Brian K. Hackett

Math 3190

Euler’s Formula

V – E + F = 2

We are going to learn about the five Platonic Solids. We are going to learn about a man who developed proof for finding any convex polyhedron. We will draw some of the polyhedron using the Sketchpad program. Then we will try to disprove his proof.
History of Platonic Solids

The five regular polyhedra were discovered by the ancient Greeks. The Pythagoreans knew of the tetrahedron, the cube, and the dodecahedron; the mathematician Theaetetus added the octahedron and the icosahedron. These shapes are also called the Platonic solids, after the ancient Greek philosopher Plato; Plato, who greatly respected Theaetetus' work, speculated that these five solids were the shapes of the fundamental components of the physical universe.

Now lets’ open Sketchpad and draw an easy three dimensional shape the cube.

How many sides does a cube have? ______

Now how many vertices does the cube have? ______

Finally, How many edges does the cube have? ______

Now try some other three dimensional shapes and see if the formula works on them. As you fine out about each solid fill in the work sheet.

Regular Polyhedra or Platonic Solids:
Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron

There are only five geometric solids that can be made using a regular polygon and having the same number of these polygons meet at each corner. The five Platonic solids (or regular polyhedra) are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

TETRAHEDRON
/
CUBE

OCTAHEDRON
/
DODECAHEDRON
/
ICOSAHEDRON

A Proof of Euler's Formula.

The followingis only one way use to prove Euler formula: for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges. Symbolically V-E+F=2. For instance, a tetrahedron has four vertices, four faces, and six edges; 4-6+4=2.

There are many ways to prove a formula. I have picked just one of them From the following Web Site

Euler's Formula, Proof 7: Dual Electrical Charge

Open one of your drawings and start labeling the vertices starting with V1. Now label the edges starting with E1, and label the faces starting with F1.Put a unit + (positive) charge at each vertex, a unit –(negative) charge at the center of each edge, and a unit + (positive) charge in the middle of each face

We will show that all but two + charges cancel. To do this, displace the charge on each edge to its right endpoint; displace the charge on each face (except the outer face) to its rightmost vertex. Each vertex receives the charges from an alternating sequence of edges and faces, canceling its initial charge. The only remaining non canceled charges are one + charge on the outer face and one + charge on the leftmost vertex.

Now with the above information and your information on the cube, input it into Euler Formula. (V – E + F = 2)

Does the formula work?

Now try some other three dimensional shapes and see if the formula works on them. As you fine out about each solid fill in the work sheet.

Platonic Solids

Final Proof of Euler Formula

V – E + F = 2

Use the information gather from your sheet to prove Euler Formula for each solid.

The Tetrahedron

The Octahedron

The Dodecahedron

The Icosahedron

Platonic Solids –Project Sheet Answers

Platonic Solid / Number of Faces / Shape of Faces / Number of Faces at Each Vertex / Number of Vertices / Number of Edges / Dual (The Platonic Solid that can be inscribed inside it by connecting the mid-points of the faces)
Tetrahedron / 4 / Equilateral Triangle (3-sided) / 3 / 4 / 6 / Tetrahedron
Cube / 6 / Square (4-sided) / 3 / 8 / 12 / Octahedron
Octahedron / 8 / Equilateral Triangle (3-sided) / 4 / 6 / 12 / Cube
Dodecahedron / 12 / Pentagon (5-sided) / 3 / 20 / 30 / Icosahedron
Icosahedron / 20 / Equilateral Triangle (3-sided) / 5 / 12 / 30 / Dodecahedron

How many sides does a cube have? _6_

Now how many vertices does the cube have? _8_

Finally, How many edges does the cube have? _12_

Now with the above information and your information on the cube, input it into Euler Formula. (V – E + F = 2)

8 – 12 + 6 = 2

Does the formula work? _Yes_

The Tetrahedron: 4 – 6 + 4 = 2

The Octahedron: 6 – 12 + 8 = 2

The Dodecahedron: 20 – 20 + 12 = 2

The Icosahedron: 12 – 30 + 20 = 2