Recursive Multi-Dimensional Graphing via Spatial Folding: A Scalable Framework for Dimensional Compression and Geodesic Shortcuts

By Alissa M.R. Eldridge

Proposed under Reality Builders Theoretical Sciences Division

Abstract

This paper presents a scalable, recursive framework for multi-dimensional graphing through spatial folding, proposing that higher-dimensional space can be constructed, visualized, and computed by stacking structured 3D graphs. The approach introduces a warping transformation modulated by gravitational potential and relativistic dilation, enabling geodesic path compression and dimensional shortcuts without violating physics. While this dissertation focuses on the 4D application for clarity, the model recursively extends to 5D, 6D, and beyond, with each dimensional coordinate represented by a grouped structure from the previous dimension. This enables real-world applications in warp navigation, quantum tunneling analogs, and advanced simulations of extra-dimensional geometries.

1. Introduction

High-dimensional physics has long pushed the boundaries of mathematical abstraction. From general relativity to string theory, the need to interpret and compute in more than three spatial dimensions is well-established, yet practical visualization and structural modeling methods are lacking.

This dissertation introduces a novel, recursive multi-dimensional graphing method, rooted in the idea that each spatial dimension beyond the third can be constructed as an ordered grouping of 3D graphs. These groupings serve as coordinate points in the next dimension. Dimensional movement is achieved not just by geometric manipulation, but through gravitationally and relativistically warped transformations that compress paths and allow for effective shortcuts across space.

This paper explores both the mathematical foundation and intuitive structure of this graphing system, defining the 4D folding process as a specific example while maintaining its extensibility to higher dimensions.

2. Core Concepts and Visualization

To conceptualize this framework, consider the following layered structure:

Each 3D graph is treated as a single coordinate along a 4th spatial axis.
A group of 3D graphs stacked by variation in the 4th axis (e.g., www) forms a 4D region.
Each 4D region becomes a single coordinate in 5D space, indexed along a 5th axis (e.g., vvv).
The process continues recursively: 5D regions (stacks of 4D spaces) are coordinates in 6D, and so on.

This method does not require direct rendering of 4D+ objects. Instead, it provides a computational and visual approach by interpreting higher dimensions through accessible groupings of 3D frames — enabling finite simulations of infinite structures.

3. Mathematical Framework

Let a point P∈R4P \in \mathbb{R}^4P∈R4 be defined as P=(x,y,z,w)P = (x, y, z, w)P=(x,y,z,w), where www is the 4th spatial coordinate derived from the stacking of 3D graphs.

We define the spatial folding transformation as:

F:R4→R4F: \mathbb{R}^4 \rightarrow \mathbb{R}^4F:R4→R4 F(P)=(x′,y′,z′,w′)F(P) = (x', y', z', w') F(P)=(x′,y′,z′,w′) x′=x⋅T(w),y′=y⋅T(w),z′=z⋅T(w),w′=w+α(x,y,z)x' = x \cdot T(w), \quad y' = y \cdot T(w), \quad z' = z \cdot T(w), \quad w' = w + \alpha(x, y, z)x′=x⋅T(w),y′=y⋅T(w),z′=z⋅T(w),w′=w+α(x,y,z)

Where:

T(w)T(w)T(w) is a warping factor based on relativistic dilation due to gravity,
α(x,y,z)\alpha(x, y, z)α(x,y,z) introduces additional curvature effects from spatial interaction.

Warping Factor:

T(w)=1−2Φ(x,y,z,w)c2T(w) = \sqrt{1 - \frac{2\Phi(x, y, z, w)}{c^2}}T(w)=1−c22Φ(x,y,z,w)

With gravitational potential:

Φ(x,y,z,w)=∑i=1NGMiri\Phi(x, y, z, w) = \sum_{i=1}^{N} \frac{G M_i}{r_i}Φ(x,y,z,w)=i=1∑NriGMi

Where:

GGG = Gravitational constant,
MiM_iMi = Mass of the ithi^\text{th}ith object,
rir_iri = Distance from point PPP to object iii,
ccc = Speed of light.

4. Recursive Dimensional Construction

To generalize this model to higher dimensions:

Let dimension DDD be constructed from groupings of elements from dimension D−1D-1D−1.
Each coordinate in dimension DDD represents a full stack or group of D−1D-1D−1 space.

This gives rise to:

3D graph→4D coordinate\text{3D graph} \rightarrow \text{4D coordinate}3D graph→4D coordinate
4D region (group of 3D graphs)→5D coordinate\text{4D region (group of 3D graphs)} \rightarrow \text{5D coordinate}4D region (group of 3D graphs)→5D coordinate
5D region (group of 4D stacks)→6D coordinate\text{5D region (group of 4D stacks)} \rightarrow \text{6D coordinate}5D region (group of 4D stacks)→6D coordinate
… and so on.

Formally:
Let GD={G1D−1,G2D−1,...,GnD−1}G^D = \{G_1^{D-1}, G_2^{D-1}, ..., G_n^{D-1}\}GD={G1D−1,G2D−1,...,GnD−1}, then

PD=(GD)∈RDP_D = (G^D) \in \mathbb{R}^DPD=(GD)∈RD

where each GD−1G^{D-1}GD−1 is itself a fully-formed dimensional object from the previous level.

This enables both stacked graph simulation and vector transformation, allowing for scalable modeling across arbitrary spatial dimensions.

5. Geodesic Compression in Warped Space

The geodesic distance (true shortest path) between points in this warped space is given by:

dgeo=∫P1P2dx2+dy2+dz2+dw2d_{\text{geo}} = \int_{P_1}^{P_2} \sqrt{dx^2 + dy^2 + dz^2 + dw^2}dgeo=∫P1P2dx2+dy2+dz2+dw2

Compared to traditional Euclidean distance:

deuclidean=(x2−x1)2+(y2−y1)2+(z2−z1)2+(w2−w1)2d_{\text{euclidean}} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 + (w_2 - w_1)^2}deuclidean=(x2−x1)2+(y2−y1)2+(z2−z1)2+(w2−w1)2

Due to the contraction introduced by T(w)T(w)T(w), it follows:

dgeo<deuclideand_{\text{geo}} < d_{\text{euclidean}}dgeo<deuclidean

This confirms the presence of a traversable shortcut through folded space — not by violating physics, but by bending space itself through higher-dimensional structure.

6. Implementation Workflow

Initialize 3D Coordinates: Begin with a standard 3D spatial data set.
Stack Along New Axis: Add www coordinate via stacking to simulate 4D.
Apply Warping: Calculate Φ\PhiΦ and use T(w)T(w)T(w) to warp dimensions.
Repeat for Higher Dimensions: Treat 4D stacks as 5D coordinates, and build upward recursively.
Analyze Path Contraction: Compare geodesic vs. Euclidean paths to verify warping effects.

7. Applications and Future Research

Practical Applications:

Warp Drive Modeling: Offers a mathematically plausible foundation for warp-based navigation.
Quantum Tunneling Analogues: Models higher-dimensional jump points as geodesic compression regions.
Topological Simulation: Enables mapping of complex multidimensional shapes through recursive layers.
Multiverse Physics: Serves as a computational scaffold for simulating alternate-dimensional frames.

Further Research:

Full generalization to Riemannian manifolds via tensor embeddings.
Use of machine learning to identify warping patterns or emergent pathways.
Visualization tools to animate higher-dimensional transitions using grouped 3D data.

8. Conclusion

This dissertation formalizes a recursive, scalable framework for multi-dimensional graphing that constructs higher dimensions from organized stacks of 3D graphs. By incorporating gravitational potential and relativistic time dilation, it enables dimensional warping that compresses geodesic distances and makes theoretical shortcuts through space plausible. The method is modular, extendable, and computationally grounded — a new tool for modeling the unreachable.

This system doesn’t just model higher-dimensional space — it builds it, one dimensional layer at a time.