Today’s star

Exponential sum of the day 10/2/2023

The star-like image above is today’s exponential sum.

The exponential sum page on my site generates a new image each day by putting the numbers of the day’s month, day, and year into the equation

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

and connecting the partial sums in the complex plane. Here m is the month, d is the day, and y is the last two digits of the year.

Some people have asked why I use American date order: month, day, year. The flippant answer is I use American date order because I’m American. But I did experiment with other date orders, and I prefer the sequence of images produced by the order above. There’s more contrast between consecutive images by associating the day with the quadratic term rather than the linear term inside the exponential.

The exponential sum page is about six years old [1], and I still enjoy checking in on it each day. Short of making the plot, it’s not possible to imagine what an image will look like based on the date, other than the very rough rule that larger numbers tend to produce more complicated images. For example, images are much more intricate on New Year’s Eve than on New Year’s Day.

The images are often highly symmetric, as today’s image is. But occasionally they have no symmetry, as will be the case on 10/10/23.

The page lets you scroll back and forth by day, but you can put in any parameters you’d like by editing the page URL. For example, the link to today’s image is

   https://www.johndcook.com/expsum/?y=23&m=10&d=2

but you can change y, m, and d to any numbers you wish. There’s nothing that constrains m, for example, to be a number between 1 and 12. You could set it to 17 if you’d like. And although thirty days hath September, you can see what the image for September 31st would have looked like.

[1] The page was launched October 9, 2017, so its sixth anniversary is a week from today.

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