faulty logic?

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faulty logic?

Postby phillip1882 » Mon May 13, 2019 6:48 pm UTC

construct a circle.
now construct a square around the circle, such that is each side is perpendicular to the circumference.
if the diameter of the circle is 1, then the perimeter of the square is 4.
if you dent in the four corners of the square, the perimeter is still four, and the shape is closer to a circle.
if you dent in the 8 new corners, again the perimeter is still 4, and the shape is closer to a circle.
now if you do it infinitely times, at each step, the perimeter stays the same, but the shape is a circle.

why isn't this a valid proof that the diameter to the circumference ratio is 1/4?
it uses the same logic as increasing the number of sides.
good luck have fun

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Re: faulty logic?

Postby DavidSh » Mon May 13, 2019 8:20 pm UTC

There is no particular reason to believe that the lengths of an arbitrary sequence of polygonal paths should converge to the length of the limit curve, absent a proof. There are a lot of non-continuous functions in the world of mathematics. And there are different sequences of non-convex paths converging to the same limit for which the lengths converge to different values.

Why, then, the traditional constructions with tangent or secant polygons? Largely because they give consistent answers, and because they agree with the physical models of chains or ropes. After all, why would an ancient Greek geometer care about the length of a curve? Because he wanted it to represent the length of some flexible object.

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Re: faulty logic?

Postby mward » Wed May 15, 2019 12:58 pm UTC

The traditional calculation involves a circumscribed polygon (whose circumference is definitely greater than that of the circle) and an inscribed polygon (whose circumference is definitely smaller than that of the circle). As the number of sides increases, the two circumferences converge to a single value. This value is therefore the circumference of the circle.

The reference to "non-continuous functions" is a red herring: it is easy to imagine a sequence of continuous functions (wavy lines) which approaches a circle (or even a straight line), but for which the sequence of path lengths approaches infinity. Imagine the waves getting smaller but the wavelength getting much, much smaller (so the waves get steeper) as the sequence progresses.

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Re: faulty logic?

Postby elasto » Wed May 15, 2019 10:00 pm UTC

Note that you could use this same logic to 'prove' that the diagonal of a unit square is of length 2...

I looked up layman's refutations of pi=4, and the most succinct seemed to me to be:

Of course, the circumference is not approximated by the sum of lengths of the lines constructed as shown, but by the sum of the hypotenuses of each of the right-angle triangles formed around the edge of the circle (forming a polygon with vertices on the circle)

(This isn't rigorous particularly, but hopefully helps illuminate the sort of mistake this proof is making...)

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