Counting Crowds at the National Mathematics Festival

By Blake Reichmuth, Ratna Khatri, Rachel Neville, and Suzanne Weekes

The National Mathematics Festival, held on April 22 in Washington, D.C., brought an incredible mixture of education, entertainment, mathematics, and science to the nation’s capital. The festival was organized by the Mathematical Sciences Research Institute, in cooperation with the Institute for Advanced Study and the National Museum of Mathematics. Events included more than a dozen short films, games and activities for younger audiences, geometric balloon bending, and over 50 presentations ranging from standup mathematics comedy and magic to quantitative analysis and hurricane storm surge modeling. However, the long duration of the event, use of multiple entrances at the venue, and lack of tracking due to free admission made estimating the number of individual attendees quite a challenge.

The Festivals Working Group of the SIAM Education Committee, comprising of three graduate student leaders of SIAM student chapters—all pursuing Ph.D.s in applied mathematics—were tasked with the responsibility of orchestrating the crowd counting. Team chair Rachel Neville (Colorado State University), Ratna Khatri (George Mason University), and Chong Wang (George Washington University) recruited graduate and undergraduate student volunteers from their three respective schools, as well as Shippensburg University. Volunteers stood out in the crowd thanks to special SIAM buttons pinned to their National Math Festival shirts.

The total count of attendees at this year’s festival was an estimated \(M\) unique individuals.¹ The team settled on this estimate using a step-by-step process. Volunteers took cell phone pictures of crowds from various vantage points with the help of 67-inch selfie sticks. They snapped photos of attendees in assigned rooms, hallways, and concourses every hour for the duration of the festival. After these pictures were collected and heads counted, the total number of heads counted, i.e., “people hours” — an approximation to \(\int_0^T \#\) people (\(t) dt\) was calculated to be 20,350.

The team then set to work determining the number of unique individuals attending the event by filtering out people accounted for in earlier tallies. During their off hours, SIAM volunteers polled the crowd to get a sense of how long attendees expected to stay. This polling information on attendance duration offered insights into the number of people who would appear more than once in the hourly photos.

Let us consider the process in more detail. Take the “hour 1” count of the number of people in attendance. The distribution of predicted attendance times allows us to estimate the number of these people who would show up in each of the later photos. The number of new people in the second round of photos is the “hour 2” head count minus the number of heads we estimate were already present in the first count, i.e., those folks who indicated they were staying at the festival for more than an hour. The number of new people at “hour 3” is the head count at hour 3 minus both those people who arrived in time for the hour 2 photos and stayed at the festival for longer than an hour, and the hour 1 people who stayed at least two hours. And so on. The number of people who attended the festival is the total of all the new people present in the hourly photos.

 In mathematical terms, let \(h_k\) be the number of individuals that have joined the crowd between hour \(k-1\) and hour \(k\), and let’s define \(\tilde{h}_k = max(0, h_k).\) Let \(H_k\) be the total number of people counted at hour \(k\). Then \(h_0 = 0,\) and \(h_1 = H_1\) is the total number of attendees that entered the festival from the time the festival began to the moment the first pictures were taken. Therefore, \(h_2 = H_2 - \delta_1 \tilde{h}_1\) where \(\delta_k\) is the fraction of people that plan to stay more than \(k\) hours, according to our polling of attendance times. Next, \(h_3 = H_3 - (\delta_2 \tilde{h}_1 + \delta_1 \tilde{h}_2)\) is the approximate number of new attendees between the second and third hour of the festival. In general, \(h_n = H_n\) \(- \Sigma^{n-1}_{i=1} \delta_{n-i}\:\tilde{h}_i\) and the total number of unique individuals who attended the festival, \(h^* = \Sigma \tilde{h}_i.\) We assume that \(\delta_k\) is not time-dependent, whereas in reality the fraction of people who plan to stay two+ hours will certainly be smaller near closing time than in the early hours of the festival. While our process is only an estimate, we do believe it provides a good start to determining the number of attendees at this year’s National Math Festival.

If you happen to be one of those individuals who attended and enjoyed the event, we hope you were able to keep your balloon-edged cube or tetrahedron safe from sharp corners, like those of your business card tetrahedra. Or perhaps you were able to meet famous mathematicians and science media experts. Either way, the festival will return in two years packed with math fun for all ages.

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¹ Since crowd counts cause so much controversy these days, we will leave this as just \(M\).

Blake Reichmuth is a master’s student in mathematical sciences at George Mason University. He is a member of the AMS, the MAA, and the AWM and SIAM student chapters. Ratna Khatri is an applied mathematics Ph.D. student at George Mason University. She is an active member of the university’s AWM and SIAM student chapters. Rachel Neville will receive her Ph.D. in applied topology from Colorado State University this June, and will begin her position as the Hanno Rund Postdoctoral Research Associate at the University of Arizona in the fall. She is chair of the Festivals Working Group of the SIAM Education Committee. Suzanne Weekes is professor of mathematical sciences at Worcester Polytechnic Institute. She is chair of the SIAM Education Committee.