calendar, not the Gregorian one with its prime 31 there to make matters so much more complicated).

For those bases off the possibly beaten track (meaning, a track not necessarily beaten but which people might seriously consider beating), I think full-blown tour posts would be overkill; perhaps an 'Esoteric Bases' tour thread would serve here.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Guys 351 was a weird base!

I am afraid I am going to be absolutely slammed in the coming fortnight, plus I will be in NYC for a bit. There might not be too much action. Am in a permission gap (two projects held) and I think both will break loose with a rapid deadline.

The American elections have likely caused a little vacuum in demand; now that these are coming up I might not be able to post quite as much. This said, the first bases on the tour were posted during heavy demand. So we'll see!

Dozens Disciple
Treisaran
Dozens Disciple
Joined: Feb 14 2012, 01:00 PM
I do hope you take those invitations as encouragement rather than pestering, Icarus. I'm looking forward to what you've got to say about base 2, because of your contention that it's an exceptional base in so many ways.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
No I do not take it as pestering, we are all curious! I am in new York thru Sunday and will see next week.

 Posts 775
Dozens Disciple
Oschkar
Dozens Disciple
Joined: Nov 19 2011, 01:07 AM
First post in four days!

I'm just going to remind Icarus of bases 22, 32, 40, 50, 56, 64, 99. Will they be published here or on your number base website?

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Oschkar,

I've been writing a very extensive compendium of definitions, which will stand as independent pages, then get boiled down into a glossary that will serve as a basis for what is the tour here. The tour pages will have expanded coverage. The intuitive divisibility tests will factor in practicality, and a full range of transparent numbers can be described. The new CSS is getting very colorful! The best thing is that diagrams will be integrated in the work, communicating concepts in more than one way. I'm also re examining nomenclature (semidivisor vs. quasidivisor, regular figure vs. regular root, etc.).

I have an active project with a deadline in a couple weeks and have had to lay it aside. I believe the American debt ceiling is damming up a great deal of demand potentially so will be blitzing the number base project in December. This has consumed my attention unfortunately. We'll see if I can put out a couple units.

The good thing is that the resources built here are applicable to the number base project.

Dozens Demigod
dgoodmaniii
Dozens Demigod
Joined: May 21 2009, 01:45 PM
Have you considered compiling all of this, and expanding it somewhat, into a book? You've got enough material for a big fat volume, the 800 (that's eight biqua, of course) pound gorilla of the field. It'd serve as the mandatory textbook for alternate base work in colleges.

I honestly think this would be a great, if heavy, project.
All numbers in my posts are dozenal unless stated otherwise.
For ten, I use or X; for elv, I use or E. For the digital/fractional/radix point, I use the Humphrey point, ";".
TGM for the win!

Dozens Disciple
Treisaran
Dozens Disciple
Joined: Feb 14 2012, 01:00 PM
icarus wrote:The new CSS is getting very colorful!
Good, hope it doesn't get too hard to distinguish between the many colours. While I'm about it, why not see to devising distinctive cross-hatch patterns? They wouldn't be needed on the Web, but they would for print (my laser printer is monochrome).