Draft:High-Level Space Fields

High-Level Space Fields (HLSFs) is a computational framework that extends graph theory by leveraging recursive patterns within regular polygons to model higher-dimensional spatial relationships. Unlike traditional geometric models that rely on Cartesian grids and Euclidean tessellations, HLSFs reveal how geometric structures inherently encode adjacency rules for multi-dimensional spatial arrangements.

HLSFs operate on the principle that regular polygons can be subdivided into triangular substructures, which recursively expand to reveal adjacency relationships that facilitate higher-dimensional spatial understanding. By subdividing regular polygons into their fundamental triangular substructures, HLSFs allow for the progressive revelation of adjacency networks and emergent spatial logic. The framework has applications in computational geometry, data visualization, urban planning, AI-assisted design, and materials science.

Origins and Development

HLSFs emerged from computational and mathematical explorations in recursive graph expansion and dimensional adjacency rules. The framework was developed as part of a broader investigation into Sphere-Based Design Theory (SBDT), which posits that regular polygons serve as the foundational units from which higher-dimensional spatial relationships can be extracted.

While SBDT provides a generalized spatial framework, HLSFs represent a specific computational approach for understanding how higher-dimensional relationships emerge from regular polygonal structures. The system was formalized through custom-built computational models in Python and Ruby, applying recursive graph algorithms to polygonal networks. These explorations demonstrated that all regular polygons inherently encode higher-dimensional adjacency information, which can be systematically extracted through recursive expansion.

Fundamental Structure: Level 0 and the Formation of a Complete Graph

HLSFs follow a structured, hierarchical model that begins at Level 0, where regular polygons are broken down into their fundamental triangular substructures. These triangles act as the atomic building blocks from which all higher-dimensional structures emerge.

Level 0: Base-Level Polygonal Subdivision

At Level 0, regular polygons are subdivided into non-repeating triangular elements consisting of one vertex connecting to half the remaining vertices - covering half the base polygon, forming the starting point for dimensional expansion. These triangular substructures are algorithmically defined—their internal angles, adjacency relationships, and relative proportions define the rules for higher-dimensional encoding.

Level 1: The Formation of a Complete Graph

A complete graph is formed at Level 1 by iteratively repeating the base level triangles until all vertices are connected within the base polygon, establishing adjacency relationships that define higher-order connectivity rules. This complete graph encodes all possible edges between vertices - all connecting lines being edges of internal, overlapping triangles - creating a fully connected structure that serves as the basis for recursive expansion. At this stage, higher-dimensional adjacency relationships emerge, allowing for recursive geometric transformations in subsequent levels.

Recursive Expansion and Higher-Dimensional Unfolding

Once Level 1 has established the complete graph, HLSFs recursively expand, revealing additional dimensional relationships at each stage.

Level 2+: Recursive Refinement and Dimensional Cascade

At each successive level, the graph expands according to the original connectivity rules set by Level 1. New adjacency structures emerge, progressively unveiling higher-dimensional geometries encoded within the base polygon. Over multiple iterations, the system moves beyond direct human visualization, but retains a logical structure that can be analyzed through graph theory.

The Role of Dimensional Angles

The dimensional angles of Level 0 triangles govern all recursive transformations. Expansion occurs within the constraints dictated by these angles, ensuring that the system maintains structural integrity as it increases in complexity. This prevents HLSFs from producing randomized structures—instead, the expansion reveals emergent order encoded within the original form.

Entities in Higher-Level Space Fields

As HLSFs expand recursively, distinct entities emerge within the growing adjacency network, forming structured patterns in entity space. These entities are defined by the repeating symmetries that arise from the interactions between isosceles triangles, base polygon edges, and the center point of the system.

Formation of Entities in Recursive Expansion

• As HLSFs progress to Level 2 and beyond, repeating isosceles triangle areas with distinct bilateral symmetry emerge within the growing graph.

These areas are defined by:

• • One edge of the base polygon.
  • The center point of the polygon.
• These isosceles substructures define the fundamental space in which higher-level entities appear, governed by symmetry constraints, representing the cross-section projection within the larger orthogonal projection of the space field.

Symmetry and Recursion in Entity Space

• Entities in higher-level space fields emerge from recursive symmetries of isosceles triangles, forming geometric structures that maintain spatial relationships across levels.
• At greater depths of recursion, these entities manifest as self-similar formations, where each subdivision preserves the proportional relationships dictated by the base polygon's dimensional angles.
• The placement and orientation of entities are influenced by the dimensional constraints of Level 0, meaning that each base polygon gives rise to a unique set of embedded structures.

Role of Entities in Dimensional Encoding

Entities within higher-level space fields serve as fundamental building blocks of dimensional encoding.

• Their recursive nature allows them to act as anchors for adjacency expansion, influencing how new vertices connect within the system.
• The emergent structures become increasingly complex at each level, leading to the progressive revelation of non-trivial spatial relationships.
• In high-level iterations, entities in space fields can be analyzed as recursively generated subgraphs, revealing the hierarchy of dimensional encoding within HLSFs.

HLSFs as an Extension and Expansion of Graph Theory

HLSFs extend graph theory by treating regular polygons as dynamic structures that encode higher-dimensional spatial relationships, rather than being static 2D objects. Unlike conventional graphs, which operate on predefined adjacency matrices, HLSFs function as self-expanding adjacency networks, where:

In HLSFs, vertices and edges are dynamically generated through recursive expansion, as opposed to traditional graphs, which define them in advance. The initial connectivity graph is only a seed, with its true structure progressively revealed through recursion. The system is neither strictly regular nor entirely chaotic—it is constrained by dimensional adjacency rules, maintaining a balance between predictability and emergent complexity.

Comparison to Traditional Graph Models

• Traditional Graphs
  • Fixed adjacency matrices
  • No inherent dimensional structure
  • Nodes and edges are predefined
  • Often abstract

• High-Level Space Fields (HLSFs)
  • Emergent, recursive adjacency
  • Encodes higher-dimensional projections
  • Nodes and edges emerge dynamically
  • Directly tied to spatial topology

Applications of HLSFs

Computational Geometry and Non-Euclidean Spaces

HLSFs provide a new method for analyzing higher-dimensional geometries, revealing non-obvious relationships within complex spaces. They serve as a tool for non-Euclidean modeling, offering alternative spatial representations beyond Cartesian grids.

AI-Assisted Architectural and Urban Design

HLSFs can be used for adaptive, multi-scale spatial layouts, particularly in terrain-sensitive urban planning. AI-assisted architectural systems can leverage HLSFs to generate optimized, emergent spatial forms, avoiding rigid, predefined constraints.

High-Dimensional Data Visualization

HLSFs offer a novel method for visualizing and interpreting high-dimensional datasets in fields such as machine learning and topology, where recursive structures provide insights into complex relationships between data points.

Structural and Material Science

HLSFs can inform next-generation lattice structures, using recursive geometry for optimal load distribution. Their triangular substructures provide high strength-to-weight efficiency, useful in biomimetic and aerospace materials.

Conclusion

High-Level Space Fields (HLSFs) is a computational framework that explores geometric relationships in higher dimensions. By treating regular polygons as encoded carriers of higher-dimensional information, HLSFs expose previously hidden adjacency rules through recursive expansion.

As an extension of graph theory, HLSFs introduce a self-expanding connectivity model, where dimensional structure is discovered rather than imposed. Through this approach, HLSFs bridge mathematical formalism with emergent geometry, offering a new pathway for exploring the hidden order within higher-dimensional spaces.