Question

Let \(G\) be a polyhedron (or polyhedral graph), each of whose faces is bounded by a pentagon or a hexagon. (i) Use Euler's formula to show that \(G\) must have at least 12 pentagonal faces. (ii) Prove, in addition, that if \(G\) is such a polyhedron with exactly three faces meeting at each vertex (such as a football), then \(G\) has exactly 12 pentagonal faces.

Short answer

The polyhedron must have at least 12 pentagonal faces and if exactly three faces meet at each vertex, then it has exactly 12 pentagonal faces.

Step-by-step solution

Apply Euler's Formula for Polyhedra

Euler's formula states that for any polyhedra, the number of vertices (V), edges (E), and faces (F) are related by the equation V - E + F = 2. The given polyhedron's faces are only pentagons and hexagons. Let the number of pentagonal faces be \(p\) and the number of hexagonal faces be \(h\). Since each pentagon has 5 edges and each hexagon has 6 edges, the total number of edges: \(E = 1/2 (5p + 6h)\). As each face is either a pentagon or a hexagon: \(F = p + h\). Substituting these variables into Euler's formula results in: \(V - 1/2(5p+6h) + p + h = 2\).

Utilize Expected Degrees of Vertices

Now it is known that at each vertex 3 faces meet. Since each pentagon shares its vertex with 1 or 2 hexagons and a hexagon shares its vertex with 1,2 or 3 pentagons or hexagons, the degree of each vertex can be calculated as: \(V = 1/3 (5p + 6h)\). Substituting this value of V into the previous formula results in: \(1/3(5p+6h) - 1/2(5p+6h) + p + h = 2\). Simplifying, minimum value of p which satisfies the equation is 12, proving (i).

Analyze Scenarios for Number of Faces Meeting at Each Vertex

In case exactly three faces meet at each vertex, each of those faces shares an edge with two others that meet at the same vertex. As such, for three hexagons to meet at a vertex, at least four faces must meet at that vertex, contradicting the given assumption. So, at least one face that meets at each vertex must be a pentagon. Thus, every vertex has a pentagon meeting at it. As each pentagon meets three vertices, there are exactly three times as many vertices as pentagons. With only pentagons or hexagons, there could not be more than three vertices per face so the number of pentagons must be: \(p = V/3 = 12\). This proves (ii).

Key concepts

Polyhedral Graph

A polyhedral graph is an important concept in geometry, particularly in the study of three-dimensional shapes. It's a graph that represents the edges and vertices of a polyhedron, which is a three-dimensional figure with flat faces and straight edges. Imagine a cube, for example; each corner is a vertex, each line connecting the corners is an edge, and each flat surface is a face. Polyhedral graphs are used to visualize and analyze the structure of polyhedra.

When it comes to polyhedra with pentagonal and hexagonal faces, the graph becomes a bit more complex but still adheres to the same foundational rules. For instance, a polyhedron that models a football, known for its pattern of pentagons and hexagons, can be represented by a polyhedral graph. In mathematics, the study of these graphs helps us understand spatial relationships and properties of three-dimensional objects.

Pentagonal and Hexagonal Faces

Diving deeper into the polyhedra in our exercise, let's look at the geometry of pentagonal and hexagonal faces. A pentagon, with its five edges and vertices, and a hexagon, with its six, are common shapes in polyhedra found in nature and man-made objects like the aforementioned football.

Characteristics of Pentagons and Hexagons

Every pentagonal face in a polyhedron shares edges with surrounding faces, and the same goes for hexagons. This interconnectedness is key to determining their arrangement in polyhedra and affects the overall shape and stability of the structure. Importantly, the number of pentagonal faces plays a significant role in the properties and classification of the polyhedron. Polyhedra with pentagonal and hexagonal faces are fascinating because they present a mosaic of these geometric shapes that tessellate to form an enclosed three-dimensional space.

Vertices, Edges, and Faces Relationship

The relationship between vertices (V), edges (E), and faces (F) in a polyhedron is governed by Euler's formula: V - E + F = 2. This equation is a cornerstone in the study of polyhedral geometry as it demonstrates a fundamental characteristic of these structures, regardless of their complexity. Let's break it down:

• Vertices (V) are the points where two or more edges meet.
• Edges (E) are the lines where two faces connect.
• Faces (F) are the flat surfaces that make up the polyhedron's skin.

Euler's formula elegantly encapsulates the balance between these elements. It allows mathematicians to predict and verify the structure of polyhedra. In the exercise, for instance, we see that a polyhedra with faces that are only pentagons or hexagons, and with three faces meeting at each vertex, must have a specific number of pentagonal faces. This discovery is not only a testament to Euler's formula's power but also highlights the intricate dance between the faces, edges, and vertices of a polyhedron.