Asymmetry Principle of Measurement

As well-known each original scientific theory is based on some deep idea, some principle determined originality and novelty of scientific theory. For example in the base of non-Euclidean geometry the new interpretation of the Euclidean axiom of parallelism underlies, Einstein's theory of relativity uses the "Principle of the light speed constancy" and so on.

To get non-trivial results in the new measurement theory we should try to formulate some fundamental measurement principle, which follows from the same essence of measurement. We mentioned repeatedly the "weighing problem" formulated Fibonacci in 13th century. "But what fundamental principle we can "extract" from this oldest problem?" - you ask. Do not hurry to answer! Let's look carefully the weighing procedure by using the balance (Fig. 1).

When we considered the classical Fibonacci "weighing problem" we emphasized that the "binary" set of the standard weights {1, 2, 4, 8, ..., 2n-1} is the "optimal" solution. And we should consider the "binary" measurement algorithm for weighing by using the "binary" weights {1, 2, 4, 8, ..., 2n-1}. The essence of the "binary" algorithm consists of consecutive adding the "binary" weights to the "free" cup of the balance starting since the biggest one.

Asymmetry principle of measurement
Figure 1. Asymmetry principle of measurement.

The analysis of the above mentioned "binary" algorithm by using the balance model (Fig.1) allows revealing one measurement property of a general character for all-thinkable measurings reduced to the comparison of the measurable value X with some standard weights.

Let's consider very carefully the process of weighing the load Q on the balance by using some binary standard weights. At the first step of the "binary" algorithm the biggest standard weight 2n-1 is put on the free cup of the balance (Fig. 1-a). In so doing the cases 2n-1 < Q (Fig. 1-a) and 2n-1 ³ Q (Fig. 1-b) can result. In the former case (Fig. 1-a) the second step is adding the next standard weight 2n-2 to the free cup of the balance. In the latter case the "weigher" has to perform two operations, i.e. to remove the previous standard weight 2n-1 from the free cup of the balance (Fig. 1-b) so that the balance should return to the initial position (Fig. 1-c). Then the next standard weight 2n-2 is put on the free cup of the balance (Fig. 1-c).

One can readily see that the both considered cases differ in their "complexity". In fact, in the former case the "weigher" fulfils only one operation, i.e. he adds the next standard weight 2n-2 to the free cup of the balance. In the latter case the "weigher's" actions are determined by two factors. First he has to remove the previous standard weight 2n-1 from the free cup of the balance and then to take into consideration the time spent to return the balance to the initial position.

The discovered property of measurement was called the "Asymmetry Principle of Measurement".

Let's introduce now the above-discovered property into the "weighing" problem suggested by Fibonacci. With this in mind let's consider the measurement as the process running during discrete periods of time; let the operation "to add the standard weight" be performed within one unit of the discrete time and the operation "to remove the standard weight" (which is followed by the returning the balance to the initial position) be performed within p units of the discrete time with p Î {0, 1, 2, 3, ...}.

It is clear that the numerical parameter p simulates the "inertness" of the balance. As this takes place the case p = 0 corresponds to the "ideal situation" when we neglect the "inertness" of the balance. This case corresponds to the classical "weighing problem". For other cases of p > 0 we have some new variants of "weighing problem".

One may show up incredible that this simple observation on the process of weighing can become the basis for development of the new mathematical measurement theory. But it is really so! And we will show in the next pages of our Museum what interesting measurement algorithms and new scientific results follow from the "Asymmetry Principle of Measurement". Follow us!