The 9-sided Enneahedron - Numberphile

Key Concepts

• Herschel Anahedron (9-sided polyhedron)
• Herschel Graph (smallest non-Hamiltonian polyhedral graph)
• Hamiltonian Graph/Cycle
• Bipartite Graph
• Polyhedral Graph
• D6 Symmetry (Symmetry of a hexagon)
• Graph of a Polyhedron
• Planar Graph

Herschel Anahedron and Herschel Graph

The video discusses the Herschel anahedron, a nine-sided polyhedron, and its corresponding Herschel graph. The Herschel graph is significant because it's the smallest non-Hamiltonian polyhedral graph. The speaker clarifies that the Herschel graph is named after Alexander Herschel, although he may not have discovered it himself.

Hamiltonian Graphs and the Herschel Graph

A Hamiltonian graph is defined as a graph where it's possible to start at one vertex, visit every other vertex exactly once, and return to the starting vertex, forming a Hamiltonian cycle. Examples given include a triangle and a square. The Herschel graph is non-Hamiltonian, meaning no such cycle exists.

The speaker demonstrates attempting to find a Hamiltonian cycle on the Herschel graph, highlighting the difficulty and the eventual dead ends. The proof that the Herschel graph is non-Hamiltonian relies on the fact that it is a bipartite graph.

A bipartite graph is one where the vertices can be divided into two disjoint sets, and every edge connects a vertex in one set to a vertex in the other set. The speaker illustrates this by circling some vertices in the Herschel graph such that every edge connects a circled vertex to an uncircled vertex. An equivalent condition is that the vertices can be colored with two colors such that adjacent vertices have different colors.

The speaker explains that a bipartite graph with an odd number of vertices cannot be Hamiltonian. This is because each time you move along an edge, you switch from one set to the other. After an odd number of moves, you must end up in the opposite set from where you started, so you cannot form a cycle.

Polyhedral Graphs and Planarity

The Herschel graph is also a polyhedral graph, meaning it represents the vertices and edges of a polyhedron. The speaker contrasts this with the graph of a cube, which is easily visualized. A key property of a polyhedral graph is that it is planar, meaning it can be drawn on a plane without any edges crossing.

Reconstructing the Herschel Anahedron with D6 Symmetry

Initially, the speaker attempted to construct a physical model of the Herschel anahedron based on the graph, but it lacked the required D6 symmetry. D6 symmetry is the symmetry of a hexagon, including rotational and reflectional symmetries.

The speaker and his colleague, Michael White, realized that the Herschel anahedron should possess D6 symmetry. The challenge was to find the three-fold rotation axis within the graph. They identified a specific vertex around which the rotation could occur.

To achieve the D6 symmetry, they assigned coordinates to the vertices, with the central vertex at Z=0. The vertices connected to it were placed at a higher level, and the top vertex at a further height. They then used algebra and vector coordinates to determine the relationships between these heights, ensuring that the quadrilateral faces were flat.

They found that the height of the top point had to be 4/3 of the height of the middle points. This left one degree of freedom, allowing them to choose how "squashed" or "stretched" the shape would be. They chose to make the middle rhombuses squares, resulting in a specific Herschel anahedron with square rhombuses in the middle and kite-shaped faces elsewhere.

The speaker demonstrates the D6 symmetry on the physical model, highlighting the three-fold rotational symmetry and the reflection symmetries. He notes that it's difficult to represent the D6 symmetry directly in the 2D graph due to the flattening required.

Conclusion

The video explores the properties of the Herschel anahedron and its graph, focusing on its non-Hamiltonian nature, its polyhedral representation, and its D6 symmetry. The process of reconstructing the polyhedron with the correct symmetries is detailed, involving algebraic calculations and geometric considerations. The final model showcases the D6 symmetry, with the speaker emphasizing the three-fold rotational symmetry.