4D Spatial Folding Theory

Recursive Multi-Dimensional Graphing via Spatial Folding

A Scalable Framework for Dimensional Compression and Geodesic Shortcuts
By Alissa M.R. Eldridge
Published under the Sci-Real Division of Reality Builders Entertainment Works (RBEW)

Abstract

This paper presents a scalable, recursive framework for multi-dimensional graphing through spatial folding. The method enables higher-dimensional space to be constructed, visualized, and mathematically operated upon using stacks of 3D spatial graphs. This recursive system introduces a warping transformation modulated by gravitational potential and relativistic dilation, allowing for geodesic path compression and effective traversal shortcuts without violating physical laws.

While the 4D application is used for demonstration, this model extends infinitely—supporting dimensional expansion to 5D, 6D, and beyond. Each new dimension is built from grouped structures of the previous one, establishing a modular method for navigating complex geometric realities.

1. Introduction

Physics has long demanded a way to explore higher dimensions, but current methods are often abstract, projection-based, or visually unapproachable. This recursive graphing framework solves that by treating each higher dimension as a collection of structured 3D graphs. The result is a visual, scalable, and computational method for interacting with N-dimensional spaces.

Spatial folding is introduced as a transformation function capable of warping traditional 3D coordinates in relation to a higher axis. This warping is shaped by gravitational influence and relativistic constraints, resulting in compressed geodesic paths through higher space.

2. Core Concept

Imagine each 3D space as a slice of paper in a stack. By varying the 4th-dimensional coordinate (w), you generate a full 4D object made up of those slices. Extend this further:

• A 3D graph becomes a coordinate in 4D space
  
  

• A stack of 3D graphs (varying by w) becomes a 4D object
  
  

• A group of 4D objects becomes a coordinate in 5D space
  
  

• This repeats recursively for 6D, 7D, and beyond
  
  

Each new dimension is built from organized groupings of the previous one. The recursive structure enables visualization and calculation through familiar 3D models—no need to directly simulate alien geometries.

3. Dimensional Folding Transformation

The spatial folding transformation is a function that maps coordinates from warped higher-dimensional space:

F(x, y, z, w) → (x', y', z', w')

Where:

• x', y', and z' are spatial coordinates modified by a warping factor
  
  

• w' is the new position along the 4th axis, optionally modified by local curvature
  
  

The warping factor is influenced by gravitational potential, defined as:

T(w) = sqrt(1 - (2 * Φ(x, y, z, w)) / c²)

Where:

• Φ is the gravitational potential at that point
  
  

• c is the speed of light
  
  

• T(w) contracts or dilates space based on relativistic effects
  
  

This transformation reshapes distances and bends dimensional orientation without breaking physical laws.

4. Geodesic Compression

True shortest paths in warped space are no longer Euclidean. They follow geodesics—the warped equivalents of straight lines. In this model, the geodesic path between two points is always shorter than the Euclidean path:

Geodesic Distance < Euclidean Distance

This is not "faster-than-light" travel—it’s shorter-path travel through folded space, caused by warping the coordinate system itself.

5. Recursive Structure

Each dimension is recursively built from the full structures of the dimension below:

• A single 4D coordinate is defined by a stack of 3D graphs
  
  

• A single 5D coordinate is defined by a group of 4D stacks
  
  

• A single 6D coordinate is defined by a collection of 5D objects
  
  

• And so on…
  
  

This makes the system infinitely extensible. All dimensions beyond the third are composed of comprehensible groupings. The result is a model of higher-dimensional reality that can be graphed, simulated, and interpreted.

6. Practical Applications

• Warp Navigation Systems: Build theoretical propulsion that uses fold-contraction pathways.
  
  

• Quantum Tunneling Models: Use geodesic compression to explain sudden state changes.
  
  

• Dimensional Physics Simulations: Construct multi-layered space-time fields in visualized form.
  
  

• Multiversal Plotting: Map fictional or theoretical realities as structured nested layers of space.
  
  

7. Conclusion

This method doesn’t just describe higher dimensions—it builds them. Using recursive stacks of 3D graphs, gravitational warping, and relativistic principles, this graphing technique creates a model for space that is extensible, visual, and fully grounded in physics.

Sci-Real isn't about pretending—it’s about proving. Through this framework, the impossible becomes structure, and the imagined becomes measurable.