Ken Kriesel, are you sure you did not mistype just ONE digit or parens in
that humongous expression at the end?
At 01:10 AM 2/10/00 -0600, Ken wrote:
>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 Kriesel, PE <[EMAIL PROTECTED]>
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