constructing pi
calvin
plot it on graph paper using compasses, without
knowing the value of the sqrt of 2.
But how could pi be constructed and plotted
using only a straightedge and compasses, without
knowing its value, only its definition?
Gerry
Pi cannot be constructed using straightedge and compass.
The reason, very roughly speaking, is this:
Straightedges make lines, which have linear equations;
Compasses make circles, which have quadratic equations;
Anything you can construct by straightedge and compass
can be expressed in terms of solutions of quadratic
equations, thus, in formulas using only square roots
(and square roots of square roots, etc.),
but (and this is the hardest part) pi is transcendental,
which means it can't be expressed as the solution of any
kind of algebraic equation with integer coefficients,
a fortiori can be expressed in terms of (iterated) square roots.
--
GM
William Elliot
IIRC, that problem is called squaring the circle,
which was much sought until it was proven impossible.
JEMebius
The Greek mathematician Archimedes ( http://en.wikipedia.org/wiki/Archimedes ) found out,
in a sense, how to construct pi using straightedge and compass only (Euclidean construction).
He posed the problem of determining the ratio of the circumference of the circle to its
diameter. Doing so he started with the inscribed regular hexagon, for which
circumference : diameter = 3 : 1 exactly.
A. knew how to construct the regular 12-gon, 24-gon etc. by Euclidean means.
He performed the accompanying calculations. AFAIK he stopped at the regular 96-gon and
concluded that
223 : 71 < circumference : diameter < 22 : 7.
So if one allows non-terminating construction processes that yield arbitrarily close
approximations to pi, along with exact constructions: yes! then pi is in this extended
sense constructible by Euclidean means.
It was not until in the 19th century that pi was proved a transcendental number, which
implies that no exact Euclidean construction for squaring the circle exists.
Good luck: Johan E. Mebius
calvin
Yes, I wondered without thinking it through if
constructing pi would be equivalent to squaring the
circle, and of course it would:
If pi could be constructed, since we know the area
of a circle is pi x r^^2, then for any radius r,
we plot the rectangle with that area. From that it
is a solved problem * to construct a square with
the same area, and thus the circle would be squared
(both the circle and the square having the same
area).
*
http://www.geom.uiuc.edu/~huberty/math5337/groupe/squarerect.html
Dave L. Renfro
> Anything you can construct by straightedge and compass
> can be expressed in terms of solutions of quadratic
> equations, thus, in formulas using only square roots
> (and square roots of square roots, etc.), but (and this
> is the hardest part) pi is transcendental, which means
> it can't be expressed as the solution of any kind of
> algebraic equation with integer coefficients, a fortiori
> can be expressed in terms of (iterated) square roots.
Proving that pi is transcendental also prevents pi from
being constructed using conic curves (which happen to be
sufficient to duplicate the cube and trisect an angle),
or even from being constructed by using algebraic curves
in general (graphs of integer-coefficient, or even of
algebraic-coefficient, polynomials in the variables x and y).
In other words, proving that pi is transcendental is a
huge over-kill for simply showing that pi is not constructible.
It's like showing a certain real number is not an integer multiple
of 8 by showing the real number under consideration is irrational.
Does anyone know whether a proof that pi isn't constructible
has been found which is shorter (or simpler) than all known
proofs that pi is transcendental? I've seen this question
posed several times in the literature from around 1890 to
the 1920s (maybe later), so I'm pretty sure no such simpler
proof that pi isn't constructible was known by the 1920s.
It seems to me that if no such simpler proof is known, then
anyone finding such a proof will be making a fairly major
contribution. Not quite as major as finding an elementary
proof of Fermat's Last Theorem, but still a major contribution.
Dave L. Renfro
Ken Pledger
<409fe57c-11d6-4947...@c2g2000yqi.googlegroups.com>,
calvin <cri...@windstream.net> wrote:
> .... we plot the rectangle with that area. From that it
> is a solved problem * to construct a square with
> the same area....
>
> *
> http://www.geom.uiuc.edu/~huberty/math5337/groupe/squarerect.html
Yes. This is Euclid II.14.
Ken Pledger.