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Old 2006-09-28, 21:51   #23
Uncwilly
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Quote:
Originally Posted by xilman View Post
Ah. In that case you should have said so.
Common usage, area of a circle refers to the area of the disk.
At: http://en.wikipedia.org/wiki/Circle they aknowledge both usages.

Food for thought, like the concentric circles, is it possible to construct polygons of the appropriate size? Or a half size circle that is bisected by a diameter?
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Old 2006-09-29, 02:08   #24
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Quote:
Originally Posted by Uncwilly View Post
Or a half size circle that is bisected by a diameter?
It is trivial to construct concentric circles of radius 1, sqrt(2), and 2.
Further, any diameter can be used to "halve" any of the circles.

But, if you use this type of construct, is not the proposed solution one with "radial" lines. (Any part of a diameter)
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Old 2006-09-29, 02:15   #25
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Quote:
Originally Posted by Uncwilly View Post
like the concentric circles, is it possible to construct polygons of the appropriate size?
It certainly cannot be done with squares. ("Squaring the circle" is a classic problem of geometry).

I think that that implies that it cannot be done with any other polygon, either.
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Old 2006-09-29, 08:35   #26
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Quote:
Originally Posted by Uncwilly View Post
Common usage, area of a circle refers to the area of the disk.
Unfortunately, you did not say "area of a circle" or "interior of a circle" or any other such phrase. You asked for the circle to be divided and it is that problem which I addressed.

Mathematics requires pedantic precision. A good case can be made that mathematics is pedantry carried to its extreme.


Paul
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Old 2006-09-29, 12:53   #27
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Quote:
Originally Posted by xilman View Post
Unfortunately, you did not say "area of a circle" or "interior of a circle" or any other such phrase. You asked for the circle to be divided and it is that problem which I addressed.

Mathematics requires pedantic precision. A good case can be made that mathematics is pedantry carried to its extreme.


Paul
I remember a similar difficulty that I had when I was a senior in high school.
The preliminary U.S. Olympiad team exam had multiple choice answers. One question asked:

What is the maximum number of regions into which a circle may be
cut by n lines?

The correct answer was not among those given. The reason is that the
one who made up the question intended "disk" instead of circle.

I left this question blank, and attached a note to the exam explaining
why. The note included a solution for the question *AS ASKED*.
The person grading the exam would not give me credit, because it was
clear to him that the question intended "disk". I argued that in mathematics
if one meant disk, one said "disk" and not "circle". The teacher grading
the exam fired back that there was *no difference* and I then fired back
that the teacher had no business teaching math because he was incompetent. This led to a discussion with the department chair, my math teacher, the high school principal, and my parents.....

The department chair gave me full credit for my answer and actually told
the teacher grading the exam that he should have listened to me; that a disk is not a circle. He also added that the teacher should have listened given who I was. (I had quite a reputation in the school math & science depts). The teacher fired back that I was "just another student, he is the teacher and was not to be argued with".

At that point, *my* teacher supported me. He said that that I was not
"just another student", that I was then taking my 3rd year of calculus and it was virtually certain that I knew more math than the grader and that he should have listened to me given who I was. Apparently the grader was a new
teacher who didn't know me from a hole in the ground.

The principal opined that what I said to the grader was rude. I agreed, but
said that I was just responding to the rudeness of the teacher in not
being willing to listen. I said that I would apologize if the teacher apologized.
The teacher refused, taking the position "I don't have to apologize to a
student". At that point I said that a teacher who was unwilling to listen to
a student should not be teaching....The teacher responded that he did
listen, decided that I was wrong (my math teacher and the department
chair rolled their eyes a bit) and that should have ended the discussion. That teacher was not teaching the following year...

I later had a private talk with my teacher and he agreed that the grader
should not be teaching given his attitude towards students and his failure
to recognize an elementary mathematical fact.
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Old 2006-09-29, 16:16   #28
ewmayer
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Quote:
Originally Posted by R.D. Silverman View Post
I remember a similar difficulty that I had when I was a senior in high school. The preliminary U.S. Olympiad team exam had multiple choice answers.
The incident you describe reminds me of a not-dissimilar one I had in the 7th grade with my math teacher at the time, regarding the nature of mathematical infinity - her concept of oo was effectively "large enough so there's no practical way of counting it, but finite". (The specific question that triggered the discussion was whether the number of grains of sand on all the world's beaches was finite or infinite.) Note these were not-terribly-well-funded public schools, and many teachers were as a consequence forced to teach subjects they really had no business handling. In my case things didn't turn quite as vituperative as in yours - she deferred to the 9th-grade advanced geometry teacher (the closest thing we had to a mathematical expert in my junior high), who gently set her straight.

Quote:
Apparently the grader was a new teacher who didn't know me from a hole in the ground.
An open hole, or one with closure? Be precise, man! ;)
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Old 2006-10-02, 17:55   #29
mfgoode
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Lightbulb 4 not so easy pieces?


Off hand I would say that this problem can be solved by using the principle of Hippocrates' lunes - a straight line, which is not radial, and a curve.
Mally
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Old 2006-10-03, 15:45   #30
mfgoode
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Lightbulb Hippocrates' lunes.

Quote:
Originally Posted by mfgoode View Post

Off hand I would say that this problem can be solved by using the principle of Hippocrates' lunes - a straight line, which is not radial, and a curve.
Mally

Splendid problem, Uncwilly, even an archaic one, solved in the time of Aristotle.

Construction : Draw a semicircle of the given circle and inscribe an isosceles triangle on the diameter. On any of its equal sides construct a semi circle.

This semi circle will be equal to the quadrant of the 1st semi circle as required. Sorry I dont have the expertise to draw it out.

This problem is based on the proof that the crescent ( lune) formed is equal to half the isosceles triangle constructed.

Mally
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Old 2006-10-03, 21:28   #31
Uncwilly
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Attached is the image that I stumbled upon whilst looking for around the i-net.

The solution look simple to construct and has no straight lines. While it has X or rotational symetry, it is not radial (star formed).
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Old 2006-10-03, 22:32   #32
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Quote:
Originally Posted by Uncwilly View Post
Attached is the image that I stumbled upon whilst looking for around the i-net.

The solution look simple to construct and has no straight lines. While it has X or rotational symetry, it is not radial (star formed).
How does one construct this with ruler and compass?

As I said earlier, 5 pieces is easier.
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Old 2006-10-03, 22:52   #33
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Quote:
Originally Posted by R.D. Silverman View Post
How does one construct this with ruler and compass?
That is left as an exercise for the pupil .... :)

Actually, it is rather easy.
First construct the diameter. Then divide it into 8 equal parts. That gives you all of the centers of the various circular segments.
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