Exponential sum for the new year

Exponential sums can make intricate patterns. Last year I made a page that displays a different page each day, using the month, day, and year as parameters in the expression below. The images plot the partial sums of this sum.

\sum_{n=0}^N \exp\left( 2\pi i \left( \frac{n}{m} + \frac{n^2}{d} + \frac{n^3}{y} \right ) \right )

This was yesterday’s image.

Image of the day, New Year's Eve.

Today’s image is surprisingly plain if we use y = 18.

This is in part because the least common multiple of 1, 1, and 18 is 18. The image could have no more than 18 vertices. In fact, it has only 6 vertices because the summand above has period 6.

But if we use y = 2018 we get something much more complex.

The Exponential Sum of the Day page will use y = 18 this year. There will be a few simple images this year but there will also be lots of surprises.

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