Random Coolness

[musik_man]

**NOTE**Nothing in the following post is in anyway cool**NOTE**
I came across an interesting math-y problem and am wondering if any more experienced mathmeticians can see if my answer is right.  The problem stems from the one given here.  Basically, prove that there is always at least one point on the Earth where the Temperature is the same as the temperature at the antipode.(if you dug a hole through the center of the earth, you'd end at the antipode of where you started digging)
https://volokh.com/posts/1142373636.shtml
The solution for this is present in the comments, but I'll repeat it.  You have to know 1st year Calculus to understand it.

Assume that Temperature is a continuous function.
Draw any great circle on the globe
Let T(Ø) represent the temperature of a point on this circle defined by its angle Ø
The temperature of its antipode would then be T(Ø+pi)
Now let's make a function f(Ø)=T(Ø)-T(Ø+pi) this gives the difference in temperature between a spot and its antipode
f(Ø)=-f(Ø+pi), since f(Ø+pi)=T(Ø+pi)-T(Ø+2pi) and T(Ø+2pi)=T(Ø), f(Ø+pi)=-[T(Ø)-T(Ø+pi)]
This means that if at any point f(Ø) is positive, there must be a negative value pi further
By the Intermediate value theorem [if a continuous function has f(a) and f(b) there must exist all f(c)'s where f(c) satisfies f(a)<f(c)<f(b)] we can prove that at some point there must be a spot where f(Ø)=0, or where a spot and its antipode have the same temperature

Now if I was just a nerd, I would've found that interesting and stopped there, but I'm some sort of super nerd.  For any sphere there are an infinite number of great circles.  The above proof works for all of them, meaning that there are an infinite number of such antipodes.  So I set out to prove that there is a line you could walk on the earth for which all points are at equal temperature with their antipodes, and that you could walk on this line to the antipode of where you started.  I managed to get a haphazard type of proof of this done with my mathmatical knowledge, but I think it could be done more elegantly.

Any in 3d space can be defined by r,ø, and Ø (basically, we're working in spherical coordinates)(r is going to stand for f)
Take the value Ø0 on the great circle from above
If you rotate the great circle about an axis defined by the points Ø0-pi/2 and Ø0+pi/2, the point Ø0 would cut out a great circle
Using the above proof that means for every Ø there exists a ø, where f(Ø0,ø)=0 (ø is the angle used to define position on the new great circle)
We can redo the situation, starting with ø and prove that for every ø a Ø exists where the antipode is of equal temperature(but I'm not gonna write it out)
Now let's go back to step two and change Ø0 to Ø0+dØ
For the new value there still must be a ø, where f(Ø0+dØ,ø)=0
Furthermore since f is a continuous function, the new value of ø0 must be equal to the old value plus dø
So f(Ø0+dØ,ø0+dø)=0
If we continued adding these infintesimal points, we'd end up with a line; Furthermore, since when f(Ø,ø)=0, f(Ø+pi,ø+pi)=0, the line would contain both the position and its antipode
This means that you could walk from a spot of equal temperature to its antipode while walking entirely on spots with equal temperatures to their antipodes 

[pianistimo]

i've just gotten in trouble for drawing a great circle on the globe.  do you think longitude is cool?  i do too.

anyways - sorry to post where i shouldn't.  i am neither a mathematician or geni - although i find the process of turning lead to gold interesting.  wondering how the nuclear power plants do it.  can you make that another lecture.  (ok.  i'm a bad girl today).

[pianistimo]

i've been pondering the ideas you put forth on antipodes and actually find it quite facinating that there's a reason for temps being consistent at an antipode point.  i mean they COULD be inconsistent - and you explained why they are consistent.  are you saying there might be a magnetic something going on? - or that it doesn't have to be soley the distance from the sun?  the earth releases heat after the day has ended, right?  so there is a flux of energy coming to and leaving the earth.

basic question - but a question - nonetheless:  does water or solid matter heat up faster?  i'd say water, then sand, then solid ground (in terms of heating from fastest to slowest)  i mean if you have the sun beating down on an area of the earth that is beachfront - it heats faster than inland areas, right?  sand heats up faster that harder soil, too, right?  so the antipode could be dependent upon what matter it is composed of?  maybe i have that backwards.  maybe it is the hardest soil that heats up fastest and yet the water would retain the energy longer and takes longer to heat up? 
guess that a few degrees of change in sea temperature greatly affects the entire globe.

with so many factors (such as weather) wouldn't the antipode change from hour to hour (or less) - or are they fairly consistent?