Graduation Subdivisions

Anyone who has thought about designing a clock face, protractor, ruler, or measuring instrument for a metrological system using a base such as twelve other than decimal is likely to have contemplated what the best way to subdivide the graduation marks would be. As such, this topic has been discussed by others before elsewhere in the context of base twelve and the marks of a straight ruler. I know of the question being asked before but not being answered satisfactorily elsewhere. The following are my opinions.

In the division of a length or angle, the first division should be by the most important and useful fraction, followed by a fraction of lesser importance. Thus, division by two before three is preferred. However, division by two in stages is a very slow way of developing a fine scale, and aggregation of two binary divisions into still subitizable quarters may be chosen. Having more levels or tiers of subdivision has the disadvantage of requiring a greater number of different lengths for distinguishing the tiers of the notches or graduations.

In the decimal scales, there is a separate binary division because there is no way to aggregate it with another binary (or even ternary) division before or as far as division to a tenth reaching as many gradations as the base. There is only one way to factorise ten into subitizable segments or sectors, and that is as two by five. If one is trying to adapt the look and feel of a decimal scale (subdivided as 2, 5) to a dozenal one, one might divide with greater prominence first into two and subsequently with lesser prominence each half into six, producing twelfths overall. This sequence of subdivision is: 2, 6. However, six is not considered to be a highly subitizable number, and since it is a composite number with prime factors, it may be subdivided in itself into two, and then each of those halves into three. This procedure thereby produces this sequence of subdivision: 2, 2, 3. Aggregating the two halvings produces subdivision first by quarters, followed by thirds, giving the subdivisional sequence: 4, 3.

These conclusions may be formulated as rules:

1. Discussion: A Subitization Rule is considered to be the most inviolable rule of subdivision (notwithstanding the fact that it can be violated, inviolable should not have a superlative grammatically) because it is the reason for subdivision in the first place.
   Weak Subitization Rule: Subdivision into a non-subitizable number of fragments should be avoided.
   Discussion: The number of subdivisions in any stage or tier should not exceed seven.
   Strong Subitization Rule: A subdivision is to be allowed if and only if it is into a subitizable number of fragments.
   Corollary: A subdivision by a composite number of fragments need not be further subdivided provided the composite number is subitizable.
   Corollary: Subdivisions by the two smallest prime numbers may be aggregated into a single subdivision provided this resulting subdivision is into a subitizable number of fragments.
   
2. Subdivision by a number of greater importance should proceed before subdivision by a number of lesser importance.
   Corollary Subdivision by a small prime number should proceed before subdivision by a larger prime number.
   Discussion: The subdivisional priority for a composite number versus another number depends on which is more important, which depends on the chosen method of ranking numbers by their importance.
   
3. A lesser number of subdivisional levels or tiers is better than a greater number achieving the same overall number of graduations.
   Discussion: The ideal number for a subdivision is a root of the base, the square root if two tiers are desired from one power of the base to the next, or the cube root if three tiers are desired between adjacent powers of the base, and so on.
   

It is not easy to determine what influence the third rule has over the first two rules, which it counteracts. For example, is it better to divide twelve by just two tiers with 2, 6, or by three tiers with smaller numbers 2, 2, 3 that are both more subitizable and more important? It might even be possible for 6, 2, despite the wrong order of divisions, to be better than 2, 2, 3. To be cautious, I take the standpoint that 2, 3 is better than 6, because we can use a slight difference in the lengths of the graduations between the tiers of 2 and 3 in such a way that they look satisfactorily not too much unlike a single division of six.

A list of the ways for subdividing twelve ranked is:
4, 3 (The most efficient and fewest number of subdivisional tiers, with the quarters having higher importance than thirds)
2, 2, 3 (The most similar to binary, sub-optimal by Rule 3)
2, 6 (The most similar to decimal. The six is not highly subitizable, almost violating Rule 1)

3, 4 (Contrary to Rule 2 if the number four is more important than the number three, which is a matter of debate.)
2, 3, 2 (Contrary to Rule 2 in the order of the second two levels; contrary to Rule 3)
3, 2, 2 (Contrary to Rule 2; contrary to Rule 3)
6, 2 (Contrary to Rule 2; almost violates Rule 1)
10; (Violates Rule 1)

If the subdivisional pattern is repeated between successive powers of twelve, the order of the numbers varies with the magnification of the scale or the initial level, such that the permutations <2, 2, 3>, <2, 3, 2> and <3, 2, 2> are part of the same scale. These then are the four different types of subdivided dozenal fractal lattice scale:

<10;>
<4, 3>, <3, 4>
<2, 6>, <6, 2>
<2, 2, 3>, <2, 3, 2>, <3, 2, 2>

Those who wanted their ruler to look more like the subdivision of inches commonly done in a binary fashion preferred the binary divisions and the sequence 2, 2, 3.

For another application, the subdivision of the twelve enumerated positions of a clock face, if this is to resemble a conventional analogue clock, the five tick marks that are found per hour could be replaced by either six or four tick marks. Four tick marks are less cluttered. For division of the hour, six marks were more beneficial for the resulting time periods and "digital" notation written dozenally being closer to that of the conventional clock. For the design of other circular dials, it seems imperative to divide the whole turn into quarters as the cardinal directions before any other division.

Diagrams of the two competing proposals may be shown as follows,
for a vertical scale:

Code:


_____   _____
_       _
__      _
_       __
__      _
_       _
____    ____
_       _
__      _
_       __
__      _
_       _
_____   _____

or if you prefer, for a horizontal scale:

Code:


|||||||||||||   |||||||||||||
| | | | | | |   |  |  |  |  |
|     |     |   |     |     |
|           |   |           |

The scheme on the left looks more similar to the decimal scales, whereas that on the right resembles the typical British Imperial or American Customary inch scales. But the left scale is also similar to the imperial or customary weighing scales. Psychologically, the left proposal may be more attractive to users of the decimal metric.

References/See also:

• https://dozenal.forumotion.com/t36-probabilities-of-primes-and-composites
  
• https://dozenal.forumotion.com/t46-optimal-analogue-clock
  

The photograph of the steel ruler shown in the post at https://dozenal.forumotion.com/t48-twelfths-metric-ruler#156 has the lengths of the gradation marks for millimetres in a style that I might call "tapered". That is, the lengths of the gradations for the millimetres increase incrementally in approaching the half centimetre marks.

In a scale of gradations for a dozenal ruler, the analogous arrangement would be:

Code:


_______
_
__
___
____
_____
______
_____
____
___
__
_
_______


Variations on this concept can be designed placing greater emphasis of length on thirds and sixths than quarters of the main unit:

Code:


______
_
___
__
____
_
_____
_
____
__
___
_
______


or vice versa with more emphasis on quarters than thirds:

Code:


_______  
_
__
_____
____
___
______
___
____
_____
__
_
_______


or:

Code:


_______  
_
___
_____
____
__
______
__
____
_____
___
_
_______


The later is an attempt to make the lengths of the graduation marks representative of the relative importance of the division. The first I know of to propose a similar concept was Michael DeVlieger.

My favorite of the tapered kinds that I propose, for simplicity of the different numbers of lengths, consistency of the tapering approaching quarters, and rough ranking of the importance of the divisions, though without distinguishing between all of the levels of importance such as between thirds and sixths, is:

Code:


_____
_
__
___
__
_
____
_
__
___
__
_
_____


The greater length for quarters represents their greater importance in metrology or measurement rather than greater frequency of the number four than the number three in random numbers.
In horizontal scales, these proposals are:

Code:


|||||||||||||   |||||||||||||   |||||||||||||   |||||||||||||   |||||||||||||
| ||||||||| |   | ||| | ||| |   | ||||||||| |   | ||||||||| |   | ||| | ||| |
|  |||||||  |   | | | | | | |   |  |||||||  |   | ||| | ||| |   |  |  |  |  |
|   |||||   |   |   | | |   |   |  || | ||  |   |  || | ||  |   |     |     |
|    |||    |   |     |     |   |  |  |  |  |   |  |  |  |  |   |           |
|     |     |   |           |   |     |     |   |     |     |                
|           |                   |           |   |           |                


References/See also:
https://dozenal.forumotion.com/t36-probabilities-of-primes-and-composites
https://www.tapatalk.com/groups/dozensonline/marking-a-dozenal-ruler-t1009.html

It is preferable for the graduation marks to have a lesser number of different kinds of lengths so that the tick marks for the next power of the base in subdivision will not become too short. The tapered design for base twelve may be simplified by making the marks for the halves be the same length as for the quarters. This produces the following scheme:

Code:


____
_
__
___
__
_
___
_
__
___
__
_
____


Code:


|||||||||||||
| ||| | ||| |
|  |  |  |  |
|          |

After the major units, this pattern promotes quarters as the first place for rounding, yet encourages the importance of sixths including thirds for the next level of rounding. I think this proposal has the best balance of simplicity regarding the numbers of different lengths and prominance to the fractions by their degrees of importance in metrology. A simpler proposal of dividing by quarters and then the quarters each into their thirds does not distinguish between the importance of sixths versus the remaining twelfths. My proposal therefore is a good compromise between those who preferred dividing first into quarters and then those quarters into their thirds versus those who preferred variations of dividing first into sixths followed by bisection of those sixths, as compared in the opening post of this topic. Division into sixths is desirable where the full resolution to twelfths is too fine to be practically executed or read.

References:
https://www.tapatalk.com/groups/dozensonline/going-to-great-lengths-for-a-measured-question-as--t1889.html
https://www.tapatalk.com/groups/dozensonline/did-a-thing-t2232.html
https://www.tapatalk.com/groups/dozensonline/a-dozenal-decimal-ruler-t2254.html
https://dozenal.forumotion.com/t13-ripples-and-awayness#36

On 1st September this year ①②⓪⑦, a topic was started on the DozensOnline forum about making dozenal rulers. In that forum, on ②⓪^th October, Paul Rapoport suggested varying the lengths of the graduation marks in a tapered style in emulation of a photographed sample of binary divisions of inches. Taking these recommendations into account, on ②①^st October, the original poster uploaded a picture of three proposals for division of length units with tapered lengths of graduations. The first of these combined an ordinary scale of quarters subdivided into their thirds parallel to a scale of only thirds and sixths with thirds longer than sixths. The second proposal was essentially the same as the one used by Michael DeVlieger that I cited on ⑩^th January this year ①②⓪⑦. That design puts greater emphasis on thirds than on quarters, and is more complicated in the number of different lengths than my proposal. The third proposal is essentially the same as the one that I proposed and preferred in my previous post here from January this year, which puts greater emphasis on quarters than on thirds. The original poster preferred the more complicated design of the parallel scales where one scale was tapered and later referred to this as an "advanced" version in images posted on ②③^rd November.

This is yet another case of an idea appearing on DozensOnline not long after it was proposed by me.

References
https://www.tapatalk.com/groups/dozensonline/physical-tools-t2410.html