Going in Circles – Part II

In Going in Circles – Part I we presented the basic reasoning underlying Q-Orders by analyzing the configurations of elements on 1 Circle and the combination of 2 circles. Already in the discussion of Axiom 4 we said informally that the combination of circles should create new ones and specified which would be permitted based on pictures.

Yet it would be a fruitless and hence useless task, to continue that way, since very rapidly we would be defeated by combinatorial explosion, as 4 elements on 1 circle already allow 3 configurations according to Axiom 1 and Lemma A 1. We need urgently rules of interference! For more than 4 elements on 1 Circle, Lemma A 2 solved the problem, as shown by Lemma A 5.

Yet for the combination of more than 2 circles –at least 3^3 = 27 possibilities- we have no rules yet and analyzing even in this case the configurations one by one more would be very clumsy, and literally impossible for only countable many elements.

A part of the problem is solved by Axiom 3.

Axiom 3  Q-Regular

It claims that

1. whenever all pairs of 3 points can be found on some circle –regardless their configurations there-, there should be a 4th element to give a complete circle, where all 4 may be positioned.
2. whenever all triplets of 4 points have their circle, the 4 points themselves may be positioned on 1 circle.

As this is a global claim, Axiom 3 is truly an Axiom i.e. a Thinking – Hypothesis that may be false in the real world (as the Parallel Axiom in non-Euclidian Geometries). Actually I’ve done some investigation on the consequences, if it were false in the frame work of Q-Orders. Due to its hypothetical character we will flag further on any proof that uses it.

Sorry, no pictures for Axiom 3! (it would have been 91 for 3.1 and 271 for 3.2) …. but whoever has the time and resources to draw them up all, including the configuration for the resulting circle, is kindly invited to provide them, or even better a small animation applet …).

Axiom 4.3 expresses a second Thinking Hypothesis as a rule of interference to allow later on for consistent orientation.

AXIOM 4.3  Q-Order

It claims that whenever a pair {c,d} is separated by another pair {a,b}, and this pair appears on the same side of some circle, then {a,b} separate {c,d} also on this circle. Or more loosely, if {a,b} separates {c,d}, then it does so anywhere or even more general: {{a,b},{c,d}} is a universal relation.

It can not be stressed enough that Axiom 4.3 is a Thinking Hypothesis, that may be false. … and has far reaching consequences as it’s implied by the concept of transitivity of partial orders and likewise by the concept of continuous functions in Real Analysis.

The Partial Order on Circles as defined by Lemma A 5 provides the arguments to include Axiom 4.3. I’ll tried to prove it’s independence from the other Axiom (1,2,4, 4.1, 4.2) but did not come up yet with a counter-example nor could I prove it’s formal deduction (May be some reader has more cleverness).

Then Lemma 7 shows that a pair {u,v} not only separates all circles that run trough {u,v} but cuts them into two equivalence classes of points- say inner-outer, left-right, upper-lower - without defining which is which.

Lemma A 7

Q-Orders sometimes have adjacent points:
Lemma A 8 Adjacent Points

Adjacent are those points which can not be separated by Q. It’s a local property, i.e. something that can’t be separated in one interval, can’t be separated in any other.

With the above, we are done with all local Axioms, remaining only the Axiom 5 Oriented that will generalize our findings.