https://github.com/ratwolfzero/emergent-dimension
Emergent Dimension
https://github.com/ratwolfzero/emergent-dimension
2d 2d-projections 3d 3d-sphere boundary-field-theories emergence emergent-behavior emergent-dimension entanglement holographic holographic-principle quantum-entanglement quantum-mechanics stereographic-projection
Last synced: 3 months ago
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Emergent Dimension
- Host: GitHub
- URL: https://github.com/ratwolfzero/emergent-dimension
- Owner: ratwolfzero
- License: mit
- Created: 2025-02-08T08:27:40.000Z (4 months ago)
- Default Branch: main
- Last Pushed: 2025-02-16T09:06:10.000Z (4 months ago)
- Last Synced: 2025-02-16T10:19:58.401Z (4 months ago)
- 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
- Language: Python
- Homepage: https://github.com/ratwolfzero/Emergent-Dimension
- Size: 276 KB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# How Stereographic Projection Can Demonstrate an Emergent Dimension?
The **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.
## Stereographic Projection Overview
The 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.
### 1. Stereographic Projection Formula
For 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:
$$
x' = \frac{x}{1 - z}, \quad y' = \frac{y}{1 - z}
$$
where \( (x', y') \) are the coordinates of the projected point on the 2D plane.
### 2. Inverse Stereographic Projection
To recover the 3D coordinates from the 2D projection, we use the **inverse stereographic projection**:
$$
x = \frac{2x'}{1 + x'^2 + y'^2}, \quad y = \frac{2y'}{1 + x'^2 + y'^2}, \quad z = \frac{1 - x'^2 - y'^2}{1 + x'^2 + y'^2}
$$
This process demonstrates how **higher-dimensional data** can be reconstructed from a **lower-dimensional encoding**, mirroring the idea of emergent dimensions.
## Demonstrating Emergence
In 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**.
However, 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.
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By 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.