Me! On Numberphile! Who would’ve thought it?
Earlier this year, Brady Haran visited Newcastle to record a video with some Leverhulme scholars. Luckily for me he had a bit of spare time to record a video with me, so we did one about the Herschel enneahedron, which I first looked at back in 2013.
There were a few common questions among the comments on YouTube. I thought I’d quickly respond to them here.
“Can you make a version of it which is a fair 9-sided die? What about if the sides all have the same area?”
It can’t be a perfectly fair die, because it’s not an isohedron: no matter how you stretch it, there’s no symmetry that maps one of the squares to one of the kites. According to Persi Diaconis’s paper Fair Dice, only isohedra are always fair, no matter the surface you roll them on.
You can get the faces to all have the same area by setting the height of the top point to $\frac{3}{8\sqrt{2}}$, if the middle vertices are arranged on a unit circle. It’s a squishy shape!
“Can you draw the graph on paper so that the drawing has $D_6$ symmetry?”
Brady asked me this in the video and I hand-waved an answer of ‘no’. A few commenters explained that you can do it in a stereographic projection, putting one of the points at infinity, but not on paper.
“Can you make the kites into squares or rhombuses?”
No, they’re always kites. Looking down on the shape, the outline is an equilateral triangle, with the kites each made of one vertex of the triangle, two points halfway along the edges, and a point in the centre of the triangle. So even when the shape is squashed completely flat, these faces are still kites.
“Where can I get that hoodie?”
You can’t! I made it as a one-off for myself. But if anyone knows a print-on-demand sublimation printing service that won’t be loads of admin for me to sell through, I’m open to suggestions!
While reminding myself of the shape’s properties, I made a little interactive 3D visualisation, which you can play with too.