Video demonstrating the successive action of four harmonic fields on the Abelian sandpile identity.
More videos

Abelian sandpiles and their harmonic dynamics

The identity of the Abelian sandpile group shows smooth 1-periodic dynamics induced by harmonic fields. This website summarizes results on the properties of these dynamics, and also provides an open-source software solution called InterPile to interactively explore and manipulate sandpiles on various domains, as well as to generate movies showing the dynamics of the sandpile identity or any other recurrent configuration induced by arbitrary harmonic functions and for arbitrarily defined domains.

Videos and Software

To access the videos showing the sandpile identity dynamics, visit our Videos section. To create own videos showing the sandpile identity dynamics, download InterPile.

Learn more

To learn more about the Abelian sandpile identity dynamics, read our preprint on this topic.


26.07.2018 – First Domain Transformation Movies

With the first beta of Interpile v1.1, we also uploaded the first movies of a new category, to which we refer as domain transformation. In these movies, the domain is e.g. slowly transformed from a square into a circle. Enjoy!

Read More

24.07.2018 – First InterPile Release

After a lot of testing, we now finally released the first version of Interpile (v1.0). There are only minor changes as compared to the last beta, and mainly bugs were fixed.

Read More

12.07.2018 – First Domain Scaling Movies and new Release of InterPile

The first videos are online showing configurations corresponding to the sandpile identity dynamics induced by a given harmonic fields at the same or scaled times for increasing domain sizes. This visualizes both the capability of our approach for renormalization, as well as the super-position of different time scales. We also released a new beta of InterPile (beta 3), with which these videos can be easily generated. Feel free to send us your best videos!

Read More

10.07.2018 – Video of fifth order harmonics

We uploaded a bunch of additional videos, including the first video showing fifth-order harmonic sandpile identity dynamics. The other videos mainly show the sandpile dynamics for lower order harmonics when starting with some interesting fractal patterns found in the videos of certain harmonics.

Read More

02.07.2018 – Even more videos

We uploaded even more videos, most of which are showing the effect when combining different harmonic fields. We also created a "promotional video" to illustrate how the different dynamics play together, called A roundtrip through the sandpile group, which can be seen on the start page of this website.

Read More

29.06.2018 – Manuscript and more videos

The preprint of our manuscript just went online, and can be accessed under https://arxiv.org/abs/1806.10823. We also uploaded several additional videos of sandpile identity dynamics, which can be accessed in the videos section of this webpage.

Read More

28.06.2018 – First videos online

The first videos showing the sandpile identity dynamics induced by harmonic fields for square and circular domains are online. Videos showing the dynamics on other domains will follow soon!

Read More

20.06.2018 – New Homepage

In preparation for the publication of the preprint of our research on the harmonic dynamics of the Abelian sandpile identity, we created this homepage. The current version of the open-source software InterPile used to generate these dynamics is already available for download. Videos showing the sandpile identity dynamics induced by different harmonics will be made available soon.

Read More

Mathematical Background

For the mathematical background on the sandpile identity dynamics, please read:

Moritz Lang and Mikhail Shkolnikov (2018). Harmonic dynamics of the Abelian sandpile. ArXiv preprint arXiv:1806.10823.

BibTeX  View

Download InterPile

All options:

Download page