Euler's Formula

by

Amanda Newton

The Swiss mathematician physicist, Leonhard Euler, is known for several findings and works in the world of mathematics and physics. In 1750, Euler derived the formula F+V=E+2 which holds for all convex polyhedra. The variables F, V, and E represent faces, vertices, and edges of a polyhedron respectively. This can be seen with ease for the five regular polyhedra shown below.

This may not be very surprising to some and may only look like a coincidence. Let's examine a few more polyhedra. First let's take a look at the truncated icosahedron, the figure is made of 20 regular pentagons and 12 regular hexagons. This may look more familar to some than the polyhedron above (it's a soccerball). The polyhedron is made by cutting off 1/3 of each edge of the regular icosahedron and as we will see, satisfies Euler's formula.

Number of faces: 32 (20 pentagons and 12 hexagons)

Number of Vertices: 60

Number of Edges: 90

F+V=E+2

32+60=90+2

90=90

EULER'S FORMULA WORKS!

Let's take a look at the truncated cube or truncated hexahedron. This is a polyhedron formed similarly to the truncated icosedron: 1/3 of each edge is cut from a cube.

Number of faces: 14 (8 equilateral triangles and 6 regular octagons)

Number of Vertices: 24

Number of Edges: 36

F+V=E+2

14+24=36+2

38=38

EULER'S FORMULA WORKS AGAIN!

Now let's try the Great Icosahedron (non-convex polyhedron):

Number of faces: 20

Number of Vertices: 12

Number of Edges: 30

F+V=E+2

20+12=30+2

32=32

EULER'S FORMULA WORKS AGAIN!

Let's take another polyhedra that is not convex. Take a rectagular prism and place a triangular prism on the top of the rectangular prism forming something similar to the picture below:

Total number of Faces: 10

6 from the rectangular prism, and 4 from the triangular prism (the bottom is no longer a face)

Total number of Vertices: 14

Total number of Edges: 21

F+V=E+2

10+14=21+2

24=23

EULER'S FORMULA NO LONGER HOLDS!

This case in itself proves that the formula cannot possibly work for *EVERY* polyhedra, but it does work for some. So then this leads us to the ultimate question: for which polyhedra does the theorem hold? An excellent book dedicated solely to this interesting formula was written by Imre Lakatos called Proofs and Limitations: The Logic of Mathematical Discovery. He considers many proofs as well as counterexamples.