Dear Karl,
Yes I can hear you.
About symmetry, I shall soon send you an explaining email, privately,
because I do not want to bother the FISers with long explanations
(unless I am required to do it).
However, I confess that many posts that I receive from the FIS list
are very hard to read, and often I do not understand their deep
content :)
In fact, that should not be shocking: few people are able to read
texts from very diverse fields (as it occurs in the FIS forum), and I
am not one of them.
Even the post of Sung was unclear for me, and it is exactly why I
asked him questions, but only on the points that I may have a chance
to understand (may be).
Best regards,
Michel.
2018-05-07 17:55 GMT+02:00 Karl Javorszky <karl.javors...@gmail.com>:
> Dear Michel and Sung,
>
> Your discussion is way above my head in the jargon and background knowledge.
> Please bear with me while a non-mathematician tries to express some
> observations that regard symmetry.
>
> Two almost symmetrical spaces appear as Gestalts, expressed by numbers, if
> one orders and reorders the expression a+b=c. One uses natural numbers – in
> the range of 1..16 – to create a demo collection, which one then sorts and
> re-sorts ad libitum / ad nauseam. The setup of the whole exercise does not
> take longer than 1, max 2 hours. Then one can observe patterns.
>
> The patterns here specifically referred to are two – almost – symmetrical
> rectangular, orthogonal spaces. As these patterns are derived from simple
> sorting operations on natural numbers, one can well argue that they represent
> fundamental pictures.
>
> The generating algorithm is 5 lines of code. Here it is.
>
> #d=16
>
> begin outer loop, i:1,d
> begin inner loop, j:i,d
> append new record
> write
> a=i, b=j, c=a+b, k=b-2a, u=b-a, t=2b-3a,
> q=a-2b, s=(d+1)-(a+b), w=2a-3b
> end inner loop
> end outer loop
>
> The next step is to sequence (sort, order) the rows. We use 2 sorting
> criteria: as first, any one of {a,b,c,k,u,t,q,s,w}, and as 2nd sorting
> criterium any of the remaining 8. This makes each of the 9 aspects of a+b=c
> to be once a first, and once a second sorting key. We register the linear
> sequential number of each element in a column for each of the 72 catalogued
> sorting orders..
>
> Do you think the idea of symmetry is somehow connected to some very basic
> truths of logic? Then maybe the small effort to create a database with 136
> rows and 9+72 columns is possible.
>
> The trick begins with the next step:
>
> We go through the 72 sorting orders and re-sort from each of them into all
> and each of the remaining 71. We register the sequential place of the element
> in the order αβ while being resorted into order γδ. This gives each element a
> value (a linear place, 1..136) “from” and a value “to”. The element is given
> the attributes: Element: a,b, “Old Order”: αβ, from place nr i, “New Order”
> γδ, to place nr. j. While doing this, one will realise, that reorganisations
> happen by means of cycles, and will add attributes :
> Cycle nr: k, Within cycle step nr:. l. This is simple counting and using
> logical flags.
>
> The cycles, that we have now arrived at, give a very useful skeleton for any
> and all theories about order. You will find the two Euclid-type spaces by
> filtering out those reorganisations that consist of 46 cycles, of which 45
> have 3 elements in their corpus, where each of the 45 cycles has Σa=18, Σb=33.
>
> The two rectangular spaces – created by paths of elements during resorting –
> are not quite symmetrical. As an outsider, I’d believe that there is
> something to awake the natural curiosity of mathematicians.
>
> Hoping to have caught your interest.
>
> Karl
>
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