We provide an open source C++ implementation which can be used to generate all unfoldings of the $n$-dimensional hypercube. Those unfoldings correspond to pairs $(T, P)$, where $T$ is a tree on $2n$ nodes and $P$ is a perfect matching in the complement of the tree. Two such pairs $(T, P)$ and $(T', P')$ are considered equivalent if there is an isomorphism between $T$ and $T'$ which is simultaneously an isomorphism between $P$ and $P'$.

This correspondence is explained in Unfolding the Tesseract by Peter Turney.

We rely on nauty by Brendan McKay and Adolfo Piperno to generate the trees and their automorphism group.

The code is used to calculate the terms of A091159 up to dimension 10; see below for the results.

Check out a video by Matt Parker on unfolding the 4d-cube.

Use ./compile.sh to compile the source code. You can then use ./test.sh to run tests, or ./compute.sh N to run the computation for a given value of N.

Compiling the code requires nauty and gtest. Running requires the nauty-gtreeng executable, as well as parallel.

To install these dependencies on a Debian-based system, you can run

sudo apt-get install libnauty2-dev nauty libgtest-dev parallel

For each $n \in {2, 3, 4, 5, 6, 7, 8, 9, 10}$ we provide a file n.cnt.txt which contains in each line a decimal number and a graph given in sparse6 format, separated by a space.

[number] [graph6 string]

For example

11704 :M`ESYOl]sLZt

The graph is a tree on $2n$ vertices and the number counts the perfect matchings in the complement of the tree up to transformations of the automorphism group of the tree. In other words, the number tells us how many different unfoldings of the hypercube in dimension $n$ exist that correspond to a certain tree. Some trees in the files don't have any corresponding unfoldings, namely the stars. They are nonetheless listed here with a count of zero.

To get the number of hypercube unfoldings we can add up all the numbers.

Update: A follow-up project uses a much better algorithm by Alex Gunning to compute many more terms in this sequence. See the blog post Calculating the number of nets of hypercubes for details.

Authors: Moritz Firsching and Luca Versari.