{"id":26046621,"url":"https://github.com/ratwolfzero/emergent-dimension","last_synced_at":"2025-03-07T21:11:49.203Z","repository":{"id":276443143,"uuid":"929305406","full_name":"ratwolfzero/Emergent-Dimension","owner":"ratwolfzero","description":"Emergent Dimension","archived":false,"fork":false,"pushed_at":"2025-02-16T09:06:10.000Z","size":283,"stargazers_count":0,"open_issues_count":0,"forks_count":0,"subscribers_count":1,"default_branch":"main","last_synced_at":"2025-02-16T10:19:58.401Z","etag":null,"topics":["2d","2d-projections","3d","3d-sphere","boundary-field-theories","emergence","emergent-behavior","emergent-dimension","entanglement","holographic","holographic-principle","quantum-entanglement","quantum-mechanics","stereographic-projection"],"latest_commit_sha":null,"homepage":"https://github.com/ratwolfzero/Emergent-Dimension","language":"Python","has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":"mit","status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/ratwolfzero.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":"LICENSE","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":"2025-02-08T08:27:40.000Z","updated_at":"2025-02-16T09:06:13.000Z","dependencies_parsed_at":"2025-02-08T09:30:50.612Z","dependency_job_id":"04c99d99-e4bb-43c9-9310-80852126e194","html_url":"https://github.com/ratwolfzero/Emergent-Dimension","commit_stats":null,"previous_names":["ratwolfzero/emergent-dimension"],"tags_count":0,"template":false,"template_full_name":null,"repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/ratwolfzero%2FEmergent-Dimension","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/ratwolfzero%2FEmergent-Dimension/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/ratwolfzero%2FEmergent-Dimension/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/ratwolfzero%2FEmergent-Dimension/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/ratwolfzero","download_url":"https://codeload.github.com/ratwolfzero/Emergent-Dimension/tar.gz/refs/heads/main","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":242461458,"owners_count":20131984,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","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":["2d","2d-projections","3d","3d-sphere","boundary-field-theories","emergence","emergent-behavior","emergent-dimension","entanglement","holographic","holographic-principle","quantum-entanglement","quantum-mechanics","stereographic-projection"],"created_at":"2025-03-07T21:11:48.713Z","updated_at":"2025-03-07T21:11:49.195Z","avatar_url":"https://github.com/ratwolfzero.png","language":"Python","readme":"# How Stereographic Projection Can Demonstrate an Emergent Dimension?\n\n![Emergent_Dimension](emergent_dimension.png)\n\nThe **stereographic projection** provides a useful analogy for the concept of **emergent dimensions**, illustrating how higher-dimensional information can be encoded onto a lower-dimensional surface. While this is not a direct physical model, it offers a way to explore the idea that the **third dimension** of spacetime might not be fundamental but rather an **emergent property** arising from a more fundamental 2D description.\n\n## Stereographic Projection Overview\n\nThe stereographic projection maps points from a **3D sphere** onto a **2D plane**, showing how 3D information can be represented in a lower-dimensional space. This projection is reversible, meaning we can reconstruct the 3D sphere from the 2D projection.\n\n### 1. Stereographic Projection Formula\n\nFor a unit sphere \\( S^2 \\) centered at the origin in 3D space with coordinates \\( (x, y, z) \\), the **stereographic projection** onto the \\( xy \\)-plane is given by:\n\n$$\nx' = \\frac{x}{1 - z}, \\quad y' = \\frac{y}{1 - z}\n$$\n\nwhere \\( (x', y') \\) are the coordinates of the projected point on the 2D plane.\n\n### 2. Inverse Stereographic Projection\n\nTo recover the 3D coordinates from the 2D projection, we use the **inverse stereographic projection**:\n\n$$\nx = \\frac{2x'}{1 + x'^2 + y'^2}, \\quad y = \\frac{2y'}{1 + x'^2 + y'^2}, \\quad z = \\frac{1 - x'^2 - y'^2}{1 + x'^2 + y'^2}\n$$\n\nThis process demonstrates how **higher-dimensional data** can be reconstructed from a **lower-dimensional encoding**, mirroring the idea of emergent dimensions.\n\n## Demonstrating Emergence\n\nIn the context of the **holographic principle**, stereographic projection serves as an analogy for how a **3D system** could be encoded in a **lower-dimensional surface**. This is conceptually similar to how **spacetime** (including the third dimension) might emerge from a more fundamental **2D quantum description**.\n\nHowever, it is important to emphasize that this is only a **classical analogy**. The actual **holographic principle** in **quantum mechanics** involves **quantum entanglement** and **boundary field theories**, which go beyond the simple geometric mapping provided by stereographic projection.\n\n---\n\nBy examining **stereographic projection** in this way, we gain intuition for how **higher-dimensional structures** might emerge from **lower-dimensional encodings**, offering a visual representation of the **emergent dimension** concept in quantum gravity.\n","funding_links":[],"categories":[],"sub_categories":[],"project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fratwolfzero%2Femergent-dimension","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fratwolfzero%2Femergent-dimension","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fratwolfzero%2Femergent-dimension/lists"}