Circle Packing and Room Capacities for COVID 19
Background: COVID 19 has required us to reconsider the maximum capacity of a room in order to maintain social distance. The capacity a rectangular room depends on how many circles with radius equal to half the social distance can be positioned so that their centers are inside the rectangular room. It's ok if a circle overlaps the wall of the room because individuals do not need to social distance from walls. Furthermore, the capacity depends on what "circle packing" pattern is used to position the circles. The two simplest circle packing patterns are known as the square packing and the hexagon packing. The former places circle centers on a square grid, the latter on a hexagonal grid. You can read more about these two packings here. In general the hexagon packing fits more circles than the square packing, but for some room dimensions, the opposite is true.
How to Use the Applet: This interactive applet lets you explore the circle packings--and the associated room capacities--of a rectangular room. To get started, enter your room's rectangular dimensions (in feet), and slide the dot to set a Minimum Social Distance (6 feet or greater). As you make changes, the program automatically calculates your room's capacity by counting the number of "X"s (circle centers) that are inside or on the boundary of the room. You can also click the button to toggle between the two aforementioned systematic circle packings. Feel free to explore. If you get lost, click the circular arrows in the top right to restore the default settings.
By default, the applet displays a 125 foot by 65 foot room, and a square packing of circles. On the right hand side, the applet has calculated that the room capacity with these settings is 231 persons. Switching to the hexagonal packing increases the capacity to 273 persons.
Note: This capacities reported by this applet are "theoretical." In particular, this applet treats people as points. In reality--of course--people are not points, and occupy space. Furthermore, people probably aren't comfortable being jammed up directly against a wall. Try increasing the Minimum Social Distance to give people some extra room.
Safety Considerations/Disclaimer: This app is only intended as an educational tool for exploring circle packings and how it is theoretically related room capacity. It should not be used as the sole tool in civic planning and decision making. This resource comes with absolutely no warranty. Nonetheless, I hope it helped you explore this interesting mathematical topic.
Contact: If you find an error, think of an improvement, or if just you want to talk more, please email me at greg [dot] petrics [at] northernvermont [dot] edu. In particular, I'm thinking of adding additional variables to treat people as non-points, and also to let the user control some "padding" from walls. I'm curious to hear what you think!