{"id":24158122,"url":"https://github.com/ulizesr/hopf-fibrations","last_synced_at":"2026-05-03T11:32:00.231Z","repository":{"id":223739806,"uuid":"727624175","full_name":"UlizesR/Hopf-Fibrations","owner":"UlizesR","description":"A simple website to visualize the Hopf Fibration, the mapping of S3-\u003eS2-\u003eR3 ","archived":false,"fork":false,"pushed_at":"2024-12-05T20:31:36.000Z","size":66,"stargazers_count":0,"open_issues_count":0,"forks_count":0,"subscribers_count":1,"default_branch":"main","last_synced_at":"2025-03-02T01:29:50.643Z","etag":null,"topics":["hopf-fibration","mathematics","topology","typescript","visualization","vite"],"latest_commit_sha":null,"homepage":"https://ulizesr-hopf-fibrations.vercel.app","language":"TypeScript","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/UlizesR.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":"2023-12-05T08:40:08.000Z","updated_at":"2024-12-05T20:31:40.000Z","dependencies_parsed_at":"2025-03-02T01:28:59.653Z","dependency_job_id":"9ce5d80c-e3fd-49d4-8e4d-38c0fe868c86","html_url":"https://github.com/UlizesR/Hopf-Fibrations","commit_stats":null,"previous_names":["ulizesr/hopf-fibrations"],"tags_count":0,"template":false,"template_full_name":null,"purl":"pkg:github/UlizesR/Hopf-Fibrations","repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/UlizesR%2FHopf-Fibrations","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/UlizesR%2FHopf-Fibrations/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/UlizesR%2FHopf-Fibrations/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/UlizesR%2FHopf-Fibrations/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/UlizesR","download_url":"https://codeload.github.com/UlizesR/Hopf-Fibrations/tar.gz/refs/heads/main","sbom_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/UlizesR%2FHopf-Fibrations/sbom","scorecard":null,"host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":286080680,"owners_count":32567187,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2026-05-03T06:36:36.687Z","status":"ssl_error","status_checked_at":"2026-05-03T06:36:09.306Z","response_time":103,"last_error":"SSL_read: 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":["hopf-fibration","mathematics","topology","typescript","visualization","vite"],"created_at":"2025-01-12T14:19:47.832Z","updated_at":"2026-05-03T11:32:00.198Z","avatar_url":"https://github.com/UlizesR.png","language":"TypeScript","funding_links":[],"categories":[],"sub_categories":[],"readme":"# Hopf Fibration Visualizer\n\nWelcome to the Hopf Fibration Visualizer! This website allows you to explore and visualize the Hopf Fibration, a fundamental concept in topology and differential geometry.\n\n## About the Hopf Fibration\n\nThe Hopf Fibration, discovered as the first example of a map from a higher-dimensional sphere to a lower-dimensional sphere that is not null-homotopic, was a groundbreaking revelation in mathematics. Contrary to the prevailing belief at the time, the Hopf Fibration demonstrated that such mappings exist, challenging the analogy with homology groups.\n\nThe Hopf map $f: S^3 \\rightarrow S^2$ arises in various contexts and can be generalized to a map $S^7 \\rightarrow S^4$. For any point $p$ in the sphere $S^2$, its preimage $f^{-1}(p)$ is a circle $S^1$ in $S^3$. Descriptions of the Hopf map, also known as the Hopf fibration, abound.\n\n## Mathematical Representation\n\nThe 3-sphere, $S^3$, defined as a submanifold of $\\mathbb{R}^4$, and the 2-sphere, $S^2$, a submanifold of $\\mathbb{R}^3$, are mathematically represented as:\n\n$S^3 = \\{(X_1, X_2, X_3, X_4) : X_1^2 + X_2^2 + X_3^2 + X_4^2 = 1\\}$\n\n$S^2 = \\{(x_1, x_2, x_3) : x_1^2 + x_2^2 + x_3^2 = 1\\}$\n\nThe Hopf map transforms points $(X_1, X_2, X_3, X_4)$ on $S^3$ to points $(x_1, x_2, x_3)$ on $S^2$ according to the equations:\n\n$x_1 = 2(X_1X_2 + X_3X_4)$\n$x_2 = 2(X_1X_4 - X_2X_3)$\n$x_3 = X_1^2 + X_3^2 - X_2^2 - X_4^2$\n\n## Usage\n\nTo use the Hopf Fibration Visualizer, simply visit the following URL:\n\n[https://hopf-fibrations-4vwvchlou-ulizesr.vercel.app](https://hopf-fibrations-4vwvchlou-ulizesr.vercel.app)\n\n## Feedback and Contributions\n\nFeedback and contributions are highly appreciated! If you encounter any issues, have suggestions for improvements, or want to contribute to the project, feel free to open an issue or submit a pull request.\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fulizesr%2Fhopf-fibrations","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fulizesr%2Fhopf-fibrations","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fulizesr%2Fhopf-fibrations/lists"}