I think the question comes down to that of whether there is some difference between the $5$-gon and $17$-gon such that the $5$-gon can be constructed with geometry alone, but the $17$-gon needs algebra. Terms such as $\cos\frac{2\pi}{n}$, $\cos\frac{4\pi}{n}$, and so on would not have been in Euclid's vocabulary. Nor would the concepts of ...

2019年11月8日 · $\begingroup$ @Landuros Ah, he did solve the equation to have the 17 solutions on the unit circle which shows that a 17-gon is constructable. But how was he able to get the whole equation with cosine? $\endgroup$

2014年1月1日 · In any case, according to A062361, the number of triangles for the regular $17$-gon would be $1054$. There is no simple formula associated with that sequence, so I guess one has to “simply” count them. Doing so correctly without interference from limited precision rounding errors can be a bit tricky.

2024年10月3日 · (Note: All images except the $13$-gon are from Antonia Buitrago's 2007 article "Polygons, Diagonals, and the Bronze Mean".) I. The golden ratio We are familiar with how $\\phi$ appears in ...

2019年4月2日 · Gauss found an unexpected link between contructible polygons and prime numbers. Gauss was so pleased with this result that he requested that a regular 17-gon be inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle.

Start the count counting the different polygons we can build from a vertex. That is, if we choose a vertex in a regular $17$-gon then we can build from here $$\sum_{k=2}^{16}\binom{16}{k}=2^{16}-17$$ different polygons, because for some set of vertex there is a unique possible polygon.

2018年12月23日 · $\begingroup$ If the 2 in front of $\Sigma \epsilon_k$ is incorrect (as I thought also), this is a typo in my version of H&W.

2019年6月3日 · First of all a picture. The given $17$-gon has in my notation the vertices $0,1,2,\dots,16$.

$\begingroup$ In Introduction To Geometry by Coxeter , there are surprisingly simple constructions shown for a side of the regular 17-gon and 257-gon inscribed in a given circle. So it's plausible that there may a short method for a side of …

How many quadrilaterals are there whose vertices are the vertices of the 17-gon and have 2 sides that are diagonals? Ask Question Asked 7 years, 7 months ago