https://github.com/ratwolfzero/emergent-dimension
Emergent Dimension
https://github.com/ratwolfzero/emergent-dimension
2d 2d-projections 2d-to-3d 3d 3d-sphere 3d-to-2d boundary-field-theories emergence emergent-behavior emergent-dimension entanglement holographic holographic-principle quantum-entanglement quantum-mechanics stereographic-projection
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Emergent Dimension
- Host: GitHub
- URL: https://github.com/ratwolfzero/emergent-dimension
- Owner: ratwolfzero
- License: mit
- Created: 2025-02-22T16:07:36.000Z (over 1 year ago)
- Default Branch: main
- Last Pushed: 2025-05-14T18:49:12.000Z (about 1 year ago)
- Last Synced: 2025-05-14T19:37:59.559Z (about 1 year ago)
- Topics: 2d, 2d-projections, 2d-to-3d, 3d, 3d-sphere, 3d-to-2d, 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: 538 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 Reveals an Emergent Dimension
## Historical Context
Although its exact origin is uncertain, stereographic projection is strongly associated with ancient Greek astronomy.
## Stereographic Projection as an Analogy
The stereographic projection offers a fascinating analogy for understanding emergent dimensions — the idea that higher-dimensional information can be encoded in a lower-dimensional surface. While not a physical model, it helps visualize the idea that the third dimension of spacetime might be emergent — not fundamental — arising from a deeper 2D description.
## Stereographic Projection: A Gateway to Dimensional Mapping
This mathematical technique maps points from a 3D sphere onto a
2D plane, demonstrating how three-dimensional data can be represented in fewer dimensions. Crucially, the projection is reversible — meaning the original 3D structure can be perfectly reconstructed from its 2D counterpart.
### Projecting 3D onto 2D
For a unit sphere **S²** 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.
### Reconstructing 3D from 2D
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 reversibility illustrates how higher-dimensional information can be encoded and decoded in a lower-dimensional framework — an essential concept in theories of emergent spacetime.
## From Math to Physics: The Emergent Dimension Analogy
The stereographic projection serves as a classical geometric analogy for the holographic principle, which suggests that our 3D universe might be encoded on a 2D boundary. However, this is just an illustration — the real holographic principle in quantum gravity involves far deeper mechanisms, such as:
- Quantum entanglement
- Boundary conformal field theories
- Non-local information encoding
Still, the stereographic projection provides valuable intuition for how extra dimensions could emerge from simpler structures.
### Why This Matters
By studying such projections, we gain insight into how:
- Higher-dimensional physics might arise from lower-dimensional rules.
- Spacetime itself could be an emergent phenomenon.
- Quantum gravity theories (like AdS/CFT*) use holographic encoding.
*Anti-de Sitter/Conformal Field Theory
While the full picture requires advanced quantum theory, the stereographic projection offers a beautiful geometric gateway into these profound ideas.