County Connectivity *1

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

*1 by Adam Helman


The question arises as to whether two groups of counties are connected to one another if they only touch at a single point. In county highpointer parlance, a group of counties with common borders all of nonzero length is a glob. Are counties "globbable" when they share but a single point?

Within the confined world of completion map painting this problem has implications for whether a set of counties, colored blue, may be connected to another set of counties, colored yellow - the home glob.

Multiple discussions have arisen in the effort to answer this admittedly silly but nevertheless interesting question. The question is silly because the answer bears little implication for our lives. The question is interesting because the answer forces one to consider issues that have plagued philosophers, mathematicians and thinkers for millennia - specifically, the concept of infinity and its inverse, zero.

Human beings find it difficult to grasp these concepts because they are absent from our collective experience. Indeed, a true infinity of objects is physically unrealizable within the confines of the known Universe. Likewise, a point of zero dimensions is physically unrealizable given our current knowledge of quantum physics.

These are purely theoretical constructs. A mathematically zero-dimensional point, e.g. the tentative point of connection for two globs, is necessarily unrealizable in the world of real political boundaries.

Has the question been settled? Not in the least. For in painting and displaying a completion map there are constructs and considerations that transcend mathematics and physics - the human sensibilities of practicality, reasonableness, and vanity. The reader can suggest others.

Is it practical to use the physical impossibility of a theoretical, zero-length boundary to resolve a question involving real borders?

Is it reasonable to allow our decisions to be based solely on mathematical and physical constraints no matter how seemingly strange or absurd the implied conclusions? Human existence is rife with illogical decisions based on suspicion, instinct, hearsay, and experience - decisions that, only after-the-fact, appear correct by some yardstick.

It is vain to paint and display one's completion map. Undeniably so. Shall we let our need to display the most impressive home glob possible taint our decision as to whether globs are connectable by a single point? We are not machines, incapable of feeling. A machine would not be vain even if it could climb. The question at hand should not be answered with some futile attempt to dismiss this aspect of our nature.

It is clear there are several criteria that one might employ, singly or in combination, to answer the county connectivity question - topology; subatomic structure and quantum physics; logical and/or practical arguments.

Indeed, a second level of inquiry exists as to which criteria should be used. The author is not bold enough to definitively answer this question - let alone answer the original question on county connectedness.

In the following sections various philosophies are presented as suggested, both tongue-in-cheek and in all sincerity, for answering the county connectivity question. The philosophies are grouped according to which criteria are used in their respective arguments.

The author then presents
his arguments, and, given all the "evidence", lets the reader decide which philosophy is most convincing, plausible, suitable, or pleasing to his taste.

A section, Random Comments and Humor, provides commentary on the detailed arguments. In a Conclusion the current state-of-affairs is summarized.

Schools of Thought - Topology

I am very happy to see that I am not the only one that has been distressed over corner points not being globbable. I was against that rule from the beginning. Now, however, it is relevant, since I got San Juan Co., UT last year and Mesa Co., CO this year. I would add two states (UT and AZ) to my western glob if I could attach at their common corner.

Fred asks for a topological interpretation. I have a masters in math with an emphasis in topology. I would glob them. The crucial point is whether they are connected. While topology has several types of connectedness, we should be interested in path connectedness. The question should be: Will counties that touch only at one point hold together? Yes, mathematically they are path connected.

I recall only three arguments against globbing. 1. Zero dimensions. The number of dimensions is not important either way. The zero-ness of something does not stop its existence. In math, a line has zero width. Does that stop us from talking about them and making important applications of them? 2. Your foot cannot completely fit into the touching states. The interest in the foot comes from an application of path connectedness in real life. While mathematically irrelevant, it suggests that path connectedness is the correct type of connectedness for our purposes. 3. Not rigidly connected. Someone said they didn't want part of their glob sagging if they held it up. This rigidity criterion is NOT a topological argument. In topology, Earth is the same as a solid cube, but neither is the same as a doughnut. This is because you can distort Earth into a cube without poking a hole in it or cutting anything off. You cannot so distort either into a doughnut. Rigid moves are congruences, whereas topology includes shrinking and expanding, whether the whole or a part. As I said earlier, topologically, corner connected counties are path connected.

The issue is far more than topological in mathematics. In calculus, the derivative of a curve at a point where the curve is continuous is the slope of the tangent line to a curve. The tangent line is the line that touches the curve at one point. Curves that cross it at two points are called secant lines. The very definition of the concept depends upon touching at one point. If things cannot touch at only one point, then there is no such thing as a tangent line. Derivatives, by the way, are essential to engineering (velocity) and business (marginal profits).

                ... Ron Tagliapietra

I'm going to place myself into the category that counties that touch at a corner do not get glob status. Ron makes a topological argument that is very reasonable. In my mind however, these are two fundamentally different "types" of touching.

If two counties share a finite, non-zero boundary, be it 50 feet long or 180 miles long, they definitely border one another. Mathematically, one can define a bijective function from one open segment to any other, essentially making any such boundary "the same" for globabulatitive* purposes. The main key here is a bijective function works both ways - i.e. its inverse is also bijective. Boundary A can be mapped to boundary B and vice versa.

If two counties touch at a point only, then we have a fundamentally different style of touching. Consider a figure-8 and a regular oval figure that has an internal boundary (e.g. separating "left half" and "right half"). One can cinch the oval shape so that the border reduces to zero, but now we have a "function" that maps all points on the internal boundary to the one point on the figure-8. This is no longer bijective. One cannot "undo" the single point and re-map it to some finite segment. The two shapes are fundamentally not the same. This involves homotopy maps but I will admit I am rather fuzzy on those details.

The famous "four-color" mapping theorem supports my somewhat shotgun argument. One can color Colorado and Arizona purple and no one gets upset. To mapmakers these two states do not share a boundary.

At issue here is simply what we define to mean by "touch". This term has not been sufficiently defined, and as a result, both Ron's argument and my argument may have merit.

(* this word (c) 2003 Scott Surgent. All Rights Reserved)

                ... Scott Surgent

As an accomplished mathematician, I have never believed that globs should be allowed to connect only at corners. Here are the classic reasons:

  1. While it's true that counties that touch only at a corner share a common POINT, the LENGTH of their common border is zero, which is the same as it would be if they didn't share any common point at all.
  2. Topologists have long considered regions that touch only at one point not to be neighbors. In the famous four-color map theorem, which deals with the problem of coloring a map so that no two adjacent regions are the same color, two regions are allowed to be the same color if they touch only at a point. Otherwise, if ten regions all converged at the same point, they would all be considered neighbors, and no two could be the same color. You'd need arbitrarily many colors to guarantee that any map could be colored, and the four-color map theorem would not be of the great historic interest that it is.

  3. In graph theory, where the plane is divided into regions, and points where their borders (which are called "edges" in this context) converge are called vertices (which can also be placed at intermediate points along a border, thus dividing the border into two edges), one can talk about the "dual" of a graph, where regions and points change places, and two regions that were separated by an edge transform to points that are connected by an edge. Thus, a glob of regions becomes a glob of points, and vice versa. Regions that touched only at a vertex on the original graph would transform to vertices that are definitely NOT be connected by an edge on the dual graph. Regions that share an edge have fundamentally different mathematical logic than those that only share a vertex.

                ... Edward Earl

...There were times when I was bothered by the phrase, "A point has zero dimensions."
A point can exist in n-dimensional space, but it has zero measure in those dimensions.

                ... David Olson

Schools of Thought - Subatomic Structure and Quantum Physics

Since we're talking theory... A few years ago, I brought boundaries up at lunch with a group of engineers and programmers. Three of the people at the table considered themselves professional mathematicians (I was working with them on floating-point math software). After starting on a mathmatical footing, the discussion took a very interesting twist and there was consensus that, in fact, this was as much a problem of physics as math. For what it's worth, the discussion went something like this:

By definition, political boundaries exist BETWEEN entities (counties, states, nations, etc.). There is no land/water that is SHARED between the counties; parcels are either in one county or the other. Therefore, political boundaries are hypothetical/theoretical lines that do not have mass/matter. Essentially, if you drill down deep enough, the boundary runs in the space between atoms/molecules. The conversation even went to the extent of estimating how wide this theoretical boundary line might actually be (for example, the distance in angstroms between water molecules at a given temp/pressure).

(Yes, there was some discussion as to whether a boundary could cut through an atom. The argument would then be similar but at the sub-atomic level.)

When I brought up the corner/point condition, they just viewed this as a subset of the longer boundary/line. At the molecular level, the distance across the corner is essentially identical to the distance across the line and they considered this distinction a non-issue. There was even some thought that you could construe situations where molecules across corners could be closer than those across the line. (In fact, there was some discussion that unless the molecules were truly organized in a grid, the matter around the boundary lines would probably not, in fact, form a "corner" in the normal 90-degree sense. And, to further complicate things, at the atomic level, the boundary is 3-dimensional since the molecules on each side have height...)

[On a side tangent, there was also some thought that "real" boundaries (ones that exist in matter) were different than "theoretical" boundaries (those that don't exist in matter). So, for example, the boundary between a plane's hull and the outer atmosphere is a "real" boundary and is simply defined by the objects involved -- that is, all matter that is not part of the plane is on the "other side" of the boundary.]

                ... Trapper Robbins

Schools of Thought - Logical or Practical Arguments

Perhaps I missed a discussion thread sometime in the past, but it seems to me that, theoretically, the shortest common border would be of length ZERO, such as the common point at the four corners of Utah, Colorado, Arizona, and New Mexico. After all, if you had visited the county highpoints of San Juan county, Utah, and San Juan county, New Mexico, but hadn't visited those of Apache county, Arizona, or Montezuma county, Colorado, couldn't you still have a connected glob at the corner? Fill me in if I missed something in the past.

                ... Dale Millsap

I'm still troubled by the group stance that counties cannot be globbed if they only have a common corner. The mathematical or geometrical arguments that I have been presented actually make me feel more strongly that they should be globbable - whether they be connected by a two dimensional line or by a one dimensional point seems irrelevant. Do those two theoretical counties actually touch at a corner, or do they not?

                ... Dale Millsap

I'm afraid Dale Millsap is incorrect in his description of a line and a point. A line has only one dimension, and a point has zero dimensions. This is why many of us do not consider that counties that "touch" only at a point really do not touch at all, and are therefore not "globbable".

This reprises a discussion from several years ago, and even some of the mathemeticians among us considered that counties "touching" at a point really do touch. I would like to get the opinion of someone with a background in advanced geometry, or perhaps topology, to offer an opinion on this.

                ... Fred Lobdell

(see the Section, Schools of Thought - Topology for Ron Tagliapietra's response)

Besides the mathematical arguments, there is also the common sense argument at four corners that NM and UT touch. This is what makes four corners interesting. If they do not touch, then why would anyone care to visit there? When I tell non-cohpers how many states are in my western glob, some have asked why I didn't count UT and AZ. I then have to explain the strange prohibition that we have. The prohibition of connecting globs across a point is against both topology, mathematics in general, and common sense.

                ... Ron Tagliapietra

Although I think that it is ultimately the decision of the person filling in their map's completed counties, I think there may be a somewhat simple way to think about whether or not you can glob cornered counties (such as the four corners area of UT/CO/NM/AZ): go ahead and glob them together if you can step from one county to the other.

It makes a little sense: If you were at four corners, you could have your foot on both counties at the same time, and they would be connected at the imaginary point in the middle: no space between (since a point has no dimensions, one could argue that there is no separation?).

Personally, I like blue better, but the yellow does give a certain sense of satisfaction.

w00t =)

                ... Ben Knorr

...and this is the one I find most compelling of all: If UT could be globbed to NM, and CO gould be globbed to AZ, you'd have a pair of "interpenetrating isthmuses" at the 4-corners point. This is totally incredulous. A "bridge" that connects UT to NM would block any connection between CO and AZ, and vice versa. You simply cannot have both connections, because they would cancel each other out. It would be hypocritical to choose just one pair to be considered neighbors, since both pairs are equivalent. The only way to resolve this controversy is to agree that neither pair is globbable.

                ... Edward Earl

Schools of Thought - Flux and Mass Transport ("Molasses") Criterion

Definition of glob:
A small drop; a globule.
A soft thick lump or mass: a glob of mashed potatoes; globs of red mud.

To find out what is globbable:

  1. Erect vertical walls along the borders of your glob to an altitude just higher than the highpoint of your glob using waterproof material. Cover the floor of your glob with a similarly waterproof material.
  2. Stand on the highest point of your glob. (Recommended but optional)
  3. Using a large container, pour molasses into the glob until the highpoint is just covered.
The glob is everywhere the molasses covers.
It will not ooze through the walls into counties connected only by points.

I do agree that touching at a point makes for a connection, but doesn't seem to honor the spirit of the word glob.

                ... Layne Bracy

This quote encapsulates what the author refers to as the transport criterion for county connectivity -

Objects A and B, each of n dimensions, are connected if a third object, also of n dimensions, can be transported from the interior of A, across the boundary, to the interior of B without deformation. Such a process requires an n-1 dimensional boundary for transport.

More generally still, the flux of any commodity suffices in the above criterion, e.g. heat, mass, momentum.

The molasses example corresponds to mass transport with n = 3. A macroscopic parcel of molasses requires a two-dimensional boundary, as the surface between "globs" A and B, for transport between the two. A one-dimensional slit connecting the two globs is insufficient for the molasses to ooze across.

A county highpoints completion map corresponds to n = 2. A pixel in county A can be dragged across a nonzero-length boundary into county B. The yellow color of county A can seep across such a boundary into the (otherwise blue-colored) county B - an example of color transport. A zero-dimensional point connecting counties A and B is insufficient for color to flow.

A zero-dimensional point as bounding surface allows for the flux of only one-dimensional objects. The transport of a (mass-free) line of infinitesimal width is one example, and corresponds to the piecewise-connectedness noted by Ron Tagliapietra in a previous section. This linear transport does not correspond to flux of a ponderable body - be it the paint of a completion map (whose universe is two-dimensional) or the molasses of a (three-dimensional) home glob.

The author is most satisified with the transport criterion for county connectivity, and is eager to use it at the exclusion of all remaining arguments.

Random Comments and Humor

(Ron Tagliapietra wrote)
The prohibition of connecting globs across a point is against both topology, mathematics in general, and common sense.
(end quote)


                ... Bob Bolton

(Ron Tagliapietra wrote)
The prohibition of connecting globs across a point is against both topology, mathematics in general, and common sense.
(end quote)

I say, Amen.

                ... Roy Wallen

(Dale Millsap wrote)
Do those two theoretical counties actually touch at a corner, or do they not?
(end quote)

When Hitler invaded Poland, he did not go pussyfooting across some 4 corners diagonal  

If I grab your Utah glob and shake it, you really don't want to see San Juan County NM, attached by only a dot, go flying off into space, do you?

And finally, for deep thinkers, my friend Guy suggests all counties are connected by a dot - at the center of the earth ...

(Ben Knorr wrote)
Go ahead and glob them together if you can step from one county to the other.
(end quote)

We hashed thru this line of argument in the memorable glob wars of days gone by.

As Dale tries to step from UT to NM, the rancher in Apache county AZ can shoot his foot off when it enters Apache county airspace, thus he will never manage to get a boot in both counties.

(Bob Bolton wrote - in favor of dot-globbing)
(end quote)

I rest my case  

                ... Andy Martin

After reading the many fascinating posts on the subject, I decided to go with the ones that support my gut instinct that corner connections are not globable, so bravo to Edward, Scott, Fred, and the rest of the nifty nonconnecters, and a loud "boo!" to the corner connection cabal.

No way the group is ever going to have a consensus on this subject, and it's fun to beat this puppy every couple of years.

                ... Mike (I have the corner on correctness) Schwartz

All - As the one responsible for starting this immense thread, I just wanted to let everyone know what I decided - and that's to trust my instinct one what's right for me. Rivers yes, bays no. Finite common borders yes, point contacts no....

Thanks for all the discussion - it's been both interesting and mind-expanding at times too!

                ... Chip Clark

Time for me to weigh in on this I reckon. I have always gone along with the group consensus that squares don't touch at corners, mostly because it was a minor issue, and it seemed to simply make sense at at the time. Now, after some truly wonderful threads from mathematicians and others, I have decided that the right answer for me is still the same; corners don't touch. We have always defined globs as two counties which touched each other via a common border. To me, two corners don't make up a border, just a single infinitely small point, which is not a border to me.

I think it's time for everyone to stop beating around the border on this. Instead of painting yourself into a corner, why don't we simply use this as an excuse to "need" another trip to another cohp? The conversation has been not only heated, but edgy. I have seen all sides of this issue touched upon in the threads, and everyone has made their point. I think we need group consensus along the line of the First Ascent and Virgin issues we have hammered into shape, so we can all be one big happy, touchy-feely family.

                ... Dave (comin' 'round the corner...) Covill  

(Dave Covill wrote)
I think we need group consensus along the line of the First Ascent and Virgin issues we have hammered into shape, so we can all be one big happy, touchy-feely family.
(end quote)

As a follow-on, may I propose two polls. The first, on the question of whether corner boundaries are to be considered contiguous, and the second, as to willingness to accept and follow the poll results. If we have a substantial minority that never accepts the poll results and continues to do their own thing, then the poll is an exercise in futility.

                ... Mike Schwartz

(Roy Schweiker wrote)
Since Bob Packard has basically sewed up all glob categories without questionable connections, there seems little reason to worry about what standards the rest of us use.
(end quote)

I am not concerned with what others want to count. All I know is that I find it unsatisfying if my globbed counties do not share a border and a I do not consider a point a border. Sharing a border is a stronger concept (as Surgent explained topologically and I feel aesthetically) than touching.

                ... Bob Packard

(in reference to Layne Bracy's molasses-based description)
Finally !! An explanation I can understand. Anyone know where I can get a good price on molasses?

                ... Dennis Poulin

(the author belatedly replies)
Smart & Final is a food chain in San Diego that sells items in bulk. From there I recently purchased
one-half gallon of molasses for $2.49 - a deeply discounted price.

* * * * * * * * * * * * * * * * * * *

Please find the time to examine the following web page in which this question of paramount importance is discussed -

I have endeavored to cast the problem of corner-touching in a broad framework.
Further, every person who has voiced an opinion on the matter is represented in the text.
This new web page is available from the top of both the alphabetized and
chronologically-ordered completion maps pages.

I hope you find the content both satisfactory and reasonably complete.

                ... Adam Helman

2 A self-reference that leads to an infinite regress reminiscent of the paradox discussed!

Read Additional Commentary after Posting of this Web Page


The question of county connectivity is unsettled because a combination of mutually irreconcilable concepts may be employed in answering it. Focusing in upon a single criterion, such as mass transport, leads to a definitive solution, namely, that counties are not globbable if they touch at but a single point. Focusing in upon a different criterion, piecewise connectivity, leads to the opposite conclusion.

It is unlikely that every person will agree to the same criteria as being relevant to the county connectivity question. The reader must seek his own answer based upon the preceeding set of arguments. Nevertheless at the time of this writing, the concensus is that counties which touch but at a single point are not connected.

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