Question

Show that the dual of the cube graph is the octahedron graph, and that the dual of the dodecahedron graph is the icosahedron graph. What is the dual of the tetrahedron graph?

Short answer

The dual graph of the cube is the octahedron, the dual graph of the dodecahedron is the icosahedron, and the tetrahedron graph is a self-dual.

Step-by-step solution

Prove the cube's dual graph is the octahedron

The cube is a regular solid with 6 square faces, 12 edges, and 8 vertices. One way to think about the dual graph is as a graph that 'inverts' faces and vertices. An octahedron has 8 faces, 12 edges, and 6 vertices - the number of vertices and faces are interchanged in comparison to the cube. Therefore, the dual of the cube graph is the octahedron graph.

Prove the dodecahedron's dual graph is the icosahedron

The dodecahedron is a regular solid with 12 pentagonal faces, 30 edges, and 20 vertices. The icosahedron, another regular solid, has 20 triangular faces, 30 edges, and 12 vertices. The number of vertices and faces are switched in comparison to the dodecahedron. Hence the dual of the dodecahedron graph is the icosahedron graph.

Determine the tetrahedron graph's dual

The tetrahedron is a regular solid with 4 triangular faces, 6 edges, and 4 vertices. If the inversion process stated above is applied to the tetrahedron, then its dual would maintain the exact same number of faces, vertices, and edges. Therefore, the dual of the tetrahedron graph is the tetrahedron graph itself.

Key concepts

Cube Graph and Its Dual

When delving into graph theory, one intriguing concept is that of dual graphs. The 'cube graph dual' illustrates this beautifully. Imagine a cube, a three-dimensional shape we're all familiar with from geometry, composed of 6 square faces, 12 edges, and 8 vertices. The idea of duality in graph theory involves creating a new graph, the 'dual graph,' by interchanging the roles of faces and vertices. It’s like looking in a mirror where faces become vertices and vice versa, while maintaining the same number of edges.

• The cube graph has 8 vertices and 6 faces.
• The octahedron graph, which is the dual of the cube graph, 8 faces and 6 vertices.

Since an octahedron consists of 8 faces and 6 vertices, each face of the original cube becomes a vertex in the octahedron, and each vertex becomes a face. The process preserves the number of edges. The realization that the cube graph’s dual is an octahedron graph can be a mind-bending but enriching concept for students to explore.

Dodecahedron Graph and Its Dual

Moving on to a more complex Platonic solid, let’s consider the dodecahedron. The 'dodecahedron graph dual' is a fascinating case in the study of dual graphs due to its intricate structure. A dodecahedron is composed of 12 pentagonal faces, 30 edges, and 20 vertices. By converting faces to vertices and vertices to faces, we obtain its dual: the icosahedron graph.

• The dodecahedron graph showcases 20 vertices and 12 faces.
• The icosahedron graph, as the dual, displays 20 faces and 12 vertices.

As with the cube and octahedron, the number of edges remains unchanged at 30. The icosahedron has 20 triangular faces and 12 vertices, exactly the characteristics needed for it to be the dual of the dodecahedron. This duality reinforces the beautiful symmetry these shapes exhibit, demonstrating how interconnected and balanced the Platonic solids are within the realm of graph theory.

Tetrahedron Graph and Its Dual

Unlike the cube and dodecahedron, the 'tetrahedron graph dual' presents a unique scenario. The tetrahedron is the simplest of the Platonic solids, consisting of 4 triangular faces, 6 edges, and 4 vertices. Applying the duality concept, where we would typically interchange faces and vertices, we find that the dual of a tetrahedron graph is itself.

• A tetrahedron has 4 faces and 4 vertices - which remain constant in its dual.

Therefore, the tetrahedron is self-dual, a fact that stands out in the study of dual graphs. This mirror-like characteristic where an object is its own dual adds another layer of wonder to graph theory and solid geometry. It posits an intriguing question for students: what does it mean for a shape to be self-dual, and how does this influence our understanding of symmetry and duality? It encapsulates the beauty and simplicity that can be found in mathematical concepts and encourages a deeper appreciation of the natural symmetry in the world around us.