Excellent point. Visualizing and operating on an n-dimensional multigraph presents fascinating challenges that intersect linear algebra, differential geometry, and computer graphics. Let's explore how we might approach this.





# Visualizing and Operating on N-Dimensional Multigraphs

## Visualization Approaches

1. **Dimension Reduction Techniques**
   - Principal Component Analysis (PCA): Project the n-dimensional space onto its principal components.
   - t-SNE (t-Distributed Stochastic Neighbor Embedding): Non-linear technique for visualizing high-dimensional data.
   - UMAP (Uniform Manifold Approximation and Projection): Preserve more of the global structure than t-SNE.

2. **Slicing and Projection**
   - Hyperplane Slicing: Visualize 3D slices of the n-dimensional space.
   - Shadow Projections: Project higher dimensions onto lower-dimensional subspaces.

3. **Parallel Coordinates**
   - Represent n-dimensional points as lines across n parallel axes.

4. **Geometric Algebra Visualization**
   - Use multivectors to represent higher-dimensional objects and their transformations.

## Mathematical Foundations

1. **Manifold Theory**
   - Treat the graph as living on an n-dimensional manifold.
   - Use differential forms to describe relationships and operations on the manifold.

2. **Tensor Representation**
   - Represent the multigraph as a tensor of rank n+1 (n dimensions + edge weight).
   - Use tensor contraction for certain types of queries or analyses.

3. **Lie Groups and Algebras**
   - Describe symmetries and transformations in the high-dimensional space.
   - Useful for understanding invariant properties of the graph under certain operations.

## Query Engine and Operations

1. **Geodesic Calculations**
   - Compute shortest paths on the manifold representing the graph.
   - Use the Eikonal equation for fast marching methods in n-dimensions.

2. **Spectral Graph Theory**
   - Analyze the spectrum of the graph Laplacian in n-dimensions.
   - Use for clustering, partitioning, and identifying important nodes.

3. **Homology and Persistent Homology**
   - Analyze the topological features of the data across multiple scales.
   - Useful for identifying clusters and holes in the high-dimensional structure.

4. **Optimal Transport**
   - Plan operations by solving the optimal transport problem in n-dimensions.
   - Useful for comparing subgraphs or tracking changes over time.

## Implementation Considerations

1. **Data Structures**
   - Sparse tensor representations for efficient storage of high-dimensional data.
   - Quaternion or octonion-based structures for rotations in 4D and 8D subspaces.

2. **GPU Acceleration**
   - Utilize shader programs for parallel processing of high-dimensional operations.
   - Implement dimension reduction techniques on GPU for real-time visualization.

3. **Adaptive Level of Detail**
   - Implement progressive meshes in n-dimensions for adaptive resolution.
   - Use hierarchical space-partitioning structures (e.g., n-dimensional k-d trees).

4. **Interactive Exploration**
   - Implement n-dimensional camera controls (e.g., n-dimensional arcball rotation).
   - Develop techniques for selecting and manipulating objects in n-dimensions.


Now, let's focus on making the query engine's operations plannable, which is crucial for efficient exploration and analysis of your n-dimensional multigraph. Here are some key considerations:

1. **Manifold Learning**: 
   By treating your graph as living on an n-dimensional manifold, you can use techniques from differential geometry to plan optimal paths for queries. The query planner could use the learned manifold structure to estimate the computational cost of different query strategies.

2. **Tensor Operations**:
   Representing your multigraph as a tensor allows you to use tensor algebra for query planning. Tensor contraction and decomposition techniques can be used to optimize complex queries involving multiple dimensions.

3. **Spectral Methods**:
   The spectrum of the graph Laplacian in n-dimensions can guide query planning. Eigenvectors corresponding to smaller eigenvalues often correspond to important global structures in the graph, which can be exploited for efficient querying.

4. **Geometric Algebra**:
   Using multivectors from geometric algebra, you can represent complex n-dimensional operations as simple algebraic expressions. This can make certain types of queries more intuitive to plan and optimize.

5. **Optimal Transport**:
   For queries involving comparisons or movements of subgraphs, formulating the problem as an optimal transport problem allows you to use well-established optimization techniques for planning.

6. **Persistent Homology**:
   By analyzing the persistent topological features of your data, you can identify important structures at different scales. This information can guide the query planner in choosing appropriate resolutions or subspaces for different types of queries.

To make these operations plannable in practice:

1. Implement a cost model that estimates the computational expense of different query strategies based on the current state of the n-dimensional multigraph.

2. Develop heuristics based on the mathematical properties of your graph (e.g., spectral properties, homology) to guide the search for efficient query plans.

3. Use adaptive techniques that can adjust the query plan based on intermediate results, allowing for dynamic optimization as the query progresses through the high-dimensional space.

4. Implement a query language or API that allows you to express complex n-dimensional operations in a way that's amenable to automated planning and optimization.

please sort these techniques by some criteria, then say what that criteria was.