Shake 1: Women give birth on time according to Benford`s Law.

Adjusted for seasonality and caesarean, births are usually considered uniformly distributed over days, weeks, months and years.

If we consider adjusted* births on day-numbers 1-31, significant deviations from uniform or equal distribution occur. Why is it so?

Benford`s Law (First Digit Law) is known to rule natural processes. That means 1 is the most frequently observed first digit and frequencies for 2-9 are falling exponentially.

For the case of this site, Benford`s Law will - not suprisingly, apply for the number of births in periodes or countries. But the power law also applies for the (rank of the) amount of births on numbers 1-31 of the day (in month, for every months every year). Well; you might say one singel day is a periode as well - and it is of course. But why do such periodes attract more births the lower the (reduced) day-number is?

Benford`s Law can be seen from the upper figure, where the algebra operation of reducing double digits of a circle group with day-numbers 1-27 to single digits and root values 1-9 (days 28-31 are left out to have equal amount of single digit generators).

Considering the start values of the charts Y-axes, births are distributed according to Benford`s Law and have uniform distribution as asymptotic limit.

It is something plastic with Benford`s Law. In fact it corresponds with the smallest real plastic number, which is the golden ratio - a point of accumulation.

Numerical precision is the very soul of science.

D'Arcy Wentworth Thompson

More on Benford`s Law - 1) A canonical ensemble keeping the world a constant size like how you see it, a grassmannian or an amplituhedron?

The professors Arno Berger and Theodore P. Hill published in May 2015 the first book on Benford`s Law. They describe the law as a surprising statistical phenomenon. This is a common opinion.

More special, Berger and Hill correct Feller`s one-liner sought to grab the essence of Benford`s Law. Obviously, it has not been easy to grab the essence or pin-point the complete general principle for Benford`s Law. Many researchers end up with the same impression as Berger and Hill in the correction of Feller:

Although many facets of BL now rest on solid ground, there is currently no unified approach that simultaneously explains its appearance in dynamical systems, number theory, statistics, and real-world data. In that sense, most experts seem to agree with that the ubiquity of BL, especially in real-life data, remains mysterious.

People tends to look at what they can see. We can easily see that magnitude of order 10..100...1.000.... favours first digit 1 in counting systems. But what is an order? The originator of the law Simon Newcomb said in “Note on the Frequency of Use of the Different Digits in Natural Numbers” (American Journal of Mathematics, 4, 1881, 39-40.)

As natural numbers occur in nature, they are to be considered as the ratios of quantities. Therefore, instead of selecting a number at random, we must select two numbers, and inquire what is the probability that the first significant digit of their ratio is the digit n.

Every number is an order in relation to another. Further in the note, we see how Newcomb nearly describes the algebra with the operation of reducing compund digits of the circle group with infinte numbers. As n increases, the distribution stabilizes (look at the birthday example as an urn). Researchers talk about better and best fit to Benford`s Law (obtained when combining all examples). I think this is a misfitting of the law. The essence of the law can rule equally strong for different slopes of distribution - just expressing different nature or qualities. The "best fit" to Benford`s Law is a point of accumulation, the golden ratio. You might take a look at the Sierpinski carpet now.

More on Benford`s Law - 2) Qualities and optimal packings

This article on place cells is interesting regarding qualitiy and optimal packing:

"Probable nature of higher-dimensional symmetries underlying mammalian

grid-cell activity patterns

Alexander Mathis, Martin B Stemmler, Andreas V M Herz

Lattices abound in nature - from the crystal structure of minerals to the honey-comb organization of ommatidia in the compound eye of insects. These arrangements provide solutions for optimal packings, efficient resource distribution and cryptographic protocols. Do lattices also play a role in how the brain represents information?

We focus on higher-dimensional stimulus domains, with particular emphasis on neural representations of physical space, and derive which neuronal lattice codes maximize spatial resolution. For mammals navigating on a surface, we show that the hexagonal activity patterns of grid cells are optimal. For species that move freely in a 3D a face-centered cubic lattice is best.

This prediction could be tested experimentally in flying bats, arboreal monkeys, or marine mammals. More generally, our theory suggests that the brain encodes higher-dimensional sensory or cognitive variables with populations of grid-cell-like neurons whose activity patterns exhibit lattice structures at multiple, nested scales."

Numbers 1-9 are mathematical canonical examples. But why do women have to follow them - as a circadian rythm or ecorithm?

Modulo 9 (reduced number) is the modulo with zero error and perfect symmetry, yielding the golden ratio and the easiest case for solving Hilbert`s 12th problem.

On

Richard Taylor ask if Babylonians knew these qualities.

Babylonian treated the ratio of two numbers (see Newcomb below) - like the length ratio of two harp strings - as an entity. “A very important step in the development of algebra” according to Otto E Neugebauer.

* Weekday effects are seasonally adjusted by trailing the weekdays on the day-numbers many more rounds than 7 when the time periode is say 50 years (one trailing round is seven years). This also adjust for caesarean births mostly planned to working days. Notice that results are consistent between different subsets of data (countries, western and non-western immigrants and time periodes ranging from each months within a year, recently years and 100 years ago). See Go safer on the top menu.

Numerical precision

How do you do?

A general, basic and theorized answere is

according to the circadian rythm to the right.

A day itself - and when in it, matter for you.

How do women do?

A general, basic and theorized answere is

according to Benford`s Law in the figure above.

But, are the answeres due to number itself or are

they rooted in earth rotation and the approx

number of moon passages? We clearly can say

yes to earth rotation.

A month is linked to the moon. It could be

because it is approximately suitable.

But why does a month itself matter for women?

Base-10 number system with modulo 9 fits nature best and

so good that it can be used as a model to induce how nature is.

In Go Safer on the top menu, you can even see significant gender differences in numbers.

A circle-group based algebra in base-10 seem to be a circadian model on all scales, since it tackles skewness

to the same degree as it is by growth most commonly (golden mean) seen in nature.

--------------------------------------------------------------------------------------------------------------------------------------------------------------

May 2015

*Source: Wikipedia*