The 9-sided Enneahedron - Numberphile
Audio Brief
Show transcript
This episode explores the Herschel Enneahedron, a nine-faced polyhedron linked to the Herschel graph and its unique mathematical properties.
There are three key takeaways from this discussion. First, the Herschel graph is a fundamental example in graph theory, representing the smallest polyhedral graph that cannot be traced by visiting every corner exactly once in a closed loop.
Second, abstract graphs can effectively represent physical 3D objects. Their inherent properties, like being bipartite or possessing specific symmetries, provide essential clues for constructing their tangible physical shape.
Third, a core proof for why a graph is non-Hamiltonian relates to its structure. A bipartite graph with an odd number of vertices, for instance, cannot have a Hamiltonian cycle because paths must alternate between two vertex sets, making a return to the start impossible after visiting every vertex.
Ultimately, this journey illustrates how mathematical discovery transforms abstract definitions into tangible, visualized structures through logical deduction.
Episode Overview
- The episode introduces the Herschel Enneahedron, a specific polyhedron with nine faces, named in connection with the Herschel family of astronomers and mathematicians.
- It delves into the properties of the "Herschel graph," which is defined as the smallest non-Hamiltonian polyhedral graph, meaning it represents a 3D shape but lacks a specific type of vertex-visiting path.
- The speaker explains the concept of Hamiltonian cycles (a path visiting every vertex exactly once) and proves why the Herschel graph, being bipartite with an odd number of vertices, cannot have one.
- The journey of translating the 2D Herschel graph into its corresponding 3D polyhedron is detailed, using its inherent symmetries to determine the final, elegant shape.
Key Concepts
- Enneahedron: A polyhedron with nine faces.
- Herschel Graph: A bipartite, undirected graph with 11 vertices and 18 edges. It is a polyhedral graph, meaning it represents the vertices and edges of a convex polyhedron.
- Hamiltonian Cycle: A path within a graph that visits every vertex exactly once before returning to the starting vertex. A graph with such a cycle is called Hamiltonian.
- Non-Hamiltonian Graph: A graph that does not contain a Hamiltonian cycle. The Herschel graph is the smallest such graph that is also polyhedral.
- Bipartite Graph: A graph whose vertices can be divided into two separate sets, where every edge connects a vertex from one set to a vertex in the other.
- Polyhedral Graph: A graph that can be drawn on a plane without edges crossing and represents the vertices and edges of a convex 3D polyhedron.
- D6 Dihedral Symmetry: A type of symmetry identical to that of a regular hexagon, which the Herschel graph possesses. This property was crucial for deducing the 3D structure of the enneahedron.
Quotes
- At 00:05 - "This isn't any ahedron. This is an enneahedron!" - The speaker introduces the topic, distinguishing this specific nine-sided polyhedron from any ordinary one.
- At 00:54 - "The Herschel graph is the smallest non-Hamiltonian polyhedral graph." - Quoting the Wikipedia definition that sparked the investigation into what this shape actually looks like.
- At 06:24 - "How did you do that? With a computer or physically?" - The interviewer expresses surprise about how the complex 3D shape was finally visualized from its 2D graph, after the speaker had struggled to build it physically.
Takeaways
- The Herschel graph is a fundamental example in graph theory because it's the smallest polyhedral graph that you cannot "trace" by visiting every corner exactly once in a closed loop.
- Abstract graphs can represent physical 3D objects, and their properties (like being bipartite or having specific symmetries) provide the necessary clues to construct their physical shape.
- A key proof for why a graph is non-Hamiltonian can be its structure; a bipartite graph with an odd number of vertices cannot have a Hamiltonian cycle because any path must alternate between two sets of vertices, making it impossible to return to the start after visiting every vertex.
- The process of mathematical discovery can be a creative puzzle, moving from a theoretical definition on a webpage to a tangible 3D model through logical deduction and visualization.
