{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/four-cubes","title":"Four Cubes","arxiv_id":"2007.03782","date":"2020-07-07","proceeding":null,"authors":["Szymon Łukaszyk"],"abstract":"A short survey on the properties of four graphs constructed in $\\{0, 1\\}^n$ Boolean space is presented. Flexible activation function of an artificial neuron in a sparse distributed memory model is defined on the basis of the Ugly duckling theorem. Cotan Laplacian on 2-face triangulation of $n$-cube has degenerate spectrum of eigenvalues corresponding to the Hamming distance distribution of $\\{0, 1\\}^n$ space. Degenerate spectrum of eigenvalues of the cotan Laplacian defined on the graph comprising $2^n$ 2-face triangulated $n$-cubes sharing common origin includes all integers from 0 to 3$n$, without the eigenvalue of 3$n$-1 (multiplicities of the same eigenvalues form A038717 OEIS sequence), while the multiplicities of the same eigenvalues $[-n\\sqrt{2}, n\\sqrt{2}]$ of the adjacency matrix of $2^n$-cube form trinomial triangle. The distance matrix of this graph, providing further OEIS sequences, as well as its relation with Buckminster Fuller vector equilibrium is also discussed.","url_abs":"https://arxiv.org/abs/2007.03782v5","url_pdf":"https://arxiv.org/pdf/2007.03782v5.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"four-cubes","repo_url":"https://github.com/szluk/FourCubes","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}