County Connectivity Addendum

How Many Angels Can Dance
on the Head of a Pin...

Adam: You have outdone yourself, especially since you have placed yourself in the ranks of the nifty non-connectors.

Speaking of consensus, when I was a very callow youth, I used to read the high school football predictions in the local newspaper. Each sportswriter's predictions were listed in a table, and the final column was labeled "Consensus." Until someone older and wiser set me straight, I wondered who had a funny name like "Consensus."

                ... Mike Schwartz

Hello Adam,

Good stuff.

I had shared this last glob-connected war, not this time:
From Advanced Calculus, by R.C.Buck, page 34 -

Definition 2: Two disjoint sets A and B, neither empty, are said to be mutually separated if neither contains a boundary point of the other. A set is disconnected if it is the union of separated subsets and is called connected if it is not disconnected.

Therefore, Jeff Davis county TX and Mexico are mutually separated because Jeff Davis ends at the bank of the Rio Grande River, and Mexico starts in the middle of the River. Neither contains a boundary point of the other. Hudspeth and Presidio counties are connected by a line of boundary points of finite, half-the-river-wide length.

We can accept that Jeff Davis and Hudspeth counties are connected, because it is trivializing to argue whether the boundary points are contained in Jeff Davis or Hudspeth or neither. But at the Four Corners a connection between San Juan UT and San Juan NM depends on one point. If we declare that THAT one point is contained in San Juan UT, then there is no connection between Apache AZ and Montezuma CO. As for the other possibility, I find it easier to declare that THAT one point unknown, hence not contained, in any of those counties, then to say that THAT one point is contained in each of those counties.

In the end, I also urge on you Dave Covill's recommendation. It is a great excuse to make another coHPing trip. Heck, I went on another trip to north Arkansas, even though Carroll AR and Taney MO have a common border a few miles long, because I wanted to "fatten" up the connection.

                ... David Olson

Hello David,

Thanks for the thoughts about corning-touching. Shall I add your comments to this new web page?

>Adam wrote, "Thereby the concept of a zero-dimensional point, e.g. the tentative
>point of connection for two globs, is necessarily unrealizable in the world of
>real political boundaries".

>I disagree. Since a political boundary is a human construct we can
>DECLARE that four boundaries meet at a point.

Mincing words, IMHO. Nevertheless for your sake I SHALL amend my statement to read, "Thereby a mathematically zero-dimensional point...etc..." . In so doing the distinction is made, as you have pointed out, between a truly infinitesimal point and the CONCEPT of such a point. The CONCEPT can yet exist in a politician's head (or that of a mapmaker) even though it is physically unrealizable... ...he simply has to DECLARE it so!!

By the way, quantum mechanics has a concept of tunneling that allows for an object to "jump" across a classically impenetrable barrier and magically appear on the other side. This has obvious implications in the current question ... but, again, the results of quantum physics must be tempered with our own human sensibilities of what is relevant (and what is not) in deciding this corner-touching issue.

                ... Sincerely, Adam Helman


Nice summary of the county connectivity issue.

My only quibble is in the conclusion section. Here, I believe you should repeat the Covill hypothesis (or perhaps this is my corollary to same) - this question is simply a temporal one, which becomes irrelevant when the subject highpointer visits the highest point in one of the adjacent counties.

Since we know that any highpointer will, in fact, have this as an eventual goal, all this fuss and bother is simply about how to deal with a temporary problem that will inevitably resolve itself over time in a manner satisfactory to those on both sides of the argument.

                ... Ken Jones

I will gladly add your comments to the web page, Ken. The dubious nature of a point-only glob connection provides impetus for additional highpointer action, so leading to an ephemeral existence for the point connection.

However there are at least two arguments which suggest refinement of your statements.

What if the highpointer dies before the true border-connection is made? His completion map will always be yellow + blue or yellow + more yellow depending on which side of the county connection fence he climbed.

This death-before-completion is the temporal analog of what happens to a completion map when the original highpoint no longer exists, i.e. Mount Saint Helens.

As a second argument, one can imagine an arrangement of objects, possibly as counties, configured so that they are NEVER connectable by more than a point. All one needs are two globs with nothing more than a pair of diamond-shaped counties joining the globs, the connection being made but by a pair of their respective corners.

Fortunately such an arrangement does not exist in the forty-eight contiguous states, and as such, does not form a practical argument that detracts from your original comment.

As indicated above, an ambiguity exists as to whether practical notions alone, or theoretical constructs alone, suffice in deciding the county connection question. Being purely theoretical, my second argument may be reasonably dismissed as irrelevant, depending, of course, upon one's criteria for deciding the connectivity issue.

However the first argument remains valid if one is amenable to considering practical issues: since we all die, eventually the argument becomes all-too-real for some unfortunate highpointer.

                ... Sincerely, Adam Helman

I gotta get my 2 cents worth in here. A glob as defined by our group is two or more counties touching. Period. Why get quantum physics into this? The only question to be posed requires a Yes/No answer. Do the counties touch? If Yes, it's part of the glob, if No, then it isn't. What could be simpler? Who cares if it touches at a point or a line or whatever? If it touches, it touches.     Amen.

                ... Bill Schuler

I'm sure I won't be the first to respond to Bill's input. Here is the direct quote from the web site:

"Glob - A collection of contiguous counties in which the highpoints have been visited.
Counties must share a border shown on USGS maps. No corner touching allowed."

That seems pretty clear to me. This definition also addresses the infamous water-crossing issue, which I plan to invoke as soon as I get Monterey County CA.

respectfully submitted,

                ... - Roy Wallen

Roy, et. al.

My point exactly. Webster defines the word "contiguous" as Being in actual contact:
touching along a boundary or at a point. So I say again, if it touches, it touches.

If you'll notice, on my map, in Idaho, I have Custer County in blue. An example of Adam's Diamond theory. This is blue only because of our silly rule about corner touching. But I know better than to suggest a change in our concrete rules.

Respectfully controversial,

                ... Bill (Schuler)

Actually, Bill, you can color Custer ID county yellow. It doesn't look this way on the crude completion maps, but Elmore county does border Custer county for about 2 miles: not enough to be visible on the completion maps, but certainly enough to connect globs. Notice that they are both yellow on my map (and I have no alternative connection).

                ... Edward "7.389056099" Earl

Thanks to Adam for gathering and summarizing the various corner issues for posterity. A few related thoughts occurred to me over Thanksgiving; and since the discussion has continued on the forum, I figured I'd share them.

First off, before this discussion started, I was completely ambivalent on globbing across corners; having read all the arguments, I remain completely ambivalent. There is ample precedent to support a decision either way on this topic.
(This isn't an emotional topic for me.)

(1) Consensus: As others have stated, if no consensus can be found, I agree that the conservative action is to not glob across corners.

(2) Math Theory:

(a) Corner Crossing Problem: Some people have indicated that the theoretical distastefulness of cross-corner "overlapping" regions is a compelling argument to disallow corner globs.

- First, this can never happen in our cohp world; i.e., you would never end up with yellow and blue globs crossing each other; instead, each corner county simply borders the other three.

- Secondly, there are plenty of real-world cases where regions DO cross each other at corners. For example, think of all the checkerboard land ownership patterns in the west.

(b) Four Color Mapping Problem: Inevitably, people bring up the 4-color mapping problem when discussing corner borders. It is important to remember that this problem doesn't "prove" that corners don't touch; rather, that is a prerequisite to constrain the problem space to one that is interesting and challenging for mathematicians. The real world contains many situations outside the 4-color map parameters. For example, before Bangladesh was independent, East and West Pakistan were discontinuous portions of a single nation and, thus, had to be the same color. Similarly, we are all familiar with scattered units that all part of a single National Forest (and, thus, need to be the same color on maps -- usually green). The checkerboard land ownership/management pattern mentioned earlier also breaks the rules of the 4-color map problems. These and other conditions that exist in our world of mapping may bum out math theorists but they post no problem or ambiguity for cartographers, governments, or land owners/managers.

Rather than be compelling arguments to prohibit corner touching, these issues instead raise a red flag to alert us that the math theory doesn't map cleanly onto our real-world problem space and, thus, the theory's relevance and applicability is in question.

[Note: I talked to a professional cartographer friend; he and his colleagues have never used the 4-color map theory to create a map. They found it theoretically interesting but of no practical use to them in their professional work.]

(3) Glob Definition: The existing glob definition can not be used to justify corner crossing arguments one way or the other. We own the definition and this discussion is, in fact, about what that definition should be.

(4) Blokus: On the lighter side, I was in "Math N' Stuff" today, a local math/puzzle/game shop. The people in the shop recommended the award winning game Blokus. From the rules on the box, each player has pieces of one color and

"The object of the game is to try and cover as much of the board as possible. Pieces of the same color can ONLY touch at the corners, not along an edge."

So if you are frustrated, buy Blokus to satisfy your corner globbing desires!

                ... hoot! trapper (Trapper Robbins)

To wring the last drops out of the county border wrangling, check out the poll we ran in 2000. Note that 73% agreed with what is in the current FAQ. This is enough to outvote a filibuster ? ;)

Question - For glob purposes, are counties connected across bodies of water?

Responses - 15 replies

Never: 0   (0.00%)
Only across fresh water: 0   (0.00%)
Across any water that can be bridged: 2   (13.33%)

Wherever borders are shown on UGSG topo maps, no corners allowed: 11   (73.33%)
Same as above, but corners are allowed: 2   (13.33%)

                ... Andy Martin

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