And now for the mathematical considerations

Today is a palindromic date: 2-22-22. While some people may view this as a lucky date, I view it as an opportunity to reflect on the humble number “two.” We all know that two is the smallest prime number and the smallest positive even number. However, here are two more mathematical facts involving two.

Picture a cube and count the number of corners (vertices), edges and flat sides (faces). You will find that a cube has 8 vertices, 12 edges and 6 faces. Notice that 8 – 12 + 6 = 2. Now picture a tetrahedron (a 3D shape with triangular faces). It has 4 vertices, 6 edges and 4 faces. And 4 – 6 + 4 = 2. In fact, any 3D shape that has polygonal faces and does not have holes will have the property that

 

Vertices – Edges + Faces = 2

 

This is an example of a mathematical invariant called the Euler Characteristic, and it is particularly relevant in the field of topology. It also plays a minor but poignant role in Cindy Neuschwander’s children’s book Sir Cumference and the Sword in the Cone.

Students who have taken Calculus 2 might recognize the following sum:

 

1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …

 

This is known as a geometric series, which is an infinite sum (as indicated by the ellipses), such that each subsequent term is a fixed multiple of the one before it. In this case, the fixed multiple is 12.

It might seem strange that you could add infinitely many numbers together and get a finite number, but indeed you can. In fact, if you add the reciprocals of powers of two, you get a convergent series. Even more,

 

1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … = 2 !

 

The notion of geometric series dates back thousands of years. This particular series is related to the well-known Zeno’s paradoxes, which math students have been talking about at parties for years. Or maybe that was just me.

Whether today is a day of luck or simply an opportunity to read a few fun facts about 2, I hope you find some joy on this special Twos-day.

Andrea Young
Interim President
and Associate Professor of Mathematical Sciences


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