Is it hot in here?

Relevant wiki: Intermediate Value Theorem

Consider taking the temperature $T_1$ and $T_2$ at two opposite points, and consider $T = T_1 - T_2$ .

Now consider taking the temperatures at their opposite points like so:

The difference of the temperature in the second case will be $-T$ . Since the temperatures and their difference varies continuously as you move the points through the surface of the earth, and it went from $T$ to $-T$ , it must have crossed zero somewhere, which would mean $T_1 - T_2 = 0$ so $T_1 = T_2$ .

Edit: A lot of people seemed to be uncomfortable with the previous proof which is fine since my oversimplification can leave out some details, but I thought I'd go a bit more in-depth on what I was doing up there. When I say, take $T = T_1 - T_2$ , what I'm actually saying is, define $T(x, y, z) = T_1(x, y, z) - T_2(x, y, z)$ where $T_1$ is the temperature at the point $(x, y, z)$ and $T_2$ is the temperature at the antipode of the point $(x, y, z)$ . Now, the temperature at a given space will be continuous, so, $T_1(x, y, z)$ and $T_2(x, y, z)$ are continuous too and since they are a continuous function $T(x, y, z)$ is continuous too.

Now, let's consider what happens if we evaluate the function at a given point $(x_0, y_0, z_0)$ , we get $T(x_0, y_0, z_0) = \alpha$ now consider what happens if we evaluate the function at this point's antipode $(x_1, y_1, z_1)$ , well we defined $T_1$ as the temperature at the point and $T_2$ as the temperature at the point's antipode, so $T_1(x_1, y_1, z_1) = T_2(x_0, y_0, z_0)$ and $T_2(x_1, y_1, z_1) = T_1(x_0, y_0, z_0)$ which means that $T(x_1, y_1, z_1) = T_1(x_1, y_1, z_1) - T_2(x_1, y_1, z_1) = T_2(x_0, y_0, z_0) - T_1(x_0, y_0, z_0) = -\alpha$ . Now we have a continuous function that goes from a negative value to a positive one or vice versa, so we can apply the intermediate value theorem and see that the function has a zero. What does it mean for this function to have a zero? It means that the temperature at that point must be equal to the temperature at its antipode.

Edit 2: For people that are still having a hard time understanding this problem, I'd like to first link to a few resources that explain it in a more visual form which helps, those are great resources and definitely worth checking, and second, I decided to go over one example here of the solution.

• Dr. James Grime video
• Math easy solution's video
• Vsauce's video
• 3Blue 1Brown's video on the Borsuk-Ulam theorem
• Discussion on Brilliant

Now for the example:

We called $T(p) = T_1(p) - T_2(p)$ with $T_1$ being the temperature at the point $p$ and $T_2$ the temperature at the point opposite to the point $p$ . So let's take point $a$ with a temperature of $\ang{20}$ and point $b$ opposite to point $a$ with a temperature of $\ang{10}$ . If we evaluate those functions we get that $T(a) = 20 - 10 = 10$ and $T(b) = 10 - 20 = -10$ , now consider moving around and evaluating this function on the points from $a$ to $b$ , no matter the shape of the function, or the shape of the earth, or where the sunlight is hitting, we have a continuous function that goes from $10$ to $-10$ and since it can't jump over the x-axis, it must cross $0$ to do so, which means that, somewhere between $a$ and $b$ , and we don't know where, there is a point $c$ where $T(c) = 0$ .

Bonus: Show that at any given time there are two opposite points with the same air pressure and air humidity.

Consider the opposite points on the globe and let's call them A and B.

If you move the two points so that A is now in B's initial position and B is on A's initial position, and were to plot the temperature of their path, the two graphs have to intersect at least once.

This works on any arbitrary path, including one that goes on every point on the globe.

The Borsuk-Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.

Let's think of that mapping process as the following:

Taking every point on Earth and plotting a scatter graph of Temperature vs. Pressure. Since both of these functions are continuous, there must be at least 1 pair of antipodal points with the same temperature and pressure.

Hence, there must be at least two opposite points on Earth with exactly the same temperature.

Earth's temperature depends on many factors and location of earth with respect to sun is the most important factor. So , imagine that there are two balls facing each other. One ball is earth and other is sun. As their centres lies on the same plane so earth's temperature will be symmetrical in both of it's halves due to sun. Hence , there must be atleast two points on earth symmetrically placed and opposite to each other which have the same temperature.

Paint the globe in the following way. Paint a point Red if it is hotter than it's opposite point. Paint it Blue if it's colder. White if it is equal. This question is equivalent to asking "Are there any White points on the globe?"

Visually, consider how this looks, there will be Red regions and Blue regions, but (assuming temperature varies continuously) there must always be a White line/region separating the adjacent Red and Blue regions. (As others have pointed out, the Intermediate Value Theorem is the rigorous way to prove this).

For those who think it is false and can still not see why, try creating a scenario where this is true. You will find that it is impossible.

There must be opposite points on the equator.

We will assume that f, where f(a) is the temperature at point a, is a continuous function from the unit circle to the real numbers. By the Borsuk-Ulam theorem, there exist antipodal points a and -a on the unit circle such that f(a)=f(-a). It follows that I points a and -a have the same temperature.

Not a proof but pragmatic: I visit the South pole (Antarctica) and my brother visits the North pole (Arctic ice pack). I'm pretty sure that we can wander around to find a spot so we are opposite each other and have the same freezing temperature. And this at any given point in time.

Following on from J Areias' solution, are there not an infinite number of such twin pairs of antipodes, since all possible great circles must contain at least one? And further, is there a path (not straight) joining these points, so that one could in theory travel along a temperature contour back to one's starting point?

On the surface of a sphere, there is an infinite number of points. Particle speed is however discreet(particles jump from one speed to another instantly at the quantum level). therefore temperature scale is a smaller infinity than the number of infinitesimal points of a sphere. (the numbers can not be mapped to each other). Therefore there is an infinite number of opposite points on the earth that are the same temperature.

Yes bcuz when f=c, they both equal to 40.

This seems to further imply that there is a connected path about the world for which the temp is the same on the other side.

There is a Vsauce video about this, the topic starts at minute 11.

https://youtu.be/csInNn6pfT4