Assume that a circle can be a disc, and that disc can be cut into sectors by the radius of the circle striking the arc of the disc.
Empirically we find that this appears to be possible 6 times, give or take a few minor sectors. Pragmatically let us define this process as sectoring the circular disc into " precisely" 6 sectors. For the Euclidesn classicists we have here the intended meaning of artios, that is precisely, not Evenly!
Empirically we might more accurately say approximately, and thus once again reveal the intended meaning of perisos, that is approximately not Oddly!
It is thus a formal act that makes a disc eternally divisible into 6 equal sectors.
Let us call the line equal to the radius displacement in length, and also as straight as the diameter line in which it proceeds from the centre to the perimeter, a chord when it strikes off a sector. In fact any straight line that strikes off any sector of a disc at the perimeter of the disc we may call a chord, (c).
Now , rather than just count the sectors let us relate each sector to a square or circular Monas or unit. Such a unit will be used to count off the Arithmos of a specific disc.
Thus for any disc, the radius chord always cuts the disc into 6 monads
Can I find a commensurable monad that counts off any general sector in comparison with another?
That is to say, given any 2 arbitrary sectors can I find a precise monad that counts them both off in whole counts? This is the classical issue of commenurability, not as we like to say " precision", except and unless we Meehan by that precisely artios.
It seems reasonable to employ the device that the count of monads( circular or square) of the sector is expressible by the count of the "radius monads",r, multiplied by the count of the "chord monads", c, and then " halved". Here the radius or chord monad is obtained by placing the square monad on the radius or chord, or equally the diameter of the circular monad on the radius or chord of the disc sector.
As the monads chosen become finer it is clear that the count becomes larger , but also the perisos monads become smaller and we approach commenurability .
Can we achieve it?
No!
The exposition is simple.
Expressing the count as rc/2 we see that for a c(hord) count that is equal to the diameter count, D, rD is the expected count for the disc.
But if I construct rD as a rectangle it is clear it is too small, coverin only a semi circle. The curved Shunyasutras outside the semi circle, can be brought within revealing 2 petal shaped areas that are not accounted for.
These areas sit within a quarter disc each, so it makes sense to again apply( iterate) the formula rc/2 but this time to use as the chord count r
)
. This comes from Pythagoras theorem, and is immediately declareable as incommensurabie.
However let us make the observation that our formula is biased toward giving too small a result.in this correction the count is clearly biased toward being too big, containing more of the circle than necessary. it is therefore reasonable to stop the iteration at this point because it is incommensurable, but also to expect a pragmatic level of resolution of the count. This means that whatever monad size we use we can expect the count to be factoriseable between the 2 arbitrary sectors by approximately this amount.
I will leave it for the interested reader to declare the result of this Fractal factorisation .
• My apologies for the Snootiness in the style. This is how we mathematicians are trained to speak or write formally. It reveals an endemic arrogance which actually is not justified or justifiable.
As agent Smith ( the Matrix gatekerper) correctly pointed out we mathematicians are viruses. We enter into the realm of the natural philosopher and completely destroy it by replicating ourselves and our own arrogances!
October 15, 2006, 09:15:13 PM
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