Algebra

# Roots of Unity Geometry

A "hop" is a movement of 4 sides at a time counterclockwise around the regular nonagon above. From the starting point, What is the minimum numbers of "hops" it will take to end up back at the starting point?

A regular hexagon has a vertex at $$(1,0)$$ and has its centroid at the origin. Which of these is another coordinate of the hexagon?

Regular polygons are placed on the coordinate plane such that they each have a vertex at $$(1,0)$$, and centroid at the origin. There is one regular $$n$$-gon for each value of $$n$$ between $$3$$ and $$50$$, inclusive.

How many values of $$n$$ are there such that the $$n$$-gon does not share any vertex other than $$(1,0)$$ with any of the other $$n$$-gons?

The point $$(5,-3)$$ is rotated $$\dfrac{\pi}{4}$$ radians counterclockwise about the point $$(2,4)$$. What is the resulting image?

Each of the partial sums $$\sum\limits_{k=1}^{n}{e^{k\pi i/3}}$$ is graphed on the complex plane for $$n\in\{1,2,3,4,5,6\}$$.

Which of the $$6^\text{th}$$ roots of unity coincides with one of these partial sums?

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