{"id":20640127,"url":"https://github.com/maresb/ice-residual-entropy","last_synced_at":"2026-03-09T17:38:18.395Z","repository":{"id":107327125,"uuid":"456284335","full_name":"maresb/ice-residual-entropy","owner":"maresb","description":"Explore the residual entropy of ice","archived":false,"fork":false,"pushed_at":"2022-03-24T21:05:53.000Z","size":321,"stargazers_count":0,"open_issues_count":0,"forks_count":0,"subscribers_count":1,"default_branch":"master","last_synced_at":"2025-12-09T01:29:43.147Z","etag":null,"topics":[],"latest_commit_sha":null,"homepage":null,"language":"Mathematica","has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":null,"status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/maresb.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":null,"code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null}},"created_at":"2022-02-06T22:06:12.000Z","updated_at":"2025-01-03T11:18:54.000Z","dependencies_parsed_at":null,"dependency_job_id":"013220d3-7d5f-468a-a257-9250c51eb1bc","html_url":"https://github.com/maresb/ice-residual-entropy","commit_stats":null,"previous_names":[],"tags_count":0,"template":false,"template_full_name":null,"purl":"pkg:github/maresb/ice-residual-entropy","repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/maresb%2Fice-residual-entropy","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/maresb%2Fice-residual-entropy/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/maresb%2Fice-residual-entropy/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/maresb%2Fice-residual-entropy/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/maresb","download_url":"https://codeload.github.com/maresb/ice-residual-entropy/tar.gz/refs/heads/master","sbom_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/maresb%2Fice-residual-entropy/sbom","scorecard":null,"host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":286080680,"owners_count":30304807,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2026-03-09T17:35:44.120Z","status":"ssl_error","status_checked_at":"2026-03-09T17:35:43.707Z","response_time":61,"last_error":"SSL_connect returned=1 errno=0 peeraddr=140.82.121.5:443 state=error: unexpected eof while reading","robots_txt_status":"success","robots_txt_updated_at":"2025-07-24T06:49:26.215Z","robots_txt_url":"https://github.com/robots.txt","online":false,"can_crawl_api":true,"host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":[],"created_at":"2024-11-16T15:28:03.457Z","updated_at":"2026-03-09T17:38:18.349Z","avatar_url":"https://github.com/maresb.png","language":"Mathematica","funding_links":[],"categories":[],"sub_categories":[],"readme":"# Residual entropy of ice\n\n## Links\n\n- [Live webpage on GitLab Pages](https://bmares.gitlab.io/ice-residual-entropy)\n- [GitLab repository](https://gitlab.com/bmares/ice-residual-entropy)\n- [GitHub repository](https://github.com/maresb/ice-residual-entropy)\n\n## [Click for interactive model:](https://bmares.gitlab.io/ice-residual-entropy)\n\n[\u003cimg align=\"center\" src=\"ice.png\"\u003e](https://bmares.gitlab.io/ice-residual-entropy)\n\n- [Live webpage on GitLab Pages](https://bmares.gitlab.io/ice-residual-entropy)\n\n## Description\n\nI'm interested in exploring [ice-type models](https://en.wikipedia.org/wiki/Ice-type_model) and their residual entropy.\n\nThere are many various [phases of ice](https://en.wikipedia.org/wiki/Ice#Phases). Currently, I have a visualization of (the oxygen atoms of) [Ice Ih](https://en.wikipedia.org/wiki/Ice_Ih), which is ordinary ice, [Ice Ic](https://en.wikipedia.org/wiki/Ice_Ic) and [square ice](https://journals.aps.org/pr/abstract/10.1103/PhysRev.162.162).\n\n\n## Notes\n\nThe oxygen atoms of each of the ice type have a crystal structure. The \"ice rule\" applies to each ice type because the underlying graph of neighboring oxygen atoms is 4-valent (each oxygen atom has 4 neighboring oxygen atoms).\n\nSince the 3D crystal structure does not involve the hydrogen, equivalent crystal structures are more often discussed in the context of different atoms in a 4-valent arrangement. For example, the oxygen atoms in cubic ice are arranged in a [diamond cubic structure](https://en.wikipedia.org/wiki/Diamond_cubic).\n\n\nEach of the underlying graphs happen to be bipartite (neighboring oxygen atoms can be given alternating colors).\n\nThe structure of the underlying graphs can be distinguished by the respective [coordination sequences](https://en.wikipedia.org/wiki/Coordination_sequence), whose _n_-th element is the number of vertices which are a distance of _n_ steps from a chosen vertex. These are:\n\n- Square: 1, 4, 8, 12, 16, 20, 24, ... ([A008574](https://oeis.org/A008574))\n- Cubic (diamond): 1, 4, 12, 24, 42, 64, 92, ...([A008253](https://oeis.org/A008253))\n- Hexagonal (lonsdaleite): 1, 4, 12, 25, 44, 67, 96, ... ([A008264](https://oeis.org/A008264))\n\n(Each coordination sequence begins with 1, 4, since the graphs are all 4-valent.)\n\nIt is also interesting to consider the [theta functions](https://en.wikipedia.org/wiki/Theta_function_of_a_lattice), which are the generating functions for the number of points of a given Euclidean squared-distance from a chosen point. These are:\n\n- Square: = 1 + 4 q + 4 q^2 + 4 q^4 + 8 q^5 + 4 q^8 + 4 q^9 + 8 q^10 + ... =  _θ₃(q)²_ ([Jacobi theta](https://en.wikipedia.org/wiki/Theta_function#Auxiliary_functions))\n- Cubic (diamond): 1 + 4 q^9 + 12 q^24 + 12 q^33 + 6 q^48 + 12 q^57 + 24 q^72 + ...\n- Hexagonal (lonsdaleite): 1 + 4 q^9 + 12 q^24 + q^25 + 9 q^33 + 6 q^48 + 6 q^49 + 9 q^57 +  2 q^64 + 18 q^72 +  + 9 q^81 + 12 q^88 + 3 q^89 + 6 q^96 + 6 q^97 + 18 q^105 + 3 q^113 + 12 q^120 + 7 q^121 + 3 q^129 + 12 q^136 + 6 q^137 + 6 q^144 + 6 q^145 + 6 q^152 + 12 q^153 + 12 q^160 + 24 q^168 + q^169 ...\n\nFormulas for the theta functions for diamond and lonsdaleite structures were worked out by [Sloane](http://neilsloane.com/doc/Me137.pdf). Diamond is ½(θ₂(q¹²)³ + θ₃(q¹²)³ + θ₄(q¹²)³, while lonsdaleite has a rather messy formula, Eq. (20).\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fmaresb%2Fice-residual-entropy","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fmaresb%2Fice-residual-entropy","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fmaresb%2Fice-residual-entropy/lists"}