Anyone who has driven across the United States from coast to coast has
most probably seen the signs proclaiming the Continental Divide, that line
or arc running north-south where any water falling on one side - the east side
- will flow to the Atlantic Ocean (most probably via the Gulf of Mexico),
and water falling on the other side - the west side - will flow to the Pacific
Ocean. These signs announcing the Continental Divide are usually found in
the Rocky Mountain states of Montana, Wyoming, Colorado or New Mexico. Drivers traveling a northern route may be surprised to see signs in North
Dakota announcing the Continental Divide. North Dakotans are not liars
nor practical jokers, but indeed a divide does pass through North Dakota, a
divide where water on one side will flow north into Canada and into Hudson
Bay while water on the other side will flow south into the Gulf of Mexico.
This line will have a generally east-west orientation, and we can well assume

this divide and the more well known divide will have a point of intersection. This point of intersection is in Glacier National Park and is known as Triple
Divide Peak. An `Ideal' raindrop falling on the very summit of Triple Divide
Peak will divide into three subdroplets. One subdroplet will flow north into
the Saskatchewan River, into Lake Winnipeg and from there down the Nelson
River into Hudson Bay. A second subdroplet will flow into the Marias River
then into the Missouri and eventually the Mississippi and the Gulf of Mexico.
The last subdroplet will flow west into the Flathead River, then the Clark
Fork and eventually the Columbia and the Pacific Ocean.

We can generalize this phenomena by saying that any point where an
ideal raindrop will split into three parts and flow into three separate drainage
basins is a Triple Divide Point. Marty Issacs, a mathematician at the University of Wisconsin at Madison, once told me that while traveling in Canada he
saw signs informing him of the North American Triple Divide Point. Indeed
a continental triple divide point would occur there as there would be a point
where water would divide three ways, one way to Hudson Bay, a second to
the Arctic Ocean via the Mackenzie River and the third to the Pacific. In
fact triple divide points are ubiquitous in North America. Wherever there is
a confluence of two streams there is a Triple Divide Point uniquely associated with the confluence. Where there is a confluence of two streams, three
basins are formed. To see this let us think of Pittsburgh Pennsylvania where
the Allegheny and Monongahela Rivers join to form the Ohio. This confluence will define a partition of North America into three disjoint regions, the
drainage basin of the Monongahela, the drainage basin of the Allegheny and
everything else. If we color the basin of the Monongahela blue and the basin
of the Allegheny red leaving the rest white we will see that there are two
points that are located on the boundary of the three colored regions. One of

the points is Pittsburgh at the Golden Triangle where the confluence of the
Monongahela and Allegheny forms the Ohio. The other point is the triple
divide point that is associated with this confluence. An arc passes from the
point of confluence to the triple divide point that is defined by the boundary
between the red and blue regions. This arc is the Monongahela - Allegheny

divide.

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The Ohio River, of course, flows into the Mississippi River near Cairo,
Illinois. This confluence will have its associated Triple Divide Point. We find
this Triple Divide Point in the same fashion as before. In this case we partition North America into the drainage basins of the Ohio (This will include
the drainage basin of the Monongahela and Allegheny Rivers as subbasins),
the drainage basin of the upper Mississippi with respect to the Ohio - this
is the river basin of the Mississippi above the confluence with the Ohio, and
lastly everything else. Again using the colors red, white and blue we discover
that there are again two points that are on the boundary of all three regions. One point is the point of confluence and the other is the triple divide point. The arc that is the boundary between the Ohio and Mississippi river basins
is the Ohio - Mississippi Divide.

Anyone familiar with Chicago Illinois is aware that the boundary of the
upper Mississippi basin is somewhat ill defined as the Chicago River flows
out of Lake Michigan and into the Illinois River and into the Mississippi.
Lake Michigan is connected to Lake Huron and the rest of the Great Lakes
which drain into the Atlantic through the St. Lawrence River. Since the flow

of the Chicago River was engineered to flow in this direction we will assume
its natural flow when forming the boundary of the upper Mississippi basin.
A similar and natural phenomena exists in Wyoming just south of the South
East corner of Yellowstone National Park where Two Ocean Creek divides
at a mountain pass into two creeks, the Atlantic Creek that is a tributary to
the Yellowstone River and ultimately the Mississippi, and the Pacific Creek
that is a tributary to the Snake River and then the Columbia. In this case we
will simply let the continental divide follow the stream bed from its source
to its point of divergence.

The Mississippi River flows into the Gulf of Mexico which we assume
to be an arm of the Atlantic Ocean. To find the Triple Divide Point for
this confluence we imagine that the oceans surrounding North America is
a stream and has a direction of flow, with a point of origin and a point of
destination. The point of origin and the point of destination will essentially
be the same point. If we choose Panama to be this point and choose the flow to be from the Atlantic side of Panama and counterclockwise around
North America to the Pacific side of Panama, then the Upper Ocean Basin
with respect to the Mississippi is that area of North America where water flows into the Caribbean Sea and the Gulf of Mexico west of the Mississippi
Delta. We thus have a partition of North America into three regions, the
Upper Ocean basin with respect to the Mississippi, the Mississippi basin and
the remaining region, which would be the lower Ocean basin with respect to
the Mississippi. Again there are two points that are on the boundary of the
three regions, one is the Mississippi River Delta - the point of confluence,
and the Triple Divide Point. This Triple divide point has been identified
as a mountain in Colorado that has just been recently named Headwaters
Hill where our ideal raindrop will split into three subdroplets, one will flow
west into the Colorado River and, if it weren't for human intervention taking
water for irrigation, eventually into the Gulf of California, an arm of the
Pacific Ocean. Another subdroplet flows into the Rio Grand and into the
Gulf of Mexico west of the Mississippi. The third subdroplet flows into the
Arkansas River and into the Mississippi.

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As we have argued a point of confluence defines a unique triple divide
point. We can also see that a triple divide point will define a point of confluence. We imagine our ideal rain drop dividing into three subdroplets flowing
into three separate basins. Eventually these three subdroplets will be recombined. At the point when the three subdroplets are rejoined, two will have
been previously rejoined. We trace the rejoined pair back to their point of
recombination and this will be the confluence point associated with the triple
divide point.

A stream may have many tributaries, but it will be a tributary to only
one stream or body of water, thus we may identify a confluence uniquely to
the tributary at the point of confluence. In the case of the Monongahela and
Allegheny we arbitrarily pick one to be the tributary of the other. The other
stream is then considered to be the extension of the Ohio river. The triple

divide point that is associated with the confluence can thus be associated
with the single stream that is the the tributary and the divide can also be
associated with the tributary. In the case of Headwaters Hill, the triple divide
point identified with the confluence of the Mississippi River and the Gulf of
Mexico, Headwaters Hill can be said to be the triple divide point for the

Mississippi River (The name seems quite appropriate as it can be said to be
the headwaters for the Mississippi river, the largest river of North America). The raindrop falling on Triple Divide Peak in Glacier National Park will
divide and flow into the surrounding oceans via the Mississippi River, The
Nelson River and the Columbia River. Since the Nelson River is between
the Mississippi and Columbia rivers with the previously given orientation of
the surrounding oceans, this triple divide point is associated with the Nelson
River.

We have established that every confluence of streams defines a unique
triple divide point. It is equally true that any two streams regardless of
how remote they are from each other will define a triple define point. To
see this let us consider Warm Springs Creek in South Central Idaho and
Otter Creek in West Central Wisconsin. We imagine two raindrops flowing

down each of these streams, the first flows from Warm Springs Creek into
the Big Wood River and then the Snake and the Columbia to the Pacific
Ocean near Astoria Oregon. The latter raindrop will flow from Otter Creek
into the Eau Claire River to the Chippewa River and the Mississippi and
into the Gulf of Mexico. With our orientation of the surrounding oceans
this latter raindrop flows around the continent and joins the first raindrop at
the mouth of the Columbia, this will be the confluence of the extensions of
the two streams and the triple divide point associated with this confluence
is the triple divide point defined by the two streams. In this case it is the
triple divide point associated with the Columbia River. This point is Three
Waters Mountain located along the continental divide in the Wind River
Range in Wyoming near the towns of Jackson on the West side and Dubois
on the East side. The Columbia river divide can also be said to be the divide
between Warm Springs Creek and Otter Creek. The Columbia River Divide
is an arc that begins at Cape Disappointment in the State of Washington,

passes over the summit of Mount Rainier and then North along the Cascade
Crest into British Columbia, where it reaches the Canadian Rockies and then
follows a southern route into Montana and then Wyoming to Three Waters
Mountain, the Columbia River Triple Divide Point. Water falling on Three
Waters Mountain flows either into the Columbia River basin via the Snake
River, the Colorado River basin via the Green River or the Mississippi River
basin via the Wind - Big Horn Rivers.

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Of course we have overlooked one small issue, and that is there are some
basins where the water does not flow to the sea. The Great Salt Lake in
Utah is such a place along with Death Valley in California and the Great
Divide Basin in Wyoming. We will "whitewash" this problem by the untrue
assumption that precipitation will exceed evaporation and percolation, thus
all basins will fill to overflowing and all water will then eventually arrive at
the surrounding oceans.

Mathematicians like to abbreviate processes, and we can define an abbreviation for finding triple divide points and divides. Let *x* and *y* be two
streams that converge to form stream *xy*. We may assume *x* is the extension of *xy*, thus *y* is the tributary. Let *U* be the basin for *x* and *V* be the basin for
y. We now indicate the boundary of and the boundary of *V* with
. The divide between the two river basins is that part of the boundaries
common to both boundaries. We indicate this by . We read this as "The boundary of the basin of *x* intersected with the boundary of the basin
of *y*." The divide can be thought of as a one dimensional region and it would
have boundaries, namely its endpoints, which are the point of confluence and
the triple divide point. We indicate these two points by Which
we read as "The boundary of the boundary of the basin of *x* intersected with
the boundary of the basin of *y*." One thing to notice here is that the unabbreviated version is ambiguous where the abbreviated version is precise. The
difference is the parenthesis. In the unabbreviated version do we take the
boundary of the boundary and then intersect the boundary, or do we take
the boundary of the intersected boundaries. The parenthesis make it clear
that it is the latter that is meant. We summarize the results of our boundary
process by . The notation {*c, t*} is read "The set containing the points *c* and *t*." In this case *c* represents the point of confluence
and *t* represents the triple divide point.

We may put an order on streams in North America. We can say stream * x* is included in stream *y* if some of the water in stream *y* originated in
stream *x*. The Ohio river is included in the lower Mississippi, but not in the
upper Mississippi, remember we are regarding the upper Mississippi and the
lower Mississippi as two separate streams. We may say that a stream can
be extended by appending streams in which it is included. The Allegheny is
extended by appending the Ohio, and further extended by including the lower
Mississippi. Now let *x* and *y* be two arbitrary streams in North America,
consider Warm Springs Creek and Otter Creek as mentioned above. Let be the maximal extension of *x* that does not contain any water from *y* and be the maximal extension of y that does not contain any water from* x*. Since by our assumption that all water must eventually converge, there must be a
confluence between . Let be the basin for and be the basin for . We can now use the boundary operation to indicate the divide and the
confluence and Triple Point for the two streams, is the divide and
is the triple divide point for streams *x* and *y*.

To determine the triple divide point for the Mississippi river we put an
orientation on the surrounding oceans of North America. We imagined the
oceans as a stream flowing from the Atlantic side of Panama counterclockwise
around North America and terminating on the Pacific side of Panama. If we
reverse the flow so that we imagine the stream flowing from the Pacific side
to the Atlantic side of Panama the triple divide point for the Mississippi
remains the same but its divide is changed. The boundary of the Mississippi
basin is a loop that begins and ends at the Mississippi Delta as it enters the
Gulf of Mexico and it passes through Headwaters Hill, the Mississippi triple
divide point. With the counterclockwise orientation of the ocean stream the
Mississippi divide, , would be that portion of the loop that would
begin west of the Mississippi Delta and pass through Texas and New Mexico
to its endpoint at Headwaters Hill in Colorado. If we choose the clockwise
orientation, then would be that portion of the loop that begins
east of the Mississippi Delta and passes through more states (and Canadian
Provinces) than we wish to list here before ending at Headwaters Hill. The
reason for this change is that the "upper" and "lower" ocean basins have
been interchanged.
The orientation of the oceans not only included a direction of flow, but
also included a point of origin and termination. We chose Panama for that
point, as it seemed to be a point most removed from the area of interest that
we have examined. Suppose that we move the point of origin and termination
from Panama to the Bering Strait, we see that the Mississippi rivers is not
between the Rio Grand and the Colorado, but rather the Rio Grand is now
the central river and thus inherits Headwaters Hill as its triple divide point.

For the Colorado to inherit Headwaters Hill as its triple divide point we
would have to choose a point along the Gulf of Mexico between the mouths
of the Rio Grand and the Mississippi rivers, perhaps Galveston Texas.

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THE TRIPLE DIVIDE POINT TOPOLOGY

The Triple Divide Point Topology is a bit of a misnomer as it is a topology
on a geographic region defined by drainage basins.

We begin by putting an order on streams by the following convention.

1. The point of origin and destination (for lack of a better term we shall
call that point the point at infinity, or the point of infinity) will be
called the category -1 stream.

2. The ocean is the category 0 stream.

3. A to a category k stream is a category *k* +1 stream.

Now let NA be the set of points of North America, and CP be the
collection of all confluence points of North America. We define maps the confluence point where a category
k stream is a tributary to a category k -1 stream where all the water from *x*
will pass through. The topology that we define on North America will be the
coarses topology that makes the collection of maps defined above continuous.
A basic open set for this topology will be , where c is a confluence point.
In other words a basic open set is the interior of a drainage basin for some
stream, or potential stream if water were present. We will assume confluence
points and triple divide points will be dense in the Euclidian Metric Topology.
To show that this defines a topology we simply verify that the intersection of
two drainage basins is a drainage basin. This is clear since drainage basins
are either disjoint or nested. Thus

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