Frustration

I spent a lot of time preparing the next post for my Four-Color series. It contained a 21 line table with headings all carefully lined up with spaces. It was saved and I left it for a while. When I returned to edit the draft, all multiple spaces had been replaced by single spaces including the double spaces that I was taught in typing class (1943?) to put at the ends of sentences. aaagh! I suppose I can use Photoshop Element's typing tool to recreate the table as a picture. Does someone know a better way?

Four-Color Problem - Part III

The last post described how when the configuration on the left in a minimal non-colorable map with one face removed could be be modified as shown in the figure. Briefly, the a-c region containing face A in the fugure on the left must also contain face C so the b-d region containing face B is surrounded thus allowing colors b and d in that region to be exchanged leading to the figure on the right. A similer argument applied to the figure on the right
allows the d-a region containing D to have its colors exchanged so face D is now colored a. This process can be repeated as show in the following table:

After 20 exchanges we have the original colors back around our five-edged face. However, it does not necessarily follow that the whole map has its original coloration. Enouch exchanges must bring it back to its original state as the map is finite and has a finite number of states and each exchange is reversable. The number of exchanges which will do this has to be a multiple of 20. [Note that if states are equated if they can be made identical by a permutation of colors this number is reduced to 5.]

One can draw paths (lines) between the faces that have to be connected by a region. Where two regions containing one different color (say a-b and a-c) the intersection must be the common color (a in this example). I drew many such lines in my diagrams labeling them and their intersections with the colors they represented trying to show that they would requiring additional faces in the map. It turned out tthis was not the case and I found a counter example. (The counter example could, of course, but the coloration cannot be found by the method described here.) I will illustrate this counter example and discuss other possible Kempe chains after I return home from Ann Arbor. Is anybody listening?

Four-Color Problem - Part II

Alfred Kempe, who showed that a minimum uncolorable map could not have a face with four edges [see Part I], gave a flawed proof that it couldn't contain a face with five edges which would then prove the conjecture. Eleven years later the flaw was discoved. His proof was an extension of his proof for four edges. Since there is one more face than colors one color must be used twice. (All four must be used at least once, of course.) In the diagram I have chosen color b. The a-d region containing A must contain D or a and d could be switched in that region leaving a for the center. Similarly for the a-c region containing A and C. Then the b-d region containing B cannot contain D and the b-c region containing E cannot contain C. The switching of b and d in the first and b and c in the second makes color b available for the center. The problem is that the first switch changes the condition required for the second switch.

In Part III I will describe my unsuccessful attempts to get around this problem.

Four-Color Problem - Part I

Preliminary:

• The four-color conjecture states that any map can be colored with four colors so that no two adjacent countries have the same color.


• It can be shown that any map can be converted to a so-called cubic map where each node terminates three edges without loss of generality.
  
• If maps exist that are not four-colorable, then one such map must be of minimum size.
  It follows that a minimum map with one country (face) removed is four colorable.


• A minimal map cannot contain a face with only one, two, or three edges for if that map with that face removed is colored there has to be at least one color left to color that face.


• A minimal map cannot contain a face with four edges: Suppose we have such a face surrounded by colors a opposite to c and b opposite to d. (see below) Consider connected regions of the map that are colored by either a or c or by b or d. If the a-c region containing A does not contain C, switch colors a and c in that region leaving color a for the center. If region a-c containing A does contain C it follows that the b-d region containing B cannot also contain D as it is surrounded by an a-c region. Then switching colors b and d in the b-d region containing B makes color b available for the center.

Two-color regions as used here all called Kempe chains. Stay tuned.

Back in Ann Arbor

For the sixth year now I attend the summer program (2nd half) of the INTER-UNIVERSITY CONSORTIUM FOR POLITICAL AND SOCIAL RESEARCH (ICPSR). My good-hearted nephew, who is an instructor there, takes me along and I sit in on one or more of the classes. We have apartments in the same building and do some things, but not everything, together. He always has lots of friends around. This year I'm "taking" Structural Equation Models w/ Latent Variables. I may add Advanced Bayesian Models for Social Scientists.

I had the date wrong so when I got a phone call from Mike saying he would there at around 10am, I quickly got my stuff together. As usual we stopped at a friend's near Paw Paw who gave us coffee and showed us the damage to her house on a small lake caused by several blown-down trees. We arrived at Maynard House around 5pm and unloaded. My furnished apartment was missing much of the furnishings because they had been cleaning the carpet and hadn't put things back. This was easily remedied. Then we went to a nearby restaurant for dinner and drinks and more conversation.

This morning at Frank's Place where Pete has reigned for year's I had my usual breakfast. I was greeted by Pete and by his son-in-law who was taking orders today. No doubt we will see the daughter, Athena, soon. She is such a pleasant person. Next to Frank's is a bicycle shop where, day after day they put about a hundred bicycles outside. In the afternoon they take them in again.

I had pizza with Mike tonight at another nearby place. We had separately found our way to a book store. I was able to have my mail forwarded on line.

Morning Coffee Thoughts

I'm disappointed in Why e = mc sqr? book. There is hand waving on how we know speed of light to so many places and how we know it is constant. I have an accumulation of books to finish. One needs to go back to library. To-do's now include call Maynard House and call
Eisenstein's office for review of meds. I broke three of the short-stem glasses on card table so returned the rest of glasses to cabinet. Small refrigerator is almost too small. I need to use camcorder now that I have it. I'm quite pleased with it so far. If time is not fixed and distance is not fixed, how can speed = distance/time be constant except by definition. Distance = speed x time. Time = distance/speed. Which is the independent variable? "Constant" is a synonym for "fixed." Which of the three statements do we say is true by definition? It's been too long since I have talked to Phyllis. I'm here rather than pulling more books out of my basement because Anna is coming to clean. I'm overdue to sort coupons. The problem with doing anything is that it keeps me from doing something else, Barbara Sher would call me a scanner. Get more exercise. Practice piano. Work on pictures. Blog. I checked on pith helmets (safari hats). The less expensive ones are only sold by the dozen. I wish Word Pad had spell check.

Digging into My Past

My latest downsizing project is attacking the boxes of paper in my basement. Ninety-five per-cent is pitched. The rest is filed by year. The process reminds me of parts of my life that I had almost forgotten (the reason for the 5 per-cent). I ran across folders about my early attempts at the four-color problem, programs in basic written for early micro-computers, flow charts for a billing process (I was a consultant), 8086 and 8087 manuals, and client notes from actuarial consulting. And I am just getting started.

Just a Few Pictures

Still having problems with placement.

The Jersey Boys

I just saw this musical (in Chicago) which is the story of The Four Seasons and their lead singer. One of their hits was Big Girls Don't Cry and there were many others. Not only was the music great but there was also a real story. They were eventually inducted into the Rock and Roll hall of fame. My significent other enjoyed it too.

Tea Leaf Saga Continues

My dully journal for today:

Wednesday, July 01, 2009 9:46am McD(N) Wt.: 178

I sent Tea Leaf pictures and had a reply with:

I’ll look forward to receiving your “TLCI Auction Consignment Form” indicating condition (damage, i.e. chips, cracks, flakes, hairlines, discoloration, and repair) as well as dimensions for each of the lots. Would you please identify the image number for each lot.

I wonder if he had talked to his wife. I haven't finished measuring.

May at the Chicago Botanic Garden

I don't know how to properly compose this page. From top to bottom it is a column heading in a decorative structure, bird feeders, and pansies.