Here's another argument for the base 8 as opposed to base 12. I still find myself quite fond of the base 8 ... although I do appreciate your arguments in favour of base 12. See link:

http://www.geocities.com/donaldsauter/base-8.htm

My original case for the base 8 is here:

http://z13.invisionfree.com/DozensOnline/i...p?showtopic=138

Any conversions?

Ah, Base 8 is beautiful, but again, it is deficient. Base 8 features the following factors: 1, 2, 4, and 8. This is as good as decimal: 1, 2, 5, 10. And while you octal people are scratching your heads about octal 121; is it divisible by what - quick! how about octal 1604, we will be sure that *69 and *630 are divisible by three. We will also be sure about your octal 100 and octal 444, being *54 and *204 being divisible by 4. (We'll both be a little set back by 5s, though. Nothing is perfect. But threes are commoner than fives.)

Because base 8 concentrates ALL its power on 2, it has little ability to accommodate other numbers, and there are other numbers. Three makes sense because after two, it is the commonest factor. One of the things human beings try to do when they consider a quantity is try to group or subdivide it so that they can better use that quantity. By placing all the power of your base in one prime factor, you limit yourself to the ability to analyze that quantity for its content of that single factor. So diversity of factors is perhaps more important than depth of analysis for a single prime factor. All your eggs are in one basket with 8.

Mr. Sauter seems to argue against himself when he talks about 3 being overrated. So then why is he employing a base of 2 to the third power? Why not go further and use hex, which even more beautifully accounts for 2 to the 2 to the 2 power?

Mr. Sauter decries 12's mult table as having 44% more figures to memorize, when I think many people memorized a multiplication table of 144 figures anyway.

Dozenal does manage to handle binary figures well, no as well as 8 but then again it is not as "focused" or "specialized" as 8 is in resolving for content of 2. The dozenal eighth, *0.16, is not unwieldy. The dozenal sixteenth, *0.09, is nice too. It's even nicer when one considers that the dozenal sixteenth is expressed conveniently in a way that allows one to divide it by three if the need arises.

Lots of "feel" talk on Mr. Sauter's site. If you want a nice expression of Mr. Sauter's binary division routine, look at Intuitor's case for hexadecimal: http://www.intuitor.com/hex/switch.html. I would say that people can indeed divide by thirds, and the use of two hands etc. is nonsense.

I've seen hundreds of steel frames with divisions into 3rds, as well as quarters and fifths, but division by 3 is a popular way to divide a span. Why is three times a charm? What's behind door number 3? In your next speech, why will you use the power of 3? Three is very important in our human culture, only seeming weird because of decimal. Mr. sauter's assessment of 3 as trumped up is a holdover, perhaps, of his decimal upbringing.

Here is an example of the power of twelve, armed with the simplest primes and some depth to analyze a quantity for 2s. There was once a poll taken in the States that determined that the common person considers percentages to be more exact than fractions (which of course is false, but people like the seeming exactitude of percentages). This was one of those Google tidbits that greets you if you have registered; I wish I could find that poll again to cite it directly. So consider percentages. People use a base's powers to extend the viability of the base to resolve fractions. Decimal percentages aren't too bad...

But even decimal is superior to eight (and sixteen!) when we examine the "percentages":
base 10's "100" 1, 2, 4, 5, 10, 20, 25, 50, 100
9 divisors. Nice.

base 8's "100"=64: 1, 2, 4, 8, 16, 32, 64
7 divisors. What happened?

base 16's "100"=256 1, 2, 4, 8, 16, 32, 64, 128, 256
9 divisors. Okay...but no bargain.

base 12's "100"=144 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
15 divisors.

Base 8 continues to lose its strength, sapped by its concentration of all its power on the prime 2:
base 10's "1000":
1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
16 divisors.

base 8's "1000"=512
1, 2, 4, 8, 16, 32, 64, 128, 256, 512
10 divisors.

base 16's "1000"= 4096=
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096
13 factors.

base 12's "1000"=1728
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108,
144, 192, 216, 288, 432, 576, 864, 1728
28 factors, almost TRIPLE octal's, and easily double hex's. Some scientific base, these binary bases, hampered by their overrepresentation of 2. Decimal, even base 14, by this measure, has a stronger "percent" than octal or hex.

So the question dozenalists might ask binary people is: "Can you keep up"? :-)

Yes, we can't keep adding diverse prime factors because the base quickly gets cumbersome. I think 12 represents the optimum mix of diverse factors, some depth of resolution for 2, and a succinct base size. With 12, you can resolve for 2 or 4, and through its powers 8, 16, etc. and also resolve for 3. With the binary bases you limit yourself to powers of 2, with little ability anywhere else.

Beautiful base, 8, but it is limited and all its power is concentrated in one place, when for just a little more, you can also account for the second most common prime factor. Eight is no bargain, and by some measures even decimal is better. And if its beauty you are after, base 16 is even more beautiful, because it is 2 to the second to the second power, even more binary than octal, which, if you are trying to avoid and deny 3, is unfortunately the 3rd power of 2. So three will haunt octal, in its logarithms, you can't get away from 3 by using 8.

Shoelace I do indeed appreciate the debate, don't want to give you the impression that your thoughts aren't appreciated, we are having a dialogue. Have a fine Friday the 13th!

The factor of 3 is very useful, and not one bit "trumped up" as the author said. Octal is a nice base, but dealing strictly in powers of 2 is a limitation. As has been pointed out, dozenal handles binary fractions quite well, much better than decimal, along with that factor of 3. It's just more flexible and useful.

Although I do like octal and hex, senary is my second favorite alternative base behind dozenal.

If I had to choose a binary-derived base, I'd surely go for hexadecimal, not octal. Hexadecimal is quintessentially two-based, because not only its only prime factor is two, also to go from two to sixteen you don't have to leave the "binary-mindset", just square two twice (2^(2^2) = (2^2)^2 = 16), while octal falls short requiring something of a "ternary-mindset" (cubing) to go from two to eight (2^3= 8) without adding the benefits of having three as a prime factor.

Packing of (cubical) items:

SENARY

Possible ways to pack 6 items in 1D, 2D and 3D arrangements:

Possible ways to pack 36 items in 1D, 2D and 3D:

36 = 6^2 = 2 * 2 * 3 * 3

2 x 18
3 x 12
4 x 9
6 x 6

2 x 2 x 9
2 x 3 x 6
3 x 3 x 4

Possible ways to pack 216 items in 1D, 2D and 3D:

216 = 6^3 = 2 * 2 * 2 * 3 * 3 * 3

2 x 108
3 x 72
4 x 54
6 x 36
8 x 27
9 x 24
12 x 18

2 x 2 x 54
2 x 3 x 36
2 x 4 x 27
2 x 6 x 18
2 x 9 x 12
3 x 3 x 24
3 x 4 x 18
3 x 6 x 12
3 x 8 x 9
4 x 6 x 9
6 x 6 x 6

Total number of possible packing arrangements for the first three powers of 6:
2 + 8 + 19 = 29


OCTAL

Possible ways to pack 8 items in 1D, 2D and 3D arrangements:

Possible ways to pack 64 items in 1D, 2D and 3D:

64 = 8^2 = 2 * 2 * 2 * 2 * 2 * 2

2 x 32
4 x 16
8 x 8

2 x 2 x 16
2 x 4 x 8
4 x 4 x 4

Possible ways to pack 512 items in 1D, 2D and 3D:

512 = 8^3 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2

2 x 256
4 x 128
8 x 64
16 x 32

2 x 2 x 128
2 x 4 x 64
2 x 8 x 32
2 x 16 x 16
4 x 4 x 32
4 x 8 x 16
8 x 8 x 8

Total number of possible packing arrangements for the first three powers of 8:
3 + 7 + 12 = 22


DECIMAL

Possible ways to pack 10 items in 1D, 2D and 3D arrangements:

Possible ways to pack 100 items in 1D, 2D and 3D:

100 = 10^2 = 2 * 2 * 5 * 5

2 x 50
4 x 25
5 x 20
10 x 10

2 x 2 x 25
2 x 5 x 10
4 x 5 x 5

Possible ways to pack 1000 items in 1D, 2D and 3D:

1000 = 10^3 = 2 * 2 * 2 * 5 * 5 * 5

2 x 500
4 x 250
5 x 200
8 x 125
10 x 100
20 x 50
25 x 40

2 x 2 x 250
2 x 4 x 125
2 x 5 x 100
2 x 10 x 50
2 x 20 x 25
4 x 5 x 50
4 x 10 x 25
5 x 5 x 40
5 x 8 x 25
5 x 10 x 20
10 x 10 x 10

Total number of possible packing arrangements for the first three powers of 10:
2 + 8 + 19 = 29


DOZENAL

Possible ways to pack 12 items in 1D, 2D and 3D arrangements:

Possible ways to pack 144 items in 1D, 2D and 3D:

144 = 12^2 = 2 * 2 * 2 * 2 * 3 * 3

2 x 72
3 x 48
4 x 36
6 x 24
8 x 18
9 x 16
12 x 12

2 x 2 x 36
2 x 3 x 24
2 x 4 x 18
2 x 6 x 12
2 x 8 x 9
3 x 3 x 16
3 x 4 x 12
3 x 6 x 8
4 x 4 x 9
4 x 6 x 6

Possible ways to pack 1728 items in 1D, 2D and 3D:

1728 = 12^3 = 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3

2 x 864
3 x 576
4 x 432
6 x 288
8 x 216
9 x 192
12 x 144
16 x 108
18 x 96
24 x 72
27 x 64
32 x 54
36 x 48

2 x 2 x 432
2 x 3 x 288
2 x 4 x 216
2 x 6 x 144
2 x 8 x 108
2 x 9 x 96
2 x 12 x 72
2 x 16 x 54
2 x 18 x 48
2 x 24 x 36
3 x 3 x 192
3 x 4 x 144
3 x 6 x 96
3 x 8 x 72
3 x 9 x 64
3 x 12 x 48
3 x 16 x 36
3 x 18 x 32
3 x 24 x 24
4 x 4 x 108
4 x 6 x 72
4 x 8 x 54
4 x 9 x 48
4 x 12 x 36
4 x 16 x 27
4 x 18 x 24
6 x 6 x 48
6 x 8 x 36
6 x 12 x 24
6 x 16 x 18
8 x 8 x 27
8 x 9 x 24
8 x 12 x 18
9 x 12 x 16
12 x 12 x 12

Total number of possible packing arrangements for the first three powers of 12:
4 + 18 + 49 = 71


HEXADECIMAL

Possible ways to pack 16 items in 1D, 2D and 3D arrangements:

Possible ways to pack 256 items in 1D, 2D and 3D:

256 = 16^2 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2

2 x 128
4 x 64
8 x 32
16 x 16

2 x 2 x 64
2 x 4 x 32
2 x 8 x 16
4 x 4 x 16
4 x 8 x 8

Possible ways to pack 4096 items in 1D, 2D and 3D:

4096 = 16^3 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2

2 x 2048
4 x 1024
8 x 512
16 x 256
32 x 128
64 x 64

2 x 2 x 1024
2 x 4 x 512
2 x 8 x 256
2 x 16 x 128
2 x 32 x 64
4 x 4 x 256
4 x 8 x 128
4 x 16 x 64
4 x 32 x 32
8 x 8 x 64
8 x 16 x 32
16 x 16 x 16

Total number of possible packing arrangements for the first three powers of 16:
4 + 10 + 19 = 33


Summing up:

Clearly, octal is a sub-optimal base for packing. Also, eight is a factor of twelve squared, which means dozenal is "8-friendly"; while twelve is not a factor of any power of eight, which means octal is "12-unfriendly". So, for example, a dozenal-minded manufacturer would be happy to produce items in 8-packs to take advantage of its 3D cube arrangement (2 x 2 x 2), and then bulk-sell them to retailers in 144-item packages (of 18 eight-packs each, which can be arranged as 2 x 3 x 3, and then these could be arranged in 1728-item cubical crates of 12 packages each, arranged as 2 x 2 x 3). But an octal-minded manufacturer would not likely be as happy to sell items in 12-packs (such as egg cartons).

Another examination of octal, and its cousin hexadecimal.

One of the things we want to do when we use a base is to discern properties of a quantity we are dealing with. This means that if we see a given set of items, we want to better understand how many items we are dealing with, and perhaps how we can divide this number of items into subsets. We'd like to understand, for instance, whether or not a set of items might be divided in half.

Base 8 does this well, so that one can analyze the quantity for divisibility in half three times. It isn't as useful if we are dividing by any other number than 2.

Another property of a base is its "totient function" (http://mathworld.wolfram.com/TotientFunction.html), a measure of how many digits are "relatively prime" out of the total digit span. This measurement may seem abstract, but it gives us a picture of how often a "strange" number crops up (here "strange" meaning a number that isn't divisible by any of the factors of the base.)

Base 2 = 1 (1) 50%
Base 6 = 2 (1, 5) 33.33333%
Base 8 = 4 (1, 3, 5, 7) 40%
Base 9 = 6 (1, 2, 4, 5, 7, 8) 66.66667%
Base 10 = 4 (1, 3, 7, 9) 40%
Base 12 = 4 (1, 5, 7, 11) 33.33333%
Base 14 = 6 (1, 3, 5, 9, 11, 13) 42.85714%
Base 15 = 9 (1, 2, 4, 7, 8, 11, 13, 14) 60%
Base 16 = 8 (1, 3, 5, 7, 9, 11, 13, 15) 50%
Base 20 = 8 (1, 3, 7, 9, 11, 13, 17, 19) 40%
Base 60 = 16 (1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59) 26%

Prime numbers greater than the base are confined to the totatives of that base, so that primes in octal will end in 1, 3, 5, or 7. This leaves the user of octal with more uncertainty about a quantity being indivisible than even the users of decimal. Hexadecimal is just as weak, despite its magnitude and the grooviness of its being the double square of 2. There are reasons why one won't turn to large bases (inability to wield them) or very small bases (inefficiency). It's interesting to observe base 6's properties, which were recently commented on by Endi in "6+ or -1". Base 12 possesses some of base 6's power. So dozenal, for instance, "rationalizes" or makes discernable a random quantity's divisibility at a higher rate than many other reasonably-sized bases. The user of dozenal knows numbers ending in 0 and 6 divide by 6; by 0, 4, or 8 divide by three, and 0, 2, 4, 6, 8, and a divide by 2. All that remains are 1, 5, 7, and b, some of which might be prime. Users of octal wonder about what might divide numbers ending in 1, 3, 5, and 7. Now this might not be as bad as I paint because there are tricks (rules of divisibility) at least in hexadecimal that mollify hex's inability to resolve content of 3 directly; by 16's proximity to 15, it can indirectly detect divisibility by three and five in a kind of cool way, like decimal can use the rule of divisibility associated with 3 (add the digits: if the digits add up to three, quantity is divisible by three). I know that octal thirds are repeating .25 and .52, and maybe that's acceptable, but it's not as clean as .4. The cool rule of divisibility 10 and 16 enjoys regarding 3 is not shared with octal.

Now none of these qualities (totative, power of magnitudes of the base, etc.) are dealmakers; I think dozenal gives us a little extra power for not much more "price" over conversion to senary. But by this quality senary is the best choice because it is the smallest base of reasonable size with the lowest totient value.

The binary bases are great for counting bits. I think the use of hex and UU and other binary bases are ideal for this, and that usage of dozenal is not as well-suited. Technology can change, along with it, perhaps counting with binary bases, if, for instance, a ternary circuit is devised and becomes popular. Usage of a base should yield a practical advantage, and I don't think you'll beat binary bases for counting bits with our current technology. But for general usage, the powers of dozenal and the other "versatile" bases (2, 6, 12, 60, 120, 360, 2520, 5040, etc, but practically only 6, 12, and maybe 60) are better suited for most everyday conditions, because everyday conditions involve random quantities that aren't always conveniently binary. Short of a perfect base, I think 12 offers the most benefits for a compact and manageable size base.

It is for this reason that I am a former proponent of adoption of hexadecimal as an everyday base. (but those rules of divisibility 16 possesses, oh how I wish dozenal had that ability to detect divisibility by five indirectly!)

What's "UU"? :unsure:

Actually, ternary computers were devised very early, but they didn't catch on. They aren't as simple to design as binary computers and ternary has important practical drawbacks, such as not allowing a finite representation for the simplest fraction 1/2 (which means it is entirely incompatible with decimal, since they don't share any common prime factor).

The reason binary is used in computers is because it isn't difficult to come up with electromagnetic devices that display two different states and because binary arithmetic is the simplest of all, so it's easy to implement it in circuitry. In fact, binary arithmetic is so simple that its addition and multiplication tables consist of solely the trivial rows (the ones for 0 and 1, which are usually obviated in the tables for other bases).

Hexadecimal is very nice as a convenient shorthand representation of binary for human use, since binary is optimal for computers but very cumbersome for humans. Octal also serves this function, with the advantage of not requiring extra digits, but is far less common since it codifies 3 bits (that is, a "non-binary" number of bits). I think there are already devices that can support 16 states to store hexadecimal digits, but to use this base internally in the ALU of computers would complicate their circuitries greatly. Same with dozenal, decimal, octal and other "human-sized" bases. The closest to having a decimal-based computer is to use BCD (binary-coded decimal), which computes with decimal digits using an underlying binary circuitry. Dozenal can be codified similarly with simpler circuitry, and hexadecimal too with a yet simpler one, but in all cases they rely on binary technology.

Icarus,
Just a minor pair of corrections to an otherwise useful reply supporting base 12. In the table above, the number of relatively prime values in base 8 is 4 and therefore the percentage should be 50%.

In the case of base 15, there are only 8 relatively prime values while you listed it as 9. The correct value of 8 gives a result of 53%.

"Just seein' if y'all was payin' attention" ;-).

Thanks for the correction. Also forgot to mention that in addition to 0, 6 divisible by six, 0, 4, 8 by 3, and 0, 2, 4, 6, 8, a by two, I forgot 0, 3, 6, and 9 by 4. Oh so very important...

We were cookin' supper yesterday and encountered a decimal quarter kilo of cheese, which we had to divide by three per recipe. 250 g divided by three. How convenient it might have been to be a dozenal quarter weight-unit. The question would've been so simple we wouldn't have to stop and think. So we used 80 g. Actually we just eyeballed it and got what we got. (And it was just fine that way).

Reminds me of the days when we used to buy blocks of ice-cream. There were five of us in the family and Dad had got division into 5 by eye down to a fine art. But I don't think he'd have managed anything higher, like 7 parts!

Reply to uaxactum's question, what is "uu". Uuencode, using a "binary" base higher than hexadecimal, like base64. There are other encoding methods that use 7 or 8 bytes. (Sorry failed to answer that last week)