Vi Hart, Andrea Hawksley, Henry Segerman, Will Segerman, Marc ten Bosch

2013 – 2015

“Monkey See, Monkey Do” was an art installation at the 2015 Bridges mathematical art exhibition:

The viewer sees, in virtual reality, monkeys arranged into various symmetry groups in four dimensional space, stereographically projected and rotated in real time. You can see it too, at

monkey-see-monkey-do works for various VR headsets in webVR, and can also be explored in a normal browser using arrow keys and WASD. Clicking the screen cycles though various 4D symmetry groups with varying numbers of elements (monkeys, in this case). Number keys 1-6 will also go to various symmetry groups. This is one of our favourite webVR scenes for phones in a google cardboard or Wearality device.

The exhibition version included 3D-printed sculptures of some of these symmetrical arrangements, by Henry Segerman and Will Segerman. Rather than changing the VR visualizations by clicking or pressing number keys, we created three touch-sensitive polyhedra out of laser-cut pieces and copper tape, which represent the polyhedral cells of three different 4D symmetry groups. Touching one of the physical polyhedra switches the VR scene to the corresponding symmetry group.

Screen Shot 2017-04-12 at 1.09.57 PM

This art project grew from earlier explorations into the quaternion symmetry group and how it could be visualized, growing to different techniques as the technology becomes available. The mathematics and techniques behind the original 3D-printed monkey sculpture, demonstrating a stereographic projection of a theoretical 4D sculpture where 8 monkeys are arranged with quaternion group symmetry, can be found in our paper “The Quaternion Group as a Symmetry Group“.

When we started working with VR, Henry created a stereo rendering of this sculpture rotating in 4D, for use in VR Video. With Emily Eifler, we created the video 4DMonkey.gif to celebrate. (Related blog post:

What we really wanted, though, was to be able to rotate and render the projections live. So when we started working with webVR, was born!

The math behind the webVR project is similar to that in Hypernom, and described in the paper “Hypernom: Mapping VR Headset Orientation to S^3” (note: Monkeys doesn’t map headset orientation to S^3, but the stereographic projection process is the same).

The code is on github.