# How many 3D nets does a 4D hypercube have?

Get to know Jane Street! https://www.janestreet.com/join-jane-street/get-to-know-us/

Which hypercube unfoldings tile space? https://whuts.org/

Yes, you can buy one of the 261 models from this video and support the channel. All hand-numbered by me and with a signed certificate of authenticity. https://mathsgear.co.uk/products/cube-model

Here is Giovanna Diaz and Joseph O’Rourke’s paper: https://arxiv.org/abs/1512.02086

A091159 Number of distinct nets for the n-hypercube. https://oeis.org/A091159

All of Moritz Firsching’s 3D models: https://mo271.github.io/mo/198722/unfoldings.html

Their post on the number of unfoldings in higher dimensions. https://mathoverflow.net/questions/300713/number-of-hypercube-unfoldings

This is the Math Overflow post which started it all: https://mathoverflow.net/questions/300713/number-of-hypercube-unfoldings

Peter Turney’s 1984 paper Unfolding the Tesseract https://unfolding.apperceptual.com/

The cubes I am using are called "mathlink" and I just bought a huge quantity from Amazon (because Learning Resources didn’t answer my emails).
US: https://amzn.to/3bsWOZd
UK: https://amzn.to/3offAsr

The unfolding animation of the ‘Dali cross’ was kindly made by my Patreon supporter John Sawyer.

I actually put the rough-cut of this video out on Patreon earlier this week so they could provide feedback and help test the whuts.org site. Thanks so much for all of your help everyone! https://www.patreon.com/posts/51144440

CORRECTIONS:
– I saw "288" at the end of the 8D number when it should be "228". The on-screen number is correct. I noticed too late to fix it!
– At 21:09 I say Diaz and O’Rourke found an unfolding of the Dali cross which tiles the plane. It’s actually a different 3D net they found which does this and the Dali is undetermined if it produces a tiling 2D net. (Thanks Dan L by email.)
– Let me know if you spot any more mistakes!

Filming and editing by Alex Genn-Bash
Maths graphics by Matt Parker
Music by Howard Carter