Maths in a minute: Equal temperatures

Maths in a minute: Equal temperatures

At every given point in time there are two points on the equator of the Earth that have the same temperature.

How do we know this? Well, here’s a proof. Let’s look at the equatorial plane which slices through the Earth at the equator. The equator is a circle which lies in that plane, and we can choose a coordinate system on the plane so that the point $(0,0)$ lies at the centre of the equator. For each point $x$ on the equatorial circle there is a point $-x$ which lies diametrically opposite $x$.

circle

Points x and -x.

Now each point $x$ on the equator comes with a temperature $t(x)$. We can assume that the function $t$, which allocates a temperature to each point, is continuous. That’s because temperature doesn’t suddenly jump up or down as you move around on the Earth.

Now consider the function

  \[ f(x) = t(x)-t(-x). \]    

It is also continuous.

If this function is equal to $0$ for some point $x$, then we are done because if

  \[ f(x) = t(x)-t(-x)=0 \]    

then

  \[ t(x)= t(-x), \]    

so the temperature at $x$ is the same as the temperature at $-x$.

If $f(x)$ isn’t equal to $0$ anywhere, then let’s assume (without loss of generality) that there is a point $x$ at which $f(x)>0,$ so

  \[ f(x) = t(x)-t(-x)>0. \]    

This implies that

  \[ f(-x) = t(-x)-t(x)=-f(x)<0. \]    

There is a result, called the intermediate value theorem, which says that if a continuous function is greater than $0$ at some point of its domain and less than $0$ at another, then it must equal $0$ at some point in between the two.

Intermediate value theorem

Illustration of the intermediate value theorem. If t(x)>0 and t(y)<0 and t is continuous, then there is a point z between x and y such that t(z)=0.

Thus, since $f(-x)<0$ and $f(x)>0$, there must be a point $y$ on the circle such that $f(y)=0$. So

  \[ f(y) = t(y)-t(-y)=0 \]    

which means that

  \[ t(y)=t(-y). \]    

So the temperature at the point $y$ is the same as the temperature at the point $-y.$

The result actually holds for any circle on the Earth, not just the equator. In fact, the result is the one-dimensional case of the Borsuk-Ulam theorem, which says that for any continuous function $t$ from the circle to the real numbers there is a point $x$ such that $t(x)=t(-x).$

The more general version of the Borsuk-Ulam theorem says that for any continuous function $t$ from the $n$-sphere to the set of $n$-tuples of real numbers there is a point $x$ such that $t(x)=t(-x)$.