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Graduation Subdivisions


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Graduation Subdivisions Empty Graduation Subdivisions

Post by Phaethon Wed Jun 15, 2022 3:07 pm

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:

<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:

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

or if you prefer, for a horizontal scale:


|||||||||||||   |||||||||||||
| | | | | | |   |  |  |  |  |
|     |     |   |     |     |
|           |   |           |
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:
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