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\begin{document}

\title{Episode 07 -- Constructible polygons}
\subtitle{European section -- Season 3}
\date{}

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  \titlepage
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\begin{frame}{Euclid}

\begin{multicols}{2}

Euclid was a Greek mathematician and is often referred to as the ``Father of Geometry.'' He was active in Hellenistic Alexandria during the reign of Ptolemy I (323-283 BC).

He knew how to inscribe a regular polygon with 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64,\ldots, sides.


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\begin{frame}{Gauss}

\begin{multicols}{2}

Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his \emph{Disquisitiones Arithmeticae}. This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons:

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\begin{frame}{Gauss' result}

\begin{block}{}
\large
A regular $n$-gon can be constructed with compass and straightedge if $n$ is the product of a power of $2$ and any number of distinct Fermat primes.
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Gauss conjectured that this condition was also necessary. It was proved by Pierre Wantzel in 1837. 

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\begin{frame}{Constructible polygons}

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Using Gauss' result, we can decude that the regular $n$-gon is constructible for the following values of $n$ :
$$3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24,$$
$$30, 32, 34, 40, 48, 51, 60, 64,\ldots$$
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\begin{frame}{Pascal's triangle and the Sierpinski sieve }

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&&    &    &    &    &  1\\\noalign{\smallskip\smallskip}
&&    &    &    &  1 &    &  1\\\noalign{\smallskip\smallskip}
&&  &    &  1 &    &  2 &    &  1\\\noalign{\smallskip\smallskip}
&&        &  1 &    &  3 &    &  3 &    &  1\\\noalign{\smallskip\smallskip}
  &    &  1 &    &  4 &    &  6 &    &  4 &    &  1\\\noalign{\smallskip\smallskip}
  & 1 &    &  5 &    &  10 &    & 10 &    & 5 &    & 1\\\noalign{\smallskip\smallskip}
1 &   & 6  &    & 15 &     & 20 &    & 15 &   &  6 &   & 1\\\noalign{\smallskip\smallskip}
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\begin{frame}{Pascal's triangle and the Sierpinski sieve }

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\begin{tabular}{ccccccccccccc}
&&    &    &    &    &  1\\\noalign{\smallskip\smallskip}
&&    &    &    &  1 &    &  1\\\noalign{\smallskip\smallskip}
&&  &    &  1 &    &  0 &    &  1\\\noalign{\smallskip\smallskip}
&&        &  1 &    &  1 &    &  1 &    &  1\\\noalign{\smallskip\smallskip}
  &    &  1 &    &  0 &    &  0 &    &  0 &    &  1\\\noalign{\smallskip\smallskip}
  & 1 &    &  1 &    &  0 &    & 0 &    & 1 &    & 1\\\noalign{\smallskip\smallskip}
1 &   & 0  &    & 1 &     & 0 &    & 1 &   &  0 &   & 1\\\noalign{\smallskip\smallskip}
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\begin{frame}{Pascal's triangle and the Sierpinski sieve }

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\begin{tabular}{cccccccccccccc}
&&    &    &    &    &  1&&&&&&& \bf 1\\\noalign{\smallskip\smallskip}
&&    &    &    &  1 &    &  1&&&&&& \bf 3\\\noalign{\smallskip\smallskip}
&&  &    &  1 &    &  0 &    &  1&&&&&\bf 5\\\noalign{\smallskip\smallskip}
&&        &  1 &    &  1 &    &  1 &    &  1&&&&\bf 15\\\noalign{\smallskip\smallskip}
  &    &  1 &    &  0 &    &  0 &    &  0 &    &  1&&&\bf 17\\\noalign{\smallskip\smallskip}
  & 1 &    &  1 &    &  0 &    & 0 &    & 1 &    & 1&&\bf 51\\\noalign{\smallskip\smallskip}
1 &   & 0  &    & 1 &     & 0 &    & 1 &   &  0 &   & 1 &\bf 85\\\noalign{\smallskip\smallskip}
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