Skip to main content
Home
plus.maths.org

Secondary menu

  • My list
  • About Plus
  • Sponsors
  • Subscribe
  • Contact Us
  • Log in
  • Main navigation

  • Home
  • Articles
  • Collections
  • Podcasts
  • Maths in a minute
  • Puzzles
  • Videos
  • Topics and tags
  • For

    • cat icon
      Curiosity
    • newspaper icon
      Media
    • graduation icon
      Education
    • briefcase icon
      Policy

      Popular topics and tags

      Shapes

      • Geometry
      • Vectors and matrices
      • Topology
      • Networks and graph theory
      • Fractals

      Numbers

      • Number theory
      • Arithmetic
      • Prime numbers
      • Fermat's last theorem
      • Cryptography

      Computing and information

      • Quantum computing
      • Complexity
      • Information theory
      • Artificial intelligence and machine learning
      • Algorithm

      Data and probability

      • Statistics
      • Probability and uncertainty
      • Randomness

      Abstract structures

      • Symmetry
      • Algebra and group theory
      • Vectors and matrices

      Physics

      • Fluid dynamics
      • Quantum physics
      • General relativity, gravity and black holes
      • Entropy and thermodynamics
      • String theory and quantum gravity

      Arts, humanities and sport

      • History and philosophy of mathematics
      • Art and Music
      • Language
      • Sport

      Logic, proof and strategy

      • Logic
      • Proof
      • Game theory

      Calculus and analysis

      • Differential equations
      • Calculus

      Towards applications

      • Mathematical modelling
      • Dynamical systems and Chaos

      Applications

      • Medicine and health
      • Epidemiology
      • Biology
      • Economics and finance
      • Engineering and architecture
      • Weather forecasting
      • Climate change

      Understanding of mathematics

      • Public understanding of mathematics
      • Education

      Get your maths quickly

      • Maths in a minute

      Main menu

    • Home
    • Articles
    • Collections
    • Podcasts
    • Maths in a minute
    • Puzzles
    • Videos
    • Topics and tags
    • Audiences

      • cat icon
        Curiosity
      • newspaper icon
        Media
      • graduation icon
        Education
      • briefcase icon
        Policy

      Secondary menu

    • My list
    • About Plus
    • Sponsors
    • Subscribe
    • Contact Us
    • Log in
    • Maths in a minute: Social distancing

      27 March, 2020
      15 comments

      Imagine lots of people in a park, sitting in the Sun. Each person wants to keep at least 2m distance to everyone else, but they want to have as many people as possible at the minimum 2m distance. How should the people arrange themselves to make this happen? And what's the number of people a person will have exactly 2m away from themselves in this arrangement?

      See here for all our coverage of the COVID-19 pandemic.

      Let's start with a single person whom we will call person A. People who want to be 2m away from person A should position themselves somewhere on the circle that has A as its centre and a radius of 2. Let's look at two people on this circle, B and C. Since B and C also want to be 2m away from each other, the triangle formed by A, B and C should be an equilateral triangle with side length 2.

      Equilateral triangle and circle

      How many such triangles can you fit around A so that they don't overlap? Well, the angles in an equilateral triangle are all 60 degrees. There are a total of 360 degrees available around A, which means that you can fit exactly 360/60=6 equilateral triangle around A. In other words, the number of people that can position themselves 2m away from A and also be 2m from each of their two neighbours on the circle is six.

      Hexagon on circle

      Now exactly the same argument holds for any other person on A's circle. The number of its closest neighbours, exactly 2m away is six, and they should be positioned around the person in the shape of a regular hexagon made up of six equilateral triangles.

      Two hexagons

      Since this works for every person the answer is that people should arrange themselves on a triangular lattice made up of equilateral triangles. Every person in the lattice is also at the centre of a regular hexagon.

      Triangular grid

      • Log in or register to post comments

      Comments

      Terence Mills

      30 March 2020

      Permalink

      I infer from this article that a safe distance in the UK is 2m; in Australia it is 1.5m; in Singapore it is 1m. Where do these numbers come from?

      • Log in or register to post comments

      Evie Winters

      18 May 2020

      In reply to Social distancing by Terence Mills

      Permalink

      Well, all those distances are what’s considered okay in those specific countries. I live in America, where we don’t really use metres, and we stay 6 feet apart, or roughly two metres. It’s only because that’s what the government of that country thinks is safe.

      • Log in or register to post comments

      Anthony

      30 March 2020

      Permalink

      This works when everything is static, but only those on the perimeter can leave. I wonder what the densest configuration would look like where anyone can leave while still maintaining a minimum distance from all others.

      • Log in or register to post comments

      Martin Douglas

      30 May 2020

      In reply to This works when everything is by Anthony

      Permalink

      Purely from the visual above I'd say take out every 3rd diagonal, like // // //, and there'd be a corridor for movement from either side, always 2m. Sorry, can't do the maths.

      • Log in or register to post comments

      python

      31 March 2020

      Permalink

      great post . but it is not realistic as it is not time varying .And doesn’t count physical obstruction such as building, road ect…. can you include this factors and post more realistic model for social distancing .thanks.

      • Log in or register to post comments

      Paul Prebble

      14 April 2020

      Permalink

      It's all very well having a 2m lattice but shouldn't it also ensure that the people's heads are on the vertices and how people would get to and from their spot? The latter would reduce the total number by a third.

      • Log in or register to post comments

      Soh

      3 May 2020

      Permalink

      In a 4x4 square, you can position 9 people at least 2m apart, but if we adopt this arrangement of equilateral triangles, you get only 8. Right!

      • Log in or register to post comments

      shane lilly

      4 May 2020

      In reply to Not quite right by Soh

      Permalink

      a 4x4 square takes up 16 square meters of area where as the hexagon with 7 people takes up less area. 10.4 square meters.
      comparing space per person that would be 1.77 square meters per person for the square vs 1.48 square meters for the hexagon.

      • Log in or register to post comments

      Mark99504

      17 May 2020

      In reply to Not quite right by Soh

      Permalink

      Got a 4x4 area, your arrangement fits 9, while triangles fit 8. Got an 8x8 area, squares fits 25, triangles fits 24. Got 12x12? squares 49, triangles 50. 16x16? squares 81, triangles 91 (I think).

      Each time you add more people past 9 or 8, you have to position them far enough away from everybody else. As the number of people increases, triangles take less area to achieve adequate distance for each person.

      • Log in or register to post comments

      Math Teacher

      22 May 2020

      In reply to Not quite right by Soh

      Permalink

      Yes, but in a 7x7 square, triangles allows for 20 people, while squares only allow for 16.

      • Log in or register to post comments

      RichCatt

      28 May 2020

      In reply to Not quite right by Soh

      Permalink

      A 4x4 square can take 9 people in an area of 16 sq. m, and the triangles can take 10 people in an area of 17.32 (10 x sqrt 3) sq. m, so the comparison depends on the shape we are trying to fill.

      • Log in or register to post comments

      Kittwit

      4 June 2020

      In reply to Not quite right by Soh

      Permalink

      In the given scenario of a large park with no discernable edges this solution would be efficient. The problems start to occur in smaller spaces when boundaries such as walls are present. In these situations there would be a lot of space lost at the edges with hexagonal packing and the conventional square packing is often more efficient. The edge effect becomes less of an issue above about 50 individuals though as the lost space becomes relatively less. So in a wide space with more than 50 people use hexagonal packing. If you're in a room a square arrangement is probably better.

      • Log in or register to post comments

      John Franklin

      14 June 2020

      In reply to Not quite right by Soh

      Permalink

      actually, in a 4x4 square not everyone is 2m apart. the person in the middle is <2m from those on the corners, I believe they are sqrt2 meters away.

      • Log in or register to post comments

      Nontokozo Mkhwamubi

      18 August 2020

      In reply to Not quite right by Soh

      Permalink

      Hi I would like know how did you figure out 9, which method or approach did you use?

      • Log in or register to post comments

      Willy

      5 January 2021

      Permalink

      Fascinating.

      • Log in or register to post comments

      Read more about...

      geometry
      tiling
      lattice tiling
      tessellation
      Maths in a minute
      University of Cambridge logo

      Plus Magazine is part of the family of activities in the Millennium Mathematics Project.
      Copyright © 1997 - 2025. University of Cambridge. All rights reserved.

      Terms