Legend:Value 1Value 2Value 3
Links:Adjacent swap 1-2Adjacent swap 2-3
Permutations, Geometry, and Distance
This explorable illustrates how permutations can be represented geometrically in different ways. Each point corresponds to a distinct permutation of a 3-element set. The Permutohedron view encodes permutation adjacency—where nearby points differ by a simple swap—while the Euclidean view maps permutations into 3D coordinates directly. These representations suggest different notions of similarity. While Euclidean distance may appear intuitive, it does not respect the structure of permutation space and can mislead clustering results. Instead, distance metrics based on permutation order (such as adjacent transpositions) offer more faithful ways to compare and group ranked data.
Projection Type
Select a projection to visualize the permutation structure:
- Permutohedron: Arranges permutations so that neighbors differ by adjacent swaps, revealing the combinatorial structure. For three elements, this forms a regular hexagon.
- Euclidean: Assigns Cartesian 3D coordinates to each permutation. While visually intuitive, Euclidean distances may not reflect true permutation similarity.
Projection:
Permutation Distances
Select two permutation nodes to view all distances.
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Permutations Table
Each row below represents a unique permutation of the three base values.
| Color | Value 1 | Value 2 | Value 3 |
|---|
| 1 | 2 | 3 | |
| 1 | 3 | 2 | |
| 2 | 1 | 3 | |
| 2 | 3 | 1 | |
| 3 | 1 | 2 | |
| 3 | 2 | 1 |
Glossary