Roots of Unity

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)(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)(1,0), and centroid at the origin. There is one regular nn-gon for each value of nn between 33 and 5050, inclusive.

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

The point (5,3)(5,-3) is rotated π4\dfrac{\pi}{4} radians counterclockwise about the point (2,4)(2,4). What is the resulting image?

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

Which of the 6th6^\text{th} roots of unity coincides with one of these partial sums?


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