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It is known from the time of Euclid, that a circle is similar to a polygon with infinite number of sides.

But this ^^ is informal.

Do you know any formalization where it appears that a circle is a polygon with infinite number of sides?

porton
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    It's called limit. – Kaster Feb 02 '16 at 08:09
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    @Kaster I know that it is a limit. But what is the topology and what is the filter on which the limit is taken? – porton Feb 02 '16 at 08:13
  • There is an old cartoon where the idea that a circle has one side leads to the invention of a triangular wheel - polygon with fewest possible sides. – Mark Bennet Feb 02 '16 at 08:13
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    One way of looking at the limit is to take a regular hexagon with the vertices one unit from the centre and bisect each of the angles, giving polygons with $12, 24, 48, 96 \dots $ vertices. The circle is the limit in the normal topology of $\mathbb R^2$. This keeps everything computable and also gives an approximation of $\pi$. An outer limit is obtained by creating a regular hexagon etc of unit distance between the centre and the midpoint of each side. – Mark Bennet Feb 02 '16 at 08:20
  • @Kaster And I prefer it to be considered not as a limit, but as a "real" polygon with infinite number of sides. For example in complex number theory the infinity is not just a limit of finite values, but a quite particular point of Riemann sphere. – porton Feb 02 '16 at 11:15
  • @porton Infinity is certainly not a real number, but there's a natural analog in the nonstandard-analysis realm. You might look into questions under this tag to see if it answers your question. – GPhys Feb 08 '16 at 16:06
  • Your question is almost a duplicate of this one. –  Jun 16 '16 at 05:45

1 Answers1

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The idea of viewing a circle as a (regular?) infinite-sided polygon goes back at least as far as Nicholas of Cusa (sometimes referred to as Cusanus). The idea was picked up by Kepler who used it in area calculations, many years before the idea was developed formally in a mathematically adequate form. The torch was picked up by Leibniz who most likely introduced the term infinitesimal so that the polygon now becomes infinitesimal-sided.

In the 18th century, Leibniz's ideas were developed by his followers like Johann Bernoulli, and his followers' followers like Leonhard Euler, who used both infinitesimals and infinite numbers in a routine way.

In the 19th century, infinitesimals were still in common use chez Cauchy, who used them to define continuity of a function $f$ as follows: $f$ is continuous if each infinitesimal increment $\alpha$ always produces an infinitesimal change $f(x+\alpha)-f(x)$ in the function.

The next generation of mathematicians developed set-theoretic foundations that ultimately formalized the real numbers, but failed to formalize infinitesimal procedures of the founders of the calculus.

The work on infinitesimals continued throughout the second half of the 19th century and the beginning of the 20th century, by people like Paul du Bois-Raymond, Veronese, Hahn, Dehn, Hilbert, and others. Skolem in 1933 introduced a model of Peano arithmetic containing infinite numbers.

It was not until the 1960s that Abraham Robinson pulled all of these strings together by creating a modern framework for working with infinitesimal and infinite numbers that meets current standards of mathematical rigor.

In Robinson's framework, one can approximate the circle by a regular polygon with $H$ sides where $H$ is an infinite hypernatural number. More precisely, the circle is the standard part of the infinite-sided polygon.

Mikhail Katz
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  • I think it would be helpful if you added some details on how this is actually done. (Assuming familiarity with hyperreals.) Another thing to note is that this infinite polygon is not unique: one can say that there are many nonstandard polygons with the same standard part equal to the unit circle. – Moishe Kohan Aug 01 '17 at 00:25