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.
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.
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:
where ( (x', y') ) are the coordinates of the projected point on the 2D plane.
To recover the 3D coordinates from the 2D projection, we use the inverse stereographic projection:
This process demonstrates how higher-dimensional data can be reconstructed from a lower-dimensional encoding, mirroring the idea of emergent dimensions.
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.
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.