Ah, the joys of the intermediate value theorem.

With what precision are you measuring the temperature. If you are measuring the temperature with sufficient precision, I would imagine the statement is never true. If you are measuring it within a degree or two then I would imagine it is often true if you are looking at the tropics over the oceans and in temperate zones during spring and fall.

Except that when it's daytime on one side, it's night on the other. So maybe not.

(And for those of you turned off by the math, I promise you don't need anything complicated. As for the Intermediate Value Theorem, you don't need anything more sophisticated than the fact that if it's 0 degrees in NY and 10 degrees in Boston then it must be 5 degrees somewhere between them. (And 4 degrees somewhere between them, and 9.99 degrees somewhere between them, and pi degrees somewhere between them, and 0.[TheNumbersFromLost] degrees somewhere between them, etc., etc.)

Temperature is not a discrete number. All you need is two antipodes with daily highs and lows where one place has a higher high and lower low and sometime during that day their temeperatures passed each other. Duplicate except hold temperature is held constant and location is varied. Is this some kind of test? What did I win?

This is technically not a "proof," but there aren't too many places where land-to-land antipodes are even possible. For example, I believe that anywhere in the lower 48 states, if you "drilled a hole through to the other side of the earth," you'd hit the Indian Ocean. About four-fifths or so of Earth's land is antipodal to water.

However, I believe parts of Argentina and Chile are antipodal to China. Also, I believe parts of Norway are antipodal to Antarctica. Furthermore, if memory serves me, parts of Spain are antipodal to parts of New Zealand as well.

Kevin, it has to be true around the Equator. Just as it has to be true that there is a place on the Arctic Circle where directly across the Arctic Circle (not going through the center of the Earth) the temperature is the same.

I like James's proof. Here's a graphic way to think about it. Let the x-axis refer to degrees around a circle. Let the y-axis refer to temperature. Draw any continuous graph that crosses the x-axis at (0,0) and (360,0)--the idea being that when you make it all the way around the circle and wind up at the same point the temperature is the same. Now draw the same-shaped graph shifted over 180 degrees to the right---i.e., the same shape, but crossing the x-axis at (180,0) and (540,0). At any point where the two graphs cross, there will be a point on the first graph, 180 "degrees" to the left, that has the same y-value (i.e., the same temperature).

I think CJ is wrong about the egg thing though. And even if s/he is right, is it in any way related to the antipode?

Ok, so any great circle has two antipodal points at the same temperature on it. That's boring first year analysis stuff.

Here's the real question: demonstrate that on the earth there are two antipodal points with the same temperature _and_ pressure!

Consider just the equator. Let the temperature fcn. at time t be defined as f ranging from 0 to 2pi. If f is constant the result is trivial. If f is not constant define f over the whole real line periodically in intervals of 2pi. Now define g(x) = f(x) - f(x+pi). f is continuous by assumption, and since g is the difference of two continuous fcns. it too is continuous. Since f is continuous but not constant, therefore g must have both pos. and neg. values. Since g is also continuous, therefore there exists x s.t. g(x) = 0. QED

In fact, there are antipodal points on the earth that have the same temperature and barometric pressure.

Let T be a temperature distribution on the earth and let U(T) be the set of points whose temperature is the same as their antipodal points.

We know from the comments above that U(T) intersects each great circle on the earth. Of course U(T) is symmetrical. In general, what properties are sufficient for a set S on the earth's surface to equal U(T) for some T?

I don't think that you are reading the quote correctly. Her first sentence is:

"There's a spot on the earth where the temperature is exactly the same as it would be if you drilled through the earth to the other side."

You could obviously say that she (or the writers) mis-spoke, but what she is claiming has absolutely nothing to do with two spots on either side of the Earth. She is saying there is some point on the Earth's surface, let's call it 'X', at which the temperature would be completely uneffected if a hole were to be drilled through the Earth all the way to the other side.

It seems to me that such a hypothetical hole would have to have SOME effect, though I suppose that would depend on it's radius. I also don't see how such a thing could be proven or disproven.

James's solution was the one I had in mind. You can actually view the whole earth instead of just the equator, but use his trick of looking at the difference between the temperature of every point on earth and the point completely opposite it. This is, of course, still a continuous function. It can't be positive everywhere (because if it's positive in NY then it's negative in the point directly opposite NY), and similarly can't be negative everywhere. So it's positive in some places and negative in others. This means that it must be 0 somewhere. Wherever it's 0, the temperature is the same at two opposite points.

Is there any sense to the assumption that temperature is continuous? Maybe there is; certainly, it makes sense to assume that there is no place where the temperature is 0 immediately next to a place where the temperature is 100.

But if you assume, as I think we are, that the notion of place is not necessarily continuous (i.e., temperature is a measure of a discrete place or field), then why would temperature be continuous?

There are an infinite number of such points. Connect any two points that are at the two ends of a diameter of the earth. Assume they are of different temperatures. Now rotate the diameter line until the ends are reversed. As each point was traversing the temperature range from what it started at to what the other end started at, the two temperatures must, at some point be the same.

I have personally balanced an egg on end during an equinox.

Being a scientist, I repeated the experiment every day for 10 days.

It's a matter of eggs and steady hands, and the equinox is irrelevant.

I don't watch that TV show, and the comment by "Will" makes no sense to me. There must be some context missing...

The trouble is the phrase "through the Earth" which can quite reasonably be taken as through the center of the Earth. If you make such a restriction that there must be a line passing through the center of the earth, then it is false that there must be a pair of points on the perimeter intersecting such a line that are of equal temperature.

It is only once the statement is relaxed to mean "any straight tunnel" that we can be assured of finding a solution.

Eng--

CJ's quote doesn't require that it hold true for every (any?) point. Rather, she says that there is "a spot on the earth" for which this will be true, given that the line in question is through the center of the earth. If the spot is on the equator, runs through the center of the earth, and hits the equator on the other side, there is probably a solution (as other posters have pointed out).

The temperature _is_ continuous, although this gets into physics, not math, and you have to measure very precisely. In any medium allowing heat diffusion, diffusion instantly smooths any transient heat discontinuity into a continuous but initially steep shape. Thus, if you have a magic wand that can create a temporary temperature discontinuity, the discontinuity will vanish right way.

Moving objects and the differences between materials complicate things, but to simplify, just replace "the earth" with "the atmosphere X meters above all buildings, ground, moving objects, and near-ground heat sources". The same argument applies to that layer of atmosphere.

As an uneducated remembrance, it was my impression that temperature and pressure were closely related, and that pressure was a sampled statistic of a point _oer time_ used to "represent" an area. Also, although there is diffusion, it is not clear that this is an instant process.

In light of these three assertions, is temperature necessarily continuous?

(Not that this in any impugns the solutions offered to the actual problem posed - just an aside that seems interesting from here.)

Maybe I'm just confused, but isn't it pretty trivial to find counterexamples to the contention that there would mathematically always be an antipode with equal temperatures?

Take the equator example. Picture the equator as a clockface--from 12 to 2 on the clockface (i.e. for 60 degrees of a circle)it is 100 degrees fahrenheit. At 2, it changes to 105. At 6 (180 degrees) it changes to 115 and at 7 is 110, and remains there until 11 when it goes back down to 100, returning to 100 at 12.

It makes more sense is you draw it on a piece of scrap paper. :)

There do not seem to be two antipodal points with the same temperature.

Juan notwithstanding the Volokh: since no cliff will be perfectly verical, there will always be some slope and thus a smooth gradient. I guess you have to assume that the earth's surface is relatively smooth (sorry, I'm sure there's a technical term for this and I'm sure I'll know it by the end of the semester when I finish my topology course).

Philistine: we assumed that the temperature was distributed continuously. You have four points of discontinuity there, which is why it doesn't work. But physically, you can't have a discontinuous temperature field for a non-infinitesimal duration.

Kevan Choset

In the case you describe, the antipodes will be between 2 and 6 and between 7 and 11. Between 2 and 6 the temperature changes from 100 to 115 and between 7 and 11 the temperature changes from 100 to 110. Somewhere in there, the points opposite will have the same temperature as each other.

I still don't see it. In my hypothetical, there could be equivalent antipodal temperatures--but only coincidentally. The vast majority of situations would not have them.

For instance--keep my hypothetical, but refine it. All temperature changes are not "instantaneous" but occur in the 1 minute (6 degrees of the circle) before they are given. (Thus from 100 to 105 occurs just before 2, 105 to 115 just before 6, 115 to 110 just before 7 and 110 to 105 just before 11.

If this is the case, I don't see antipodal equivalents. Am I missing something?

--Philistine

I do not believe that any of the given math proofs work. They make too many assumptions about the physics of temperature distributions that are not true. They assume that because one spot is temperature A and its antipode is temperature B, the difference A-B can be brought to zero by rotating around a plane that passes through the center of the earth. As others have noted, this assumes no temperature discontinuities. The only way that we can guarantee no temperature discontinuities is if we assume that the spots at which temperatures are measured are infinitely small, a physical impossibility (because temperature requires moving molecules, and molecules have size). Another false assumption is that the A-B difference as we rotate around the bisecting plane is linear. Why should it be? If the gradient is non-linear, then there may never be a place where the antipodal temperature difference is zero.

SLS - Concedering arguendo that temperature becomes undefined when you try to consider it on a small enough scale (i.e., for a point as opposed to for a volume), why would you assume that "continuity is a reasonable physical approximation"?

(Again, this is not to detract from the elegance of the proof adduced above by many.)

Nope, uniformity is not required.

SLS: The normative judgment "bad physical approximation" is appropriate or inappropriate depending on the context, obviously, and, for many purposes, the assumption of continuity renders feasible many calculations and understandings with few errors or other problems.

Here, however, the proofs all depend on the fact of continuity. So, it is a reasonable to question to ask. Just saying that it is a good assumption does not make it so (especially where we've already discussed that temperature is a statistic, not an attribute, as far as the temperature of a particular point at a particular time).

Siona: By connected, I meant path connected. Sorry.

Let U be a path connected subset of R^2 \ {0} s.t. if p \in U, then -p \in U.

Pick a point p. Then, by assumption, there exists a path q(t) in U s.t. q(0) = p and q(1) = -p.

Define the following path

r(x) = q(2x) for x \in [0,1/2]
r(x) = -q(2x - 1) for x in [1/2,1]

By assumption again, r(x) is in U. It's easy to see that r is continuous and that r(0) = r(1), so it is a circle. It can be seen to be noncontractible in R^2 \ {0} by looking at polar coordinates, for example.

From the "problem" :"You can assume that temperature is continuous (i.e., there's no spot on the earth where the temperature just "jumps" from 0 degrees at one spot to 100 degrees at a spot infinitesimally close to it)."

So let’s leave out of it the "real" world for the sake of the conditions in the problem to solve. I don't have a problem with the mathematical proof given so far for opposite points from an equal midpoint. Dr.T adds some excellent points I think. The thing is (and I wish someone could put this in mathematical terminology for me) that proving that is not enough. The problem calls for equal temperatures as well as antipodes.
The problem calls for a cross correlation of "same area" between the math proof given for the equidistant points AND the ranges where the temperatures are the same at any given instant. It is worth noting that it doesn’t say the antipodes(as far as space-time are concerned)must remain in any sort of continuity. They could exist in any of possible area that the conditions are able to be satisfied in at any given instant.

I think the key to the problem is given in the final statement about the temperature being continuous. It then even defines continuous for us by saying you cannot find a place to measure temperature, "at no spot on the earth", where a "spot infinitesimally close to it" has a non-continuous jump of value.

What does this mean if not that two points(or the range of area defined by the points in the surface of a sphere intersecting with the surface of the earth) are connected by a continuum of temperature? That you can not MEASURE any two spots "infinitesimally close", or more to the point, any two points NOT that close(range of min/max variable in earth topology) without finding a gradient of values between the two points. It doesn't just jump from 0 to 100, 25 to 75, or anything to any other thing without being able to MEASURE a gradient between the two points in space.

I am sure there is a mathematical way to formulate this. To me it means since there are an infinite number of gradients between two points equidistant from each other then measuring for value in those gradients(temperature) can give infinite results. Somewhere in infinity will be a matching condition at any given instant?

I also think you would never be able to prove this type of "problem" by observation of nature. It's mostly mental masturbation.

Antipodes, islands, New Zealand - rocky uninhabited islands, 24 sq mi (62 sq km), South Pacific, c.550 mi (885 km) SE of New Zealand, to which they belong. Explored by British seamen in 1800, the Antipodes are so named because they are diametrically opposite Greenwich, England.

Prove to me that Greenwich and The Antipodes never have the same temperature.

I agree with Kevan that James' explanation was best, and I think Kevan's illustration is best.

"If the difference-function is nonzero at some point (call it A), then the difference-function is the negative of that nonzero value at the exact antipode of A."

I missed that insight, which makes his solution much more elegant than mine. I proved it by showing that:

1 if your difference function ever returns 0, then the points match

2 if it's always negative, that means temp always decreases as you go around the circle, but that's no good because if you go around and around multiple times it keeps getting colder--but at 0* it must be the same temperature as at x* when x approaches 360.

3 if it's always positive, that means that means the temp always increases as you go around, with the same problem.

4 if it's both positive and negative at times, then it must pass through the origin (be equal to zero at some place), because it is continuous. (this is where the Mean Value Theorum actually comes into play)

But my proof is better, because it can be expanded to show to show that for ANY continuous cyclical path with continuous temperatures, for ANY distance between the two points (not necessarily "halfway" but 10%, or 10 miles, etc.) you will be able to find some two points that satisfy the distance requirement AND are at equal temperature. So there :-)

By definition the "antipode" of a point on a sphere is the point defined by the intersection of a line through the original point and the center of the sphere and the surface of the sphere and that is not the original point. All arguments using the Intermediate Value Theorem, Mean Value Theorem, Rolle's Theorem or any variation just show that at least two points on a circumference of a sphere have the same value for any continuous function mapping the surface of the sphere to the real number line. None of these proofs demonstrates that these two points are antipodes except in very special cases. The property everyone is attempting to prove is a global property of the sphere and dependent on the sphere's topology. What everyone's trying to prove must be a corollary of the Brouwer Fixed Point Theorem: probably a proof by contradiction involving a constructed pull-back function. I don't have time to work out the details. By the way, there is an elementary introduction to category theory -- a mathematically inclined high school student could handle it -- that contains a proof of the fixed point theorem using only category theory. Very neat stuff. I didn't see a proof of Brouwer's theorem till my junior year at university.

I was wrong. Aaron Bergman has convinced me the Mean Value proofs work.

The Earth is tilted, so that the poles are in complete darkness or complete light at certain times of the year. The only times that the poles may have the same temperatures is when the Earth's tilt is tangental to the Sun -- this happens twice a year.

Also, Bergmann, I am only picking on you because you seem to be one of only about 5% of the posters on this thread who do not clearly have no idea whatsoever what mathematics or a mathematical proof even is; the remaining 95% of the posts cannot even be discussed.

I had no idea so many other math people read the VC!

/feels a little better now

Siona Sthrunch:

Perhaps it's just for the 2-dimensional case? There are a number of cute ways of proving it that don't require very much machinery at all: Sperner's lemma, the game of Hex, some easy covering space arguments to get the fundamental group of the circle, etc.

BobGo--

I believe it is correct that there are not only at least two such points, but there are an infintite number of points (in fact, I think that these points must form a continuous path, that crosses the equator at least twice--and so it also crosses any circumference--but I could be wrong about that. In fact, there may be more than one such path, but there is at least one.)

The trick to thinking about it is, the points on this path need not all be the same temperature. We're only talking about equivalent temperatures--but any temperature can have an equivalent. We're also assuming that a temperature scale is smooth, so you can slice it up into infinitely fine degrees. This also means that it's not necessarily true for every single temperature on the scale--in other words, it might only be true at any given time for the range between 50F and 51F, for example.

It's mathematically true, but in this case the mathematics can reflect the real world.

Since this does require continuity of both the temperature and the surface of the earth, wouldn't a discontinuity allow for the conjecture to fail?

Continuity of temperature requires a careful definition. It's true that at the sub-molecular level temperature loses it meaning. So we can reasonbly say that the temperature of a point in space is defined as the average for a sphere centered on that point. (Pick any size sphere, as long as it contains enough vibrating molecules to make the concept of temperature meaningful.)

But ...

In the real world there are truly discontinuities in the surface. The obvious example is an OVERHANG, noted by Dick above. If your great circle passes over an overhang, you truly get a discontinuity in elevation. And therefore you truly get a discontinuity in temperature.

Say the overhang is 10 feet high and in the sun. Say you step over the edge to the bottom, which is in the shade. The temperature change may be continuous over this 10-foot drop, but that 10-foot path is not part of the great circle.

So does this allow the conjecture to possibly fail??