Hi everyone,

for several reasons, you need a sub-base in number systems with base>6:

• grouping tally marks
• Roman numerals
• number names for high numbers: zyriad (10^4) or zillion (10^6)
• money: coins and notes; here one could also use two sub-bases
• abacus: it's still used, especially in Asia
• scales

Sub-Base 4 would be logical, because it's near to the square root of a dozen.
Sub-Base 6 would be similar to the role of 5 in the decimal system; but it's perhaps somewhat difficult to distinguish 6 consecutive lines (in scales and when dealing with tally marks).
Therefore with base 6, you could use also 3 as an additional sub-base; that would be similar to finger counting (3 finger segments=1 finger).

Opinions?

So should we use the thruppenny bit or the groat?

Either would be easier than having 5p, 2p and 1p. The test is how many coins would it take to make up any particular amount against different types of coin in circulation.

Tally mark would seem best to proceed

1.

2.

3.

4.

5.

6.

I think Endi's suggestion is a good one.

A sub-base of six could work: I, II, III, IIII, IV, V, VII, VIII, VIIII, IX, X. But then it might be more convenient to use double subtraction: I, II, III, IIV, IV, V, VI, VII, VIII, IIX, IX, X.

Undecided.

I'd recommend a 1-2-6 system for Europe (as a parallel to the current 1-2-5 system) and a 1-3 system for America (5/6 dimes, quarters, and dollars).

I've heard that they're popular among blind people too.

What kind of scales? Musical scales? The scales on maps? Weighing devices? Judging on a scale from 1 to 10?

Honestly, I didn't know how to properly translate the German word Skala (which means the distribution of lines on measuring devices).
I thought of scales on measuring devices: ruler, thermometer, and so on. For example rulers: usually there is a longer line at 5mm and an even longer one at 10mm. In dozenals, the same system would give a longer line at 6mm; but that leaves five consecutive lines inbetween, which makes it difficult to decide the exact length without looking very closely. So here, the best choice would be 4 as sub-base. Same on temperature devices, mileometers, slide rules ... 3-6 would be also possible, but, I think, unnecessary complicated.

I think, if we take a sub-base, it should be the same one as often as possible. Money would be an exception, if we want to have two subdivisions here for convenience. I would prefer the 1, 4, 6 distribution, for reasons explained here.

what do you mean you "need" a sub-base ?

why?

I'm not sure you need to call it sub-base, but you need some subdivision of 12 for the reasons explained above. In order to get consistency, it would be desirable to have the same "sub-base" as often as possible. That's why I opened this thread.

Of course, I do not mean sub-base in a sense that you would write numbers in another way, an idea that you would reject, as far as I know you... ;-)

Ah, that kind of scale.

My suggestion would be to have two levels of binary divisions (as on inch rulers), and then a ternary division, like this:

The problem with the 1-4-6 denominations is that the optimal change can't be determined by a greedy algorithm. If you use a greedy algorithm, then the optimal denominations are either 1-2-5, 1-3-5, 1-3-7, or 1-3-8. However, if you restrict the denominations to factors of twelve, then 1-4-6 is indeed one of the optimal combinations, along with 1-2-4, 1-2-6, 1-3-4, and 1-3-6.

This is exactly what I have on an old school ruler where the inches are divided into twelfths - small marks for the twelfths, longer for the quarters and a longer one for the half-inch.
(The ruler has twelfths and sixteenths, tenths etc). The ruler I was issued with at school had inches and tenths on one scale on the front, with dm/cm/mm on the other, and the eighths twelfths and sixteenths on the back. We used inches and tenths in Geometry.

Do you mean like this?



This one has an extra binary division splitting the twelfths down to 24ths. There are also sacales showing 1/2s, 1/4s and 1/8s. The latter two go down to 48ths.
An added feature of all 4 scales is that they are numbered from left to right AND right to left.

I think I can go one better.
Among my collection of rulers and tape measures I have two steel rules that go down below *1/40.

Here's a shot of one of them, (which has inch and tenths, inch and twelfths on the front and metric with inch and sixteenths on the back) showing part of the front scale.

Note how the inch and tenth goes down to hundredths and the inch and twelfths to eightdozenths.

Shaun,

Sorry but I did not intend my last post to imply a "Mine's bigger than your's" or in this case smaller, type of contest. I was just attempting show an example of a ruler that had the characteristics proposed by Dan. The further comments about other scales was only additional information, not a challenge. I just can't be bothered to go rummaging in the garage!

By the way, the ruler I illustrated is made of wood and what looks like ivory. I don't think it is plastic as it has very fine longtitudinal striations only just visible to the eye. I don't think plastic would show this feature, but I don't want to do the heated wire test as I don't want to mark the ruler especially as it is in very good condition.

Any suggestions for a non-destructive test would be appreciated. Anyone?

Nor did I - it was just to demonstrate the scales that were available.
It makes sense to combine the usual binary divisions with subdivision into three.

With those subdivisions (halving the units first, then halving the halves, and finally "thirding" the quarters), it is not immediately apparent where the thirds and sixths go, because only the binary side of twelve's divisibility is emphasized.

I've tried several other schemes, including these:

But the one I like the most is this:

That is, halving the units first, then "thirding" the halves, and finally halving the sixths. I think it's the one that makes it the easiest to find where all the natural subdivisions of a dozen (into units, halves, thirds, quarters, sixths and twelfths) go.

What about using both sides of the ruler? Halves and quarters on one side, thirds and sixths on the other.

Super!!!

Yes, if your ruler allows for that; but many rulers have a shape adapted to measuring on one side only, and there are contexts where you simply don't have two edges available. Thus, a scheme that shows both the binary and the ternary side of twelve on the same side is preferrable as a general solution, IMHO.

For scales, I think all we really need is an extended halfway mark, à la the typical centimetre division. I've made and printed off numerous dozenal scales (usually spans) with digits numbered and an extended mark only at the middle of the digits. It's plenty easy to read and it looks clean.

It's not hard to quickly see that a mark is one or two beyond or shy of a longer one, and it's not hard to see that a mark is directly halfway between longer ones, which is all you need for reading a dozen divisions.

Like this?

It's readable, but I think an extra subdivision would help.

Yes, like that.



Readable, attractive and tidy.

These tally marks could be the basis of a numeral system

1 = I
2 = ||
3 = |||
4 = ||||
5 = / (abbreviation for a slashed group of tally marks)
6 = X (abbreviation for a double-slashed group of tally marks)
7 = XI
8 = XII
9 = XIII
:A = X||||
:B = X/
*10 = XX (we'd probably want to introduce a special symbol here)

Or, we can write the I's connected to each other, getting

1 = I
2 = V
3 = N
4 = W
5 = /
6 = X
7 = XI
8 = XV
9 = XN
:A = XW
:B = X/
*10 = XX

Perhaps we can abbreviate them further by replacing the first X by an underline:

1 = I
2 = V
3 = N
4 = W
5 = /
6 = X
7 = I
8 = V
9 = N
:A = W
:B = /
*10 = X

The "/" would probably be written as "L". The underline of the "I" is hard to see; we could write "⊥" but there's still a good risk that the underline would sloppily be written asymmetrically and confused with "L", so perhaps we should use an overline instead to better distinguish it, and write seven as "T".

Now that we've got a single symbol for each whole number up to twelve, we're almost ready for a positional system. Just need a zero.

0 = O
1 = I
2 = V
3 = N
4 = W
5 = /
6 = X
7 = T
8 = V
9 = N
:A = W
:B = L

I don't find it easy to read the thirds off that ruler, because it forces me to actually count a certain amount of marks to find where they go. There are too many undifferentiated marks in a row, more than the subitizing range (thus forcing actual counting, and therefore not as readable compared to other possible options). All patterns in twelve below the half get blurred, veiling the beauty of twelve's divisibility, as if it were another mediocre number like ten, there being so little visual difference between that design and the decimal-ruler one. In short, I find it a sub-optimal design on all accounts. Twelve allows and deserves better.

A side-by-side comparison of all the "regular" divisions.

Undivided

It's all we need. Each bundle of undifferentiated marks only contains one more than do those on a usual metric scale. Two marks away from the middle mark is something that registers pretty automatically, so I can't see how thirds would present any difficulty. And actually, a hairy assortment of graduations of different lengths, in addition to looking like dung, can have an effect opposite that of making reading easier.

Not that I'm saying there shouldn't be further investigation. I'll make up some scales with the various divisions you presented and go around measuring. :lol:

It might even depend on the size of the unit. A digit-sized unit looks good/reads well with an extended mark only at 0'6, but a unit the size of 0'01 yard is more readable with six main divisions (along with an extended 0'6), then a really short one in between those.

Rulers can be made in many different formats, each suited to a specific set of uses. Besides the regular flat rulers with two sides, there are also rulers with a triangular cross section giving six edges for measuring/ruling. Each end of one side can also show different subdivisions.

This is not a new idea. I have an example made of wood. The scales are not what I would suggest for a dozenal version, but the opportunity to rule each edge to show different dozenal subdivisions and even binary divisions of an inch would seem to provide sufficient scope for variations. As one example, on one side you could have 96 divisions of an similar to that of Shaun's post of Apr 5 2006 9:35PM. On one edge, subdivisions could be 1/2, 1/3s, 1/6s, 1/12s, 1/24s, 1/48s and 1/96s. The other side could show the same smallest subdivisons (i.e. 1/96s), but the other divisions could be 1/2, 1/4s, 1/8s, 1/16s, 1/32s, 1/48s and 1/96s.

Other arrangements could be used on other rulers, but of course, always as dozenal fractions.