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 $t(x)=t(-x)$.

Comments

I'm sure I am missing something, but for this continuous function there is 'a' maximum value and 'a' minimum value, (1st moment =o) this is only at a single (x) value?

The theorem states that there are two points on the equator with the same temperature. It does NOT state that for any given temperature found on the equator there is another at that same temperature.