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.

2019年11月8日 · I am currently very interested in the derivation of the constructability of the 17-gon by Carl Friedrich Gauß.

2014年1月1日 · 4-dimensional analogue of centered polygonal numbers. Also number of regions created by sides and diagonals of n-gon in general position. The latter is what you've been asking about, except for the distinction between triangles and non-triangular faces. So the answer to your question about the $17$-gon would be 2500.

And since perpendicular radii have been constructed, the $20$-gon follows just as easily. Inscribing a regular $17$-gon in a given circle also entails constructing perpendicular radii, one of them through the given vertex. Thus the $34$-gon and $68$-gon follow just as easily.

2018年12月23日 · The point of that last equation is that $$\sum_{k=1}^{16} \epsilon_k = -1.$$ This follows most easily from the fact that the sum of all the $17$ th roots of unity is $0$, and you've omitted $\epsilon_0 = 1$.

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.

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

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, 10 months ago