| Topic: |
Science > Physics |
| User: |
"Alan Horowitz" |
| Date: |
09 Feb 2004 10:59:24 AM |
| Object: |
construct a circle of unit area? |
how to construct a circle with area = exactly one unit?
any applications for it outside of probability?
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| User: "Sam Wormley" |
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| Title: Re: construct a circle of unit area? |
09 Feb 2004 01:28:29 PM |
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Alan Horowitz wrote:
how to construct a circle with area = exactly one unit?
Circle:
http://mathworld.wolfram.com/Circle.html
set equation (5) A = 1 and solve for r
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| User: "Uncle Al" |
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| Title: Re: construct a circle of unit area? |
09 Feb 2004 01:38:01 PM |
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Alan Horowitz wrote:
how to construct a circle with area = exactly one unit?
any applications for it outside of probability?
1) Define an arbitrary circle to have unit area.
2) Pi is irrational and transcendental. There is no way to
construct a "unit area circle" by Greek rules (straightedge and
compass, linear and/or quadratic functions).
3) Construct a unit square. Convert to a unit circle using a
quadratrix curve. There are lots! Alas, no quadratrix curve can be
constructed by Greek rules.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
"Quis custodiet ipsos custodes?" The Net!
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| User: "Arturo Magidin" |
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| Title: Re: construct a circle of unit area? |
09 Feb 2004 11:22:18 AM |
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In article <1e3670a7.0402090859.4d5faa80@posting.google.com>,
Alan Horowitz <alanh_27@yahoo.com> wrote:
how to construct a circle with area = exactly one unit?
Depends on (a) you mean by "construct"; and (b) what you mean by
"unit".
For example, by "construct" mathematicians usually mean "using only
compass and straightedge", which is equivalent to saying that given
two points, you can draw the line between them, given a point and a
distance, you can draw a circle with center at the given point and
radius equal to the given distance; and that you can find
intersections of constructed lines and circles.
One way to construct a circle of "exactly one unit" in area would then
be to construct any circle whatsoever, and then declare its area to be
the equal to one unit of whatever you are measuring. But that would be
deemed as "cheating" by many.
So, presumably, you mean: given a specific distance to be used as a
"unit", to construct (perhaps in the sense above) a circle whose area
is equal to the area of the square whose sides are one unit
long. (Such as square can be constructed using only compass and
straightedge).
If so, the answer is "no": a circle of area exatly one unit square
would have to have a radius equal to 1/sqrt(pi). Which means that in
order to construct such a circle, you must be able to construct two
points whose distance is exactly 1/sqrt(pi). If you could do that,
then you would be able to construct (through well known methods), two
points whose distance is exactly the reciprocal of that, sqrt(pi); and
from that, again through well known methods, you would be able to
construct two points whose distance is the square of sqrt(pi), which
is pi.
However, pi is a transcendental number, and it is known that the only
distances that are constructible (in the sense above) are those
real algebraic numbers alpha which can be obtained from rational
numbers by taking a sequence of square roots; say, things like 3 +
sqrt(2+sqrt(4-sqrt(2+3sqrt(7)))). So it is impossible to construct two
points whose distance is pi (this is also why it is impossible to
square the circle: that means, given a circle, construct a line
segment whose length squared has the same area as the given circle;
the circle of radius 1 one would require you to construct sqrt(pi)).
Therefore, it is impossible to construct a circle of area exactly 1
unit, in the sense above (i.e., given the unit first, and allowing the
use of only a compass and straightedge).
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu
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| User: "" |
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| Title: Re: construct a circle of unit area? |
09 Feb 2004 08:31:31 PM |
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In article <1e3670a7.0402090859.4d5faa80@posting.google.com>, Alan Horowitz
<alanh_27@yahoo.com> writes:
how to construct a circle with area = exactly one unit?
any applications for it outside of probability?
That depends: One unit of what?
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| User: "Robert J. Kolker" |
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| Title: Re: construct a circle of unit area? |
09 Feb 2004 10:02:17 PM |
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wrote:
That depends: One unit of what?
One unit of area. The problem comes down to squaring the circle.
Construct a circle whose area equals that of a unit square. This cannot
be done with ruler and compass used in the classical Euclidean fashion.
Bob Kolker
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| User: "Mark Mallory" |
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| Title: Re: construct a circle of unit area? |
10 Feb 2004 05:26:22 AM |
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wrote:
how to construct a circle with area = exactly one unit?
****
That depends: One unit of what?
One unit of AREA, stupid sHead.
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