Flag Day celebrates the adoption of the United States' flag on this day in 1777. The symbol for unity, spread over thirteen stripes and fifty stars, stands tall as a momentous proclamation of the United States' values. Over the years, with the growth of our country, the flag of the U.S. has also changed, from thirteen stars to fifty. With each revision of the flag's design, a great deal of thought goes into the arrangement of our star spangled banner; and while mathematics is not always considered in this process, we know math is capable of bringing to our attention beauty, so today we will consider how math could play into our flag.

Read more after the break.

We will first start with an important facet of mathematics, packing problems. These problems are geometric in nature, yet closely related to covering problems, which are topological at heart. Here, we will take an informal look at how math could develop new ways of designing a flag through packing problems.

Packing problems include trying to optimize an arrangement of shapes (stars) given a certain space (a blue rectangle). It is important to note that many outcomes of trying to fit a number of disks (which we can think of as our stars) into a rectangle will not always yield a visually pleasing result, but we will continue our exploration anyway!

On an infinite plane, disks can be packed freely, and the most efficient way is by using a hexagonal lattice, which makes the disks take up 91% of the space. Most U.S. flag designs are a variation of this lattice structure, but let's dive into what we get when looking for the most efficient way to pack a certain number of disks into a finite space.

Using Wolfram Alpha's packing functionality, we can easily experiment with the parameters. What happens when the parameters are relatively prime? ... when one of them is pi? When the disk is only a fraction smaller than a factor of the square length? Have fun with your packings, and maybe you'll find the next design for the flag of the United States!

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