HDCP

This is a database that contains the critical configurations of hard disks on the hexagonal torus, where the number of hard disks is $n = 2 \ldots 12$. A critical configuration is one where disk contacts prevent any collective motion that would allow the radius of all disks to increase linearly with the extent of the motion. Equivalently, a critical configuration is one that has a mechanically-balanced contact graph. The torus is the regular hexagon of edge length $1 / \sqrt{3}$ with opposite edges identified.

Security Statement

Connecting to a Jupyter notebook server running on your local machine can provide many benefits. With these benefits come serious potential risks. By connecting to a local runtime, you are allowing the Colaboratory frontend to execute code in the notebook using the local resources on your machine. This means that the notebook could:

• Invoke arbitrary commands (e.g. "rm -rf /")
• Access the local file system
• Run malicious content on your machine

Before attempting to connect to a local runtime, make sure you trust the authors of the notebook and ensure you understand what code is being executed. For more information on the Jupyter notebook server's security model, consult Jupyter's documentation.

Usage

The capabilities of this notebook are contained in two functions, import_database and plot_crit. These are defined in the Define necessary functions cell (double clicking expands the cell if you would like to examine the code). Run this cell first.

The next cell imports the database of critical points for the desired number of disks, passed as the single argument to import_database. Critical point databases are available for $n=2 \ldots 12$. Running this cell displays the (truncated) table, which you can search for particular values of the index or radius by pressing the Filter button. The truncated columns contain information about the positions of the disks and the contacts between the disks.

The next cell displays the adjacency matrix and disk configuration of a selected critical point, passed as the second argument to plot_crit. The value of this argument must be an integer that appears in the crit_number column of the preceeding data table.

For example, to plot the second critical point for seven disks:

• Run the Define necessary functions cell (only needs to be done once)
• Change the argument of import_database in the next cell to 7 and run the cell
• Change the second argument of plot_crit in the next cell to 2 and run the cell

References

Research into the topology of the configuration space of hard disks is ongoing, but the references below provide some context for this project.

• G. Carlsson, J. Gorham, M. Kahle, J. Mason, Computational topology for configuration spaces of hard disks, Phys. Rev. E 85 (2012): 011303.
• Y. Baryshnikov, P. Bubenik, M. Kahle, Min-type Morse theory for configuration spaces of hard spheres, International Mathematics Research Notices 2014.9 (2014): 2577–2592.
• H. Alpert, M. Kahle, R. MacPherson, Configuration spaces of disks in an infinite strip, arXiv:1908.04241.

Contributors

The contributors to this project are (by alphabetical order of last name):

• Ozan Ericok ([email protected])
• Matthew Kahle ([email protected])
• Jeremy Mason ([email protected])
• Katherine Ritchey ([email protected])

Please contact Matthew Kahle ([email protected]) or Jeremy Mason ([email protected]) for more information.

License

The contents of this repository are licenced under the GNU General Public License, Version 3.

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. 1839370. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.