Sandpile monomorphism and harmonic dynamics

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


The abelian sandpile model is a cellular automaton defined on a domain of the standard square lattice. It serves as the archetypical model for self-organized criticality, and important process in physics, biology, geology and other fields. The recurrent configurations of the sandpile model form an abelian group, the sandpile group. This website summarizes our results on this group.

Harmonic sandpile dynamics

The identity of the sandpile group shows smooth 1-periodic dynamics induced by harmonic fields. To analyze these harmonic dynamics, we provide 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.

Sandpile monomorphisms

The concept of criticality is strongly related to scale-invariance, i.e. certain properties of a critical system, like the distribution of avalanche sizes for the sandpile model, remain qualitatively the same under spatial-temporal scaling. We recently showed that also the sandpile group itself is susceptible to such scaling relationships in the form of families of monomorphisms from each sandpile group on an NxM rectangular domain to all sandpile groups on s(N+1)-1xs(M+1)-1 domains.

Videos and Software

To access the videos showing the harmonic sandpile dynamics, visit our Videos section. To test the sandpile monomorphisms or create own videos of custom harmonic sandpile dynamics, download InterPile.

Learn more

To learn more about the harmonic sandpile dynamics, read our PNAS paper on this topic. For the sandpile monomorphisms, we refer to our recent preprint.


30.04.2019 – Preprint online

The preprint of our new article on families of sandpile monomorphisms between each NxM rectangular domain and all s(N+1)-1 x s(M+1)-1 domains went online today. Check it out at

Read More

28.04.2019 – New beta and upcoming paper

We released beta 4 of Interpile 1.1, just in time for our upcoming paper. The main new functionality is that the families of sandpile monomorphisms from every NxM domain to all s(N+1)-1xs(M+1)-1 domains are now completely implemented and bug free (hopefully, at least, that's why it's still only a beta). What are these sandpile monomorphisms we are talking about? Check the theoretical background in our upcoming paper "Sandpile monomorphisms and harmonic functions" from Moritz Lang and Mikhail Shkolnikov, which should be available at arXive at Tuesday, 04/30/2019. We keep you updated as soon as the paper is online, and we have the full reference!

Read More

27.02.2019 – New beta and more press

We released beta 3 of Interpile 1.1, which now includes the first glimpse on some new functionality: unique coordinates identifying recurrent configurations, as well as group homomorphisms between sandpile groups on certain NxN domains. All that is however yet pretty preliminary, extremely slow and potentially even buggy...well, it's only a first glimpse. Furthermore, there was some more press coverage of our PNAS paper in,, and

Read More

14.02.2019 – Press coverage

The APA (Austria Presse Agentur) has covered the release of our PNAS paper in a small article. Check it out under (in German).

Read More

06.02.2019 – Paper out

Our paper on the harmonic sandpile dynamics is now officially online at PNAS, and can be accessed under

Read More

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 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

Harmonic sandpile dynamics:

Moritz Lang and Mikhail Shkolnikov (2019). Harmonic dynamics of the abelian sandpile. Proceedings of the National Academy of Sciences (online first). DOI: 10.1073/pnas.1812015116

BibTeX  View

Sandpile monomorphisms:

Moritz Lang and Mikhail Shkolnikov (2019). Sandpile monomorphisms and harmonic functions. ArXiv preprint arXiv:1904.12209.


Download InterPile

All options:

Download page