edge
Let an agent define edges in geometry and their counting relations, relate them to edges in graph theory, help compute edge counts and lengths, and separate the concept from unrelated aliases.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent define edges in geometry and their counting relations, relate them to edges in graph theory, help compute edge counts and lengths, and separate the concept from unrelated aliases.
In geometry, a line segment joining two vertices of a polygon, polyhedron or higher polytope and forming part of its boundary, as the sides of a triangle or the twelve edges of a cube; edges are counted in relations such as the Euler formula, are generalised in graph theory to connections between nodes, and the registry entry also gathers aliases for elementary particle names and a baseline, which are unrelated.
What it is for: Describing the boundaries of polygons and polytopes.
It can be define edges and their properties; relate to graph theory edges; compute edge counts and lengths; separate unrelated aliases.
Distinguishing features
One-dimensional boundary element
Joins two vertices
Counted by Euler relations
Generalised in graphs
What it looks like
A straight segment where two faces of a solid meet or between two vertices of a polygon.
Physical character
edges of a cube: 12 count
Euler formula for convex polyhedra: V - E + F = 2 relation
How it is recognised
Segment between vertices on a boundary
Sides of polygons, edges of polyhedra
Faces are two-dimensional; vertices are points; graph edges may not be geometric
Related models
is a kind of - in registry terms
is part of - in registry terms
joins - two of them
is generalised by - connections between nodes
In practice
Families and kinds
sides of polygons
edges of polyhedra
edges of higher polytopes
edges in graph theory
edges in meshes and computer graphics
Standards and regulation
ISO 80000-2 notation for geometry
Failure modes and hazards
Confusing edges with faces or vertices
Sense confusion with unrelated aliases
Miscounting edges in complex polytopes
Also called
Where this came from
wikidata · CC0 1.0
Also registered as vr.tr.edge-artifact
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Define What an edge is.
Definition.
Definition
Definition.
Definition
Definition.
- What is an edge in geometry, and how does it relate to vertices and faces? definition
- Is the question about geometric edges, graph edges or an unrelated sense? boundary
Graphs
Edges in graphs.
Graphs
Graphs.
- How does the graph theory notion of edge generalise the geometric one? definition
- Which entry fits graph theory? action
Compute Counting and measuring.
Computation.
Counting
Counting edges.
Counting
Counting.
- How are edges counted using the Euler formula and combinatorial reasoning? action
- What is the count for the polytope in question? measurement
Lengths
Edge lengths.
Lengths
Lengths.
- How are edge lengths and angles computed for regular and general solids? action
- Which entry fits polyhedron? action
Apply Applications.
Application.
Graphics
Meshes and graphics.
Graphics
Graphics.
- How are edges represented in meshes and computer graphics? provenance
- Which entry fits polygon mesh? action
Structures
Structures.
Structures
Structures.
- How do edges figure in crystallography and structural frameworks? provenance
- Which references are standard? provenance
Context History and teaching.
Context.
History
History.
History
History.
- How did Euler and later mathematicians develop edge counting relations? provenance
- Which entry fits the history of topology? action
Teach
Teaching.
Teach
Teaching.
- How are edges, faces and vertices taught, and which are confused? provenance
- Which sources are cited? provenance
What the second pass must settle
- Should graph edge be a separate entry?
- How should geometry references be linked?
- The registry entry has merged aliases for particle names and a baseline; should they be removed?