faulty logic?

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phillip1882
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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

DavidSh
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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.

mward
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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.

elasto
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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...)

thefargo
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Re: faulty logic?

Postby thefargo » Thu Jun 20, 2019 9:13 pm UTC

Elasto brings up an interesting way of viewing it that clarifies the problem for me in a way it hasn't before. In order for this proof to be true, you'd have to essentially argue that for a right triangle (one where a^2+b^2=c^2) then if you make the triangle small enough (that is to say take the limit), then suddenly it is true that a+b=c.

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wraith
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Re: faulty logic?

Postby wraith » Fri Jun 21, 2019 6:03 am UTC

thefargo wrote:... you'd have to essentially argue that for a right triangle (one where a^2+b^2=c^2) then if you make the triangle small enough (that is to say take the limit), then suddenly it is true that a+b=c.


This is true, though.
0²+0²=0²
It's only when you look at an ant through a magnifying glass on a sunny day that you realize how often they burst into flames

thefargo
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Re: faulty logic?

Postby thefargo » Fri Jun 21, 2019 3:43 pm UTC

But that's not what a limit is. it is small, but not zero.

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Bloopy
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Re: faulty logic?

Postby Bloopy » Sun Jul 21, 2019 10:53 am UTC

The shape may appear close to a circle from a distance, but every dent just adds smaller and smaller zigzags. It'd be a fractal you can zoom in on forever, and would never have the smooth curve of a circle. Also compare to the coastline paradox.

benkoren
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Re: faulty logic?

Postby benkoren » Mon Jul 22, 2019 3:18 pm UTC

You incorrectly called this “puzzle“. We are not talking about faulty logic - we are talking about faulty conditions of the puzzle, which is a jumble of absurd statements:
“if you dent in the four corners of the square, the perimeter is still four...“ - nonsense . “if you dent in the 8 new corners, again the perimeter is still 4...“ - nonsense again and so on.
I propose to call it FAULTY-PUZZLE. But this, I believe, is not for this forum?

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Sizik
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Re: faulty logic?

Postby Sizik » Mon Jul 22, 2019 4:05 pm UTC

benkoren wrote:You incorrectly called this “puzzle“. We are not talking about faulty logic - we are talking about faulty conditions of the puzzle, which is a jumble of absurd statements:
“if you dent in the four corners of the square, the perimeter is still four...“ - nonsense . “if you dent in the 8 new corners, again the perimeter is still 4...“ - nonsense again and so on.
I propose to call it FAULTY-PUZZLE. But this, I believe, is not for this forum?


In case you're not understanding what they're proposing:
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she/they
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Yes. And if wishes were horses, wishing wells would fill up very quickly with drowned horses.

benkoren
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Re: faulty logic?

Postby benkoren » Tue Jul 23, 2019 6:12 pm UTC

Unfortunately, the conditions of the pazzle were not clear and I understood that it is about the simplest way to make two corners of one, i.e. just to cut it off with one line. In any case pardon my style.

Regarding the pazzle. It is true that the game of scales creates illusion of the length of a broken line approaching the length of circumference. Nevertheless, this is an illusion, and that's why.
I propose to imagine that every time we delete a corner we double the scale and observe a single fragment. What do we see?
We see that this fragment remains a right-angled triangle in which two sides are straight line segments, and the third side is an arc that with each step closer to a straight line. The triangle is undergoing deformation, but in the direction of becoming a normal right triangle whose sum of the two sides is always greater than the third side.
Just increase the scale and observe.


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