At 08:35 PM 2/9/2000 -0500, <[EMAIL PROTECTED]> wrote:
>It would take infinite area of an infinitesimally thin layer of paint, which
>would have no volume due to its thinness. Since paint can't be infinitely
>thin,
>this also means you can't actually fill the object with paint, because there
>will be volume in areas into which paint molecules can't fit.
>
>Mike
Filling the horn with paint has a couple additional problems:
since it is an infinitely long capillary, filling time would be infinity^4
or so (laminar flow conductance being proportional to diameter^4
and inversely proportional to length)
A realizable section of Gabriel's horn would necessarily be lumpy
when constructed of real material. Think of a tube constructed
of soccer balls glued together. If the horn inner diameter is a kilometer,
great, it looks pretty smooth. (Say for the sake of argument the
diameter of these soccer balls is 3 decimeters.)
But further along, where the inner
diameter has fallen off to one meter, it's beginning to look pretty
lumpy already, and when inner diameter drops to 1 decimeter,
the tube roughness is very significant.
Now move out to where the inner diameter is 1 Angstrom,
and the atoms of which the wall is constructed are 3 Angstroms
diameter, and it looks the same.
I'm surprised noone responded about continued fractions to
Ian Halliday:
At 10:42 PM 2/9/2000 +1300, [EMAIL PROTECTED] wrote:
>Over history, there have been numerous other approximations to the value
>of pi. Our current culture seems to favour 22/7 as an approximation, and
>the Biblical approximation above suggests 333/106. However, this is not
>the best available in three digits, which is, so far as I know, 355/113,
>which is correct to an astonishing one part in ten million. I understand
>that in certain quarters, 3 1/7 was not in vogue, with 3 1/8 favoured.
>What, argued these particular mystics, could be a better number than
>five squared shared by two cubed? N P Smith asked whether we should be
>more concerned by those who serious propose answers which are clearly
>wrong or by those who spend time in repeatedly refuting these spurious
>claims.
PI~=3.1415926535897932384626433832795
subtract the integer part, take the reciprocal of the rest, and iterate, to
produce the continued fraction's coefficients.
Reassemble successively increasing numbers of terms,
until the rational number obtained is sufficiently accurate.
This is an effective method of determining gear ratios approximating
arbitrary reals.
3+ 1 / (7 +
1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/(2+1/(1+1/(3+1/(1+1/(14+1/(2+1/(1+...))))
)))))))
3= 3
4= 3 +1
3.14 2857142857... =3+1/7 = 22/7
3.1 25 = 3+1/(7+1) = 25/8
3.1415 09433962264150943396226415... =3+1/(7+1/15) = 3 + 15/106 = 333/106
3+1/(7+1/(15+1/)) =355/113, see below
3.141592 9203539823008849557522124... =3+1/(7+1/(15+1/1)) = 3 + 1/(7+1/16)=
3+1/(113/16) = 3+ 16/113 = 355/113
3+1/(7+1/(15+1/(1+1)))= 3.1415 525114155251141552511415525
3+1/(7+1/(15+1/(1+1/292))) = 103993/33102 = 3.141592653 0119026040722614947737
3+1/(7+1/(15+1/(1+1/293)))= 3.141592653 9214210447087159415927
3+1/(7+1/(15+1/(1+1/(292+1/(1+1/1))))) = 3.141592653 4674367055204547853492
3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1)))))) = 3.141592653
6189366233975003014106
3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1))))))) = 3.1415926535
583573009183052053374
3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/2))))))) = 3.14159265358
10777712044193065819
3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/(2+1)))))))) = 3.1415926535
914039784825424142193
3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/(2+1/(1+1)))))))))= 3.14159265358
70561991705458087813
3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/(2+1/(1+1/3)))))))))=3.14159265358
9 3891715436873217069
3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/(2+1/(1+1/(3+1))))))))))=3.1415926
53589 8153832419437773074
3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/(2+1/(1+1/(3+1/2))))))))))=3.14159
2653589 6274836288508219852
3+1/(7+1/(15+1/(1+1/(292+1/(1+1/(1+1/(1+1/(2+1/(1+1/(3+1/(1+1/14)))))))))))=
80143857/25510582=3.14159265358979 26593756269457122
Ken
Ken Kriesel, PE <[EMAIL PROTECTED]>
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