Circle Packing = Social Distances

I just finished attending the Virtual NCTM meeting this year and I loved being able to attend talks without leaving home.  In addition, it is great I'll have access to recordings because I can see the talks that I missed due to all being at the same time.  One of the talks, a short 30 minute one, looked at circle and sphere packing and it's applications to the modern world.

The one thing the speaker said throughout the talk is that once people figure out the math, they never know all of the idea's possible applications at the time.  He began with the circle inscribed inside the square.  The square was say one unit by one unit while the square had a radius of 1/2. 

He showed how the area of a circle is about 78% of the area of the square.  I gather this is an old problem whose new application only became apparent during this pandemic with social distancing.

With today's 6 foot distancing mandates, offices often have to figure out how to arrange workers so companies can have as many workers as possible while still maintaining social distancing.  I used the basic idea when I arranged my classroom for my biggest class while trying to maintain the appropriate distances.  

The specific idea being used is called circle packing which is where circles are arranged within a certain area so that many of the sides are tangent to each other but do not overlap at all.  Most people think of packing circles within a square area by doing rows upon rows, so everything lines up such as in the picture of flowers.  

This is not the best arrangement for circles because there is still quite a lot of space available between the circles.  This is one of the usual arrangement teachers choose for their classroom, rows of seats all neat and orderly.  As stated earlier, this is not the most efficient arrangement.

After a lot of exploration, mathematicians discovered the best arrangement is actually hexagonal with a density of almost 0.91.  I didn't realize the hexagonal arrangement was the most efficient when I arranged my classroom. I just eyed things and put the chairs into this orientation because it was the only way I could figure how to arrange all the chairs to maintain distancing.  Although it seems that it might be a good idea to pack circles in side a circle, it really is not the most efficient.  

In addition, the idea of circle packing can be extended to sphere packing which is the idea that one can arrange spheres in the most effective arrangement.  This problem was first proposed in 1611 and Kepler came up with a solution but it was until around 1998 that it was proven the best arrangement is just the way they stack oranges in a pyramidal shape at the grocery store. Others researched the best arrangement up to 24 dimensions and found one that works.  Sphere packing is used currently for data transmission and error correcting code. So we have something first proposed in 1611 whose applications are extremely important in today's society.

I found this extremely interesting.  Math first proposed centuries ago that is now playing a huge part of our daily lives.  Let me now what you think, I'd love to hear.  Have a great day.