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).

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

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.

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

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.