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This question already has an answer here:

Is this "proof" just a mathematical joke or might there be some deeper truth in it, eventhough the theorem is obviously false?

Definition: Let a regular $n$-gon be a geometric figure $f^d_n$ that

  • is uniquely defined by

    1. one integer-valued parameter $n$ (the figure's number of corners)

    2. one real-valued parameter $d \neq 0$ (the figure's diameter)
      (Let's assume $d=1$ and drop the parameter in the following.)

  • has an area $\neq 0$

Lemma 1: A regular $n$-gon $f_n$ is a regular $m$-gon $f_m$ iff $n=m$.

Theorem: $0 = \infty$

Proof: A circle is a regular $0$-gon $f_0$ because it's defined by its diameter and its number of corners, which is 0.

On the other side, a circle is a regular $\infty$-gon $f_\infty$ being defined by its diameter and its number of corners, which is $\lim_{n\rightarrow \infty} n = \infty$.

From Lemma 1 it follows that $0 = \infty$. $\blacksquare$

Lemma 2: There are no regular $1$- and $2$-gons.

My questions are:

  1. Which are the obvious errors in this proof?

  2. Is there some deeper truth in this proof?

  3. If it's just a mathematical joke: Who told it first?

  4. If it contains some deeper truth: Who told it first?

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marked as duplicate by Asaf Karagila Aug 25 '18 at 12:52

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    $\begingroup$ First of all, $\infty$ is not a number. Next, a $0$-gon makes no sense. According to this "proof", a circle has no and simultaneously infinite many corners, which is of course impossible. In fact, this "proof" is a mathematical joke, but I would not consider it a good "fake-proof. $\endgroup$ – Peter Aug 25 '18 at 12:48
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    $\begingroup$ I don't think a circle is a polygon, a limit of polygons could be a definition. I don't think polygons can have 2 or less corners $\endgroup$ – DWade64 Aug 25 '18 at 12:49
  • $\begingroup$ @Peter: I've foreseen your objections, thanks for making them explicit. Might you want to explain, which kind of "fake-proofs" you consider good ones? $\endgroup$ – Hans-Peter Stricker Aug 25 '18 at 12:53
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    $\begingroup$ Why do you say it's far fetched: In my mind I can rather easily imagine a circle both as a 0-gon and as a $\infty$-gon - switching between the two pictures, like for the Necker cube.. And I'm sure I'm not the only one. $\endgroup$ – Hans-Peter Stricker Aug 25 '18 at 12:59
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    $\begingroup$ If someone insists that $\infty$ IS a number and a $0$-gon DOES make sense, you can finally explain : That a distinct number of corners arises from different definitions, does not make the distinct numbers equal. $\endgroup$ – Peter Aug 25 '18 at 13:06