After playing with the "dig a hole through the earth" tool built on
google maps (http://www.digtootherside.cjb.net/), I wondered in what %
of dry land can you dig and come up in dry land?

It's been a while since statistics class (and there was only one, so
very long ago), so if you could also include the math/work behind it,
I'd appreciate it.

The following is not as precise as detailed statistics, but it was
enough to satisfy my curiosity.

End result:

Approximately 10% of land should be opposite land if you dig straight
through the earth.

Methodology:

The point opposite a point north of the equator is always a point
south of the equator and vv.

Since there is less land south of the equator, it is easiest to
estimate using those land masses; anything north of the equator will
match water unless it matches one of the southern land masses.

Using three points for Africa south of the equator (a point near the
Cape of Good Hope and two points along the equator at the east and
west ends of Africa) shows that this part of Africa matches an area in
the Pacific Ocean.

Using four points to define Australia as a rectangle shows Australia
matches an area in the Atlantic Ocean.

Most of Antarctica is very far south;  the only north-of-the-equator
area of any size that matches it is most of Greenland and some
neighboring Canadian islands.

South America from its southern tip to about the top of Chile, then
slanting southeast, matches land area in Asia.

Area data used: 
(from http://www.cia.gov/cia/publications/factbook/geos/ci.html and
the corresponding fact book pages for other countries and for the
entire world)

Greenland	2,166,086
Argentina	2,736,690
Chile	           748,800
Paraguay	  397,300
Uruguay	  173,620
Bolivia     1,084,390

sum of these areas	7,306,886 sq km

land area of the world     148,940,000 sq km

7,306,886 / 148,940,000 = 4.9% for the areas listed

The total land area opposite other land area equals the areas listed
plus the matching areas on the opposite side of the earth - which
doubles the amount and percentage.

That's a neat analysis, brix24-ga!

Since I once lived there, I checked out Bogor, Indonesia (south of
equator, on the island of Java) and found that it is (roughly) on the
opposite side of the world from Bogota, Colombia.  The Indonesian
archipelago won't add a lot of surface area to your totals, but if we
were weighting things by population density, it would make a splash!

regards, mathtalk-ga

Thanks, mathtalk-ga!

Since Indonesia is so spread out, it was easy to forget that its area
is considerable (1,826,440 sq km). It looks like the end regions of
Indonesia are opposite an ocean (western Sumatra opposite an area of
the Pacific and Irian Jaya opposite an area in the Atlantic). About
half of Indonesia, though, looks like it matches parts of northern
South America. So, since we need to include both these central parts
of Indonesia and the matching parts of northern South America, I would
like to revise my estimate to about 11%. (1,826,440 / 148,940,000 =
1.2%)

That's a great way to think about it. I'd done simple rough
percentages and came up with 9%, and didn't even think about trying to
estimate it this way.  Great answer! thanks!

I get that 13.5% of land is opposite land.  MATLAB has a built in
1-degree resolution earth topography matrix.  If you sum up all the
'above sea level' elements which are opposite other 'above sea level'
elements, and normalize by the cosine of the latitude and the total
number of 'above sea level' elements, you get 13.5%.  Of course, there
are some lakes whose bottoms are above sea level, and dry valleys
below sea level, but I think the biggest of these are opposite ocean
anyway, so don't matter.  Also, 1 degree is quite coarse, but since
the sum is taken over so many elements, the error associated with
graininess should be smallish.  I think +/- 1% are very conservative
error bars on the 13.5% estimate.

Cambodia and parts of Vietnam and Thailand are opposite Peru and
Ecuador.  A good chunk of New Zealand is opposite Spain.

If you have MATLAB, the following code makes a pretty picture:

load topo
topo2=flipud(topo);
topo=[topo(:,181:360) topo(:,1:180)];
lat=-89.5:89.5;
lon=-179.5:179.5;
tblue=topo;
tblue(tblue<0)=0;
tblue(tblue~=0)=1;
tred=topo2;
tred(tred<0)=0;
tred(tred~=0)=2;
tboth=(tred+tblue)+1;
colormap([0 0 .2; 0 .3 .1; .1 .3 .6; .6 .6 .2]);
image(lon,lat,tboth)
daspect([1 1 1]);
set(gca,'xtick',[-165:15:165],'ytick',[-75:15:75],'yDir','normal');
grid

From 

http://www.ngdc.noaa.gov/mgg/gdas/gd_designagrid.html

I downloaded 10-minute global topography and repeated the above
calculation.  13.50% of the earth's surface that is above sea level is
opposite a spot that is also above sea level.  Land below sea level
and water above sea level introduce an unknown but small error.

Wow. Now I wish I had MATLAB or access to it at least...

Well, I know I've seen people post links to images.  Do you know how
to do that?  If you tell me, I'll post a figure.

http://viewmorepics.myspace.com/index.cfm?fuseaction=viewImage&friendID=42779117&imageID=357544301&Mytoken=14B951EF-C179-D34F-4DC08722AEC2B96F98390042


Hope that works.  You will probably have to copy and paste the
multiple lines of the url...

Does myspace use cookies? All I see is a dot and the comment/email
buttons. Email it to yourself and post that url?
Thanks for putting the picture up in the first place!

FWIW, it worked when I tried it a while ago, but now it's not showing up.

You will like the visualization rracecarr created, but I had to take a
little time to get used to the superimposition at first.

I once printed out two copies of a blank world map, inverted one,
placed them on top of one another, but somehow my paper
superimposition was off according to "digthroughtheearth." rracecarr's
is not off.

It works in Firefox. WOW that is really cool. Thanks!

racecarr - that picture is fantastic. I can't thank you enough.

Hey, no problem.  Glad you like it.

brix, you did an excellent job too. I could not be happier! 
Originally I was just toying with the probabilities

Roughly, the earth is 30% land, 70% water.
So I figured that the (lower bound) chance for starting on land and
endiung up on land was only 9% (30% * 30%, doing the intersection
based on independent factors).

I was pleased to get way better answers.

Thanks!

Thanks for the kind words! The problem is interesting in itself, but
I'm also glad to hear that my efforts helped.

Does this mean that 13.5% of the land on the 2D map is acrood from
land, or on a 3D globe?

I'm aware that the way the maps were superimposed on eachother is
consistant with a spherical globe, but the areas (especially near the
poles)are stretched on the map.  Also...where did the north pole go?

If the question was asking what the odds are that picking a point ON
GOOGLE MAPS that digs from land to land, the 13.5% sounds good.  If
the question is asking what the odds are of a line connecting land on
opposite sides of the globe, brix24 may be closer to a solution, but
is still a rough estimate.

The 13.50% is for the world, not a map.  That's what I meant by
normalizing by the cosine of the latitude.  I suppose it's possible to
calculate the percentage for umpteen different flat projections
(cylindrical, mercator, etc. etc.) but that doesn't seem very
interesting.

fractl-ga asked: where did the north pole go?

Excellent question.

There is a polar ice cap but no land at the north pole:

[Beyond "Polar Express": Fast Facts on the Real North Pole]
http://news.nationalgeographic.com/news/2004/11/1108_041108_north_pole.html

"There is no land beneath the ice of the North Pole. The Arctic ice
cap is a shifting pack of sea ice some 6.5 to 10 feet (2 to 3 meters)
thick..."

The northernmost point of land known is a tiny island off the coast of
Greenland, at latitude 83° 40? 34.8" N:

[New Islands, New Northernmost Point of Land on Earth
 Discovered off Northern Coast of Greenland]
http://www.dpc.dk/PolarfrontenDPC/3_01/IslandsReport.html

There is land (Antarctica) at the south pole, although technically the
polar ice cap there has perhaps depressed much of the land to below
sea level:

[British Antarctic Survey -- School Pack]
(see esp. page 3, Cross-section through Antarctica)
http://www.antarctica.ac.uk/Resources/schoolzone/resources/SchoolsPack/01nature_dl.pdf


regards, mathtalk-ga

For anyone who's interested, a friend had this kid's book: How to Dig
a Hole to the Other Side of the World by Faith McNulty and Marc
Simont. It details the difficulty of digging through extreme heat at
the center of the earth. It was fun.