Alejandro
Oct25-04, 08:09 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\n\nFunny enough, it seems that HdV formulas could hide spinors after all.\nIn a third set of guesses, Hans de Vries has pointed out another\nmnemotechnic for leptons. But this one asks for the Pythagorean\nrelationship, which as we know is the hallmark of spinors. They should\nbe Pythagorean Spinors ( http://math.ucr.edu/home/baez/week196.html ),\nbecause they are integers.\n\nPoint is, consider the only two triples having consecutive integers,\nthis is, (3,4,5) and (-1,0,1). Given any three numbers a+b=c, we can\nuse them to build a unit-independent relation between quantities x, y\n,z:\na ln x + b ln y = c ln z\nie, if we change units x --> k x, y -->k y, z--> k z, the relation\nstill holds.\n\nNow we used both triples, in a not very straightforward way, to build\none relation of this kind:\n\n(3^2+0^2) ln x + (4^2 + (-1)^2) ln y = (5^2 + 1^2) ln z\n\nWhich holds very fairly when [x,y,z] = mass of [electron, tau, muon]\nrespectively.\n\nWith a bit of algebraic manipulation the equation can be written\n\nln z/y\n--------- = 3^2 / (4^2+1)\nln x/z\n\nand then we can compare it with the same quotient from the first set\nof HdV guesses:\n\nln z/y pi^2 -1\n-------- = --------------\nln x/z pi^2 -3\n\nOn a first view, it seems a failure. But noticing that pi^2 can be\nexpanded as one sixth of the sum of inverse squares (Zeta(2), so to\nsay), we see that the first term of this expansion is not far away\nfrom the one in the new guess. In fact they coincide if we "correct" a\nbit the formula to be expanded:\n\nln z/y pi^2 -1 -1/2\n-------- \\approx ---------------------\nln x/z pi^2 -3 -1/2\n\nwhich makes oneself to feel more motivated about the existence of a\nmeaning for this family of approximations.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Funny enough, it seems that HdV formulas could hide spinors after all.
In a third set of guesses, Hans de Vries has pointed out another
mnemotechnic for leptons. But this one asks for the Pythagorean
relationship, which as we know is the hallmark of spinors. They should
be Pythagorean Spinors ( http://math.ucr.edu/home/baez/week196.html ),
because they are integers.
Point is, consider the only two triples having consecutive integers,
this is, (3,4,5) and (-1,0,1). Given any three numbers a+b=c, we can
use them to build a unit-independent relation between quantities x, y
,z:
a ln x + b ln y = c ln z
ie, if we change units x --> k x, y -->k y, z--> k z, the relation
still holds.
Now we used both triples, in a not very straightforward way, to build
one relation of this kind:
(3^2+0^2) ln x + (4^2 + (-1)^2)[/itex] ln y = (5^2 + 1^2) ln z
Which holds very fairly when [x,y,z] = mass of [electron, \tau, muon]
respectively.
With a bit of algebraic manipulation the equation can be written
ln z/y
--------- = 3^2 / (4^2+1)ln x/z
and then we can compare it with the same quotient from the first set
of HdV guesses:
ln z/y \pi^2 -1
-------- = --------------
ln x/z \pi^2 -3
On a first view, it seems a failure. But noticing that \pi^2 can be
expanded as one sixth of the sum of inverse squares (\Zeta(2), so to
say), we see that the first term of this expansion is not far away
from the one in the new guess. In fact they coincide if we "correct" a
bit the formula to be expanded:
ln z/y \pi^2 -1 -1/2
-------- \approx ---------------------
[itex]ln x/z \pi^2 -3 -1/2
which makes oneself to feel more motivated about the existence of a
meaning for this family of approximations.
In a third set of guesses, Hans de Vries has pointed out another
mnemotechnic for leptons. But this one asks for the Pythagorean
relationship, which as we know is the hallmark of spinors. They should
be Pythagorean Spinors ( http://math.ucr.edu/home/baez/week196.html ),
because they are integers.
Point is, consider the only two triples having consecutive integers,
this is, (3,4,5) and (-1,0,1). Given any three numbers a+b=c, we can
use them to build a unit-independent relation between quantities x, y
,z:
a ln x + b ln y = c ln z
ie, if we change units x --> k x, y -->k y, z--> k z, the relation
still holds.
Now we used both triples, in a not very straightforward way, to build
one relation of this kind:
(3^2+0^2) ln x + (4^2 + (-1)^2)[/itex] ln y = (5^2 + 1^2) ln z
Which holds very fairly when [x,y,z] = mass of [electron, \tau, muon]
respectively.
With a bit of algebraic manipulation the equation can be written
ln z/y
--------- = 3^2 / (4^2+1)ln x/z
and then we can compare it with the same quotient from the first set
of HdV guesses:
ln z/y \pi^2 -1
-------- = --------------
ln x/z \pi^2 -3
On a first view, it seems a failure. But noticing that \pi^2 can be
expanded as one sixth of the sum of inverse squares (\Zeta(2), so to
say), we see that the first term of this expansion is not far away
from the one in the new guess. In fact they coincide if we "correct" a
bit the formula to be expanded:
ln z/y \pi^2 -1 -1/2
-------- \approx ---------------------
[itex]ln x/z \pi^2 -3 -1/2
which makes oneself to feel more motivated about the existence of a
meaning for this family of approximations.