6. Chains


Reminder: Step 5:

Those sorting orders, which to each addition assign identical sequential numbers, build a common ‘clan’ together. In the version of the table presented here, 20 clans are visible. Members of a clan can differ on their number of teeth. The place of an element within a tie is not quite indeterminable but is rather dependent on how finely the preceding sorting order was sorted.

Step 6:


After having eliminated the easy cases, where no reordering is needed, we now confront the mechanics of transforming the sequence αβ into the sequence γδ. (V[αβ, γδ]=.f.) This procedure is called ‘reordering’ and as its effect, an element j that previously had the sequential place p1 has now the sequential place p2. (p1 {=|≠} p2)

Of specific interest is the case, wherein during a reorder, several elements have to move together in the course of a reordering. This is the main concept of the Twelve Steps.

Explaining the main concept:

The central concept can be highlighted using deictic methods. Before doing so, let us try to clarify the idea in colloquial speech. One knows from everyday life that a change of places may be an intricate process, as oftentimes person 1 has to first vacate the place where person 2 will come to sit. Person 1 must ask person 3  to liberate his place in turn, and so on. This can get quite complicated, but normally people don’t give it much thought.

Rubik’s cube shows a specific instance of the central concept at work. (Please see 6.graph. Rubik). The concept presented here is similar to that made visible in the cube. Six planes are given and 24 elements move while 6 are fixed; the task is to deduct from the known results of the planes, the collection and the sequence of the repetitive procedures (operations) that will result in the goal being achieved. Here, we have 136 elements, none of them fixed and the number of planes can be a subject of a spirited debate. The task here is to deduct the appearance of the planes after having gone through all repetitive procedures (operations). In the case of the cube, the pleasure of having solved the puzzle comes as soon as one understands the procedures that result in planes. Here, success comes from having understood the planes that result from the applications of the procedures.

In logistics one would speak of ‘merchandise in transit,’ where one will use effective and expected matches between material and spatial references.

Names for the main concept:

Wittgenstein calls the idea discussed in the present context a Sachverhalt (please see in the Tractatus here around 2.01). It is acceptable to interpret that the Sachverhalt is ‘amount is on place p, while the Zusammenhang is that ‘j moves together within {j,j’,j’’,...} during a reorder from αβ into γδ.’

Heraclit has predicted the dynamic interdependence among realisations of order without giving a specific name to it. He points out the ‘upward-downward path’ (http://en.wikipedia.org/wiki/Heraclitus#Panta_rhei.2C_.22everything_flows.22).

Here, we may make use of the connotations of a convoy, chain, string, filament or rhythm for the Zusammenhang and step, or tact, for the Sachverhalt.


The deictic definition is done by presenting a fragment of table T (please see 6.num). Each line in table T is one step in the process of reordering. We publish – next to the permutation of the first-level arguments that were used at the creation of the table – order αβ (the previous order), order γδ (the new order), arguments (a,b), the sequential number for the Zusammenhang i, the step number j within string number [tact j in rhythm i], and sequential place previous (from) and sequential place new (to).


There is quite a wide range of properties for the chains. Interactive Figure 6.graph.1 shows the properties of convoys within a reorder to discuss how many strings are necessary for a reorder and how the number of steps in the chain (tacts of the rhythm) are distributed; 6.graph.2, in the form of a circle, shows which places are connected into a string during a reorder;

One may select and deselect specific chains to see the additions that are connected by a Zusammenhang in the form of points in the plane built by the axes X: αβ, Y: γδ (6.graph.3).

We keep track of the ‘carry’ in the form of ‘carry_a’ and ‘carry_b’ (not shown here). The carry is the sum of the argument over all j in chain i.

In the next step we shall propose chains to be used as standard units. In steps eight and nine we build spaces with rectangular axes by using properties of the standard chains.

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