Hi Quantum Quack, handsa, Tach, rpenner, everyone.

This particular (immediately-below-quoted) exchange demonstrates where the problem may lay when trying to bring the discussion into the real physical world from the seemingly 'uncaring' mathematical assumptive world where the actual meanings of things like "point" and 'number 0" are unwittingly treated as a 'given' even though there is a fundamental question hanging over the competency of the current/conventional 'mathematical axiomatic' system to deal with these things at present stage of the mathematics formulation/practice:

Tach to handsa said:

handsa said:

You can divide a circle into three equal parts. Draw three radius from the center of a circle, which are 120 degrees apart. This will divide the circle into three equal parts.

This is one way of doing it, not the most elegant. How would you do it with a ruler and a compass only?

rpenner to someguy1 said:

someguy1 said:

That's harder than it looks. How do you account for each of the points on the boundary?

Those points have measure zero, so they really don't count, but a simple procedure of assigning each point to it's clockwise-adjacent arc will split the circle into N congruent parts.

Tach to rpenner said:

rpenner said:

Those points have measure zero, so they really don't count, but a simple procedure of assigning each point to it's clockwise-adjacent arc will split the circle into N congruent parts.

"Sharing" the endpoints by congruent arcs accomplishes the same result.

That incompetency is highlighted whenever (in the usual exchanges like that quoted above), there are effectively UNEXPLAINED assumptions made about the "0" and the "point" as for that 'proof of division into equal parts' in the case of a real 'pie' and not just an abstract 'circumference line for a circle. Now...

**Consider the reality requirement for division of MD's real (not abstract) PIE; namely:**
In the real world example of MD's PIE having a certain AREA in reality, the division has to INCLUDE the 'partition sector' VALUES so that they add together to the WHOLE starting area value.

**Consider that such 'division' cannot be accomplished in reality, because:**
There is no way in mathematics/reality to treat the CENTRE POINT which in handsa's 'method' is the "origin" point FOR the 'process' of division he suggested would do the trick.

**Consider further the other 'methods' suggested earlier:**
These too are EFFECTIVELY 'mute' on such matters as to what to do with the CENTRAL point (not circumferential points) in an AREA of points constituting a REAL pie DISC area (and not just the abstract notion of a 'circumference' LINE of POINTS isolation from that line's enclosed AREA which has to be SECTOR partitioned...in which case, as rpenner just pointed out...

rpenner said:

Those points have measure zero, so they really don't count,...

So, guys, this above observation, together with the earlier observations of such problematic "0", "superposition" and "origin points" as applies in the CIRCUMFERENTIAL ONLY context of MODULO MATHS in the 12-hour Clock context (please see my post to rpenner #146), CLEARLY demonstrates that conventional maths axiomatic treatments are not fully competent to EXPLAIN and consistently treat such "SINGULAR" cases where "0" and "point" is effectively associated with/constitutes NOT a "UNIQUE NUMBER" ON the number line, BUT SOMETHING ELSE entirely having contextual meanings/properties which BRIDGE ACROSS the limited treatments/domains of applicability of the current matyhs formalism as axiomatically constructed TO DATE.

Hence the need for further contextual axiomatic EXRENSION/REFINEMENT to ENHANCE the competency of the mathematics to handle such things that are BOTH/MANY 'things/values/states' SIMULTANEOUSLY as they REPRESENT TRANSITION states/points etc, and NOT just 'a number'. Hence the need for something like an INFINITESIMAL entity which can be associated with these states/transitions/values/superposition 'points' of CENTRE, ORIGIN, SINGULARITY characteristics/properties/effect IN REALITY and the MATHS both!

Then we may be able to obviate these constant Maths-versus-Reality cross-purpose arguments brought about by the current incomplete contextual capability of maths/physics systems/models to reflect/treat these things without 'outputting' such meaningless/unsatisfactory INCOMPLETE results as "UNDEFINED", or "INFINITY" or "LIMITED TO CHOSEN BOUNDARY CONDITIONS" etc.

Again, everyone, please read this post in conjunction with my earlier post to rpenner (

**#146**) and consider carefully the subtle but important points I raised supported by examples/observations already familiar to us all by now. Please don't kneejerk; and do please read carefully, and without any emotional reaction at what you THINK you 'see' "at first blush".

**No rush, guys!**
I can wait until tomorrow or later for your replies. No sweat! So please maybe read this post (together with post #146) a couple times, and then sleep on it overnight and read it again. Then make your calmly considered and unemotional comments/responses. Thanks.