I write, therefore I am
With this variation on a famous statement by the philosopher Descartes, I would like to express that writing about what happens in my life is important to me.
Triangulation of regular polygonsThe number of ways a regular polygon with n-2 sides can be dissected into triangles (also known as triangulation) is given by the Catalan number Cn. Every pair of triangulation for a given regular polygon has a number of (internal) lines in common. I wrote a number to count these numbers and the results are given in the table below. The results are given in the table below. The second column gives the Catalan number for the value given in the first column. The following numbers give the number of pairs that have the number of lines in common equal to the number given above the column. The total number of pairs is equal to Cn(Cn-1)/2. The diagonal sequence 1, 5, 21, 84, 330, and so on, seems to be equal to OEIS sequence A002054.
n C 0 1 2 3 4 5 6 7 ----------------------------------------------------------------- 2 2 1 3 5 5 5 4 14 34 36 21 5 42 273 308 196 84 6 132 2436 2928 1992 960 330 7 429 23391 29898 21555 11220 4455 1287 8 1430 237090 321490 244420 135080 58630 20020 5005 9 4862 2505228 3594756 2872694 1670812 773773 292292 88088 19448
Another dead-endLast Wednesday, I found an interesting question on MathOverflow related to the the Four colour problem by the user Robert Carlson (maybe Bob Carlson from the University of Colorado). The question Are all Hamiltonian planar graphs 4 colorable? Does this imply all planar graphs are colorable? talks about how a Hamiltonian cycle on the faces of a planar graph, would split that graph in two tree graphs. It talks about 'diamond switches' and states:
4-connectedThe paper A theorem on planar graphs by William Thomas Tutte, which appeared in Trans. American Math. Soc. 82: pages 99-116, states in Theorem II (on page 115): Let G be any 4-connected planar graph having at least two edges. Then G has a Hamiltonian circuit. 4-connected means that it is not possible to separate the graph into two parts by removing 3 or less vertices. The dual graph of interesting graphs with respect to the Four colour problem have only faces with three edges and all vertices have degree five or higher, meaning that they have five or more edges. Because of the triangle faces (faces with three edges connecting three vertices), every minimal set of vertices that separates the dual graph must be a cycle. If this is not the case, it means that there are two vertices on the sequence of vertices that separate the two components on the plane. These two vertices will be connected to both vertices on of the two components, and because there are only triangle faces, it means that there must be an edge between vertices in the separate components, which would be a contradition. Now suppose that the dual graph is 3-connected, then the original graph can be split in two parts that are connected with three edges. It is obvious that this graph can be reduced. This means that the dual graph of all non-reducable colouring problems must be at least 4-connected. And from Theorem II it follows that the dual graph must have a Hamiltonian cycle.
Double tree graphsSome weeks ago, I realized that all possible expansions of the 00 sequence according to the rules I described on November 23 (of the same length) must have some sequence in common. What I mean is that if you start with the sequence, choose an expansion sequence consisting of n successive locations, and applies this to the sequence 00, resulting in 2n sequences (when expanding with 1 and 2 at each location), that this collection of sequences always has a sequence in common with any other expansion sequence. Yesterday, evening, I thought about the idea, that maybe I could proof this by studying the properties of the graph you get, when you glue two tree graphs (represeting each expansion pattern) together and see if it has some special properties. With some smaller tree graphs, I got the idea that they always contain a face with three or four vertices and that such a face could be easily removed. But after some calculations, I concluded that there was no evidence for such a property. This morning, I decided to look for the smallest counter example of a combined graph with only faces with five or more vertices. That is when I made the above drawing of the dodecahedron graph and drew a line with pencil visiting all faces once, and thus spliting the graph into two tree graphs. The digits give (except for some errors) the number of vertices of the face on given side of the pencil line. The pencil line is actually a Hamiltonian cycle in the dual graph. Next, I realized that maybe the dual graph of every interesting graph with respect to the Four colour problem has a Hamiltonian cycle. According to Tutte, each 4-connected planar graph has a Hamiltonian cycle. If this is the case for all interesting (non-reducable) graphs, then it would be sufficient to proof what I wanted to proof in the first place to proof the Four colour theorem as well. That would simplify the matter a lot, except for the fact that it will problably be very difficult to proof that all expansions (with the same number of expansions) have at least one sequence in common.
Flowers in magnoliaI noticed some flowers in our magnolia. Many of them are in new branches. It also needs to be pruned again. It seems it is growing higher and higher each year. Annabel is feeding branches to her rabbit, who seem to like it very much. The little magnolia plant now has nine leaves, but the last three leaves have brown edges and they appear to be smaller than earlier leaves. bookshop Broekhuis for € 8.95. The book is about some popular math topics at the M.Sc. level.
Saturday, July 5, 2014
Toki Pona: The language of goodThis evening, I got the book Toki Pona: The language of good by Sonja Lang (formerly Sonja Elen Kisa), ISBN:9780978292300 from bookshop Broekhuis, which I had ordered, and paid € 19.99. In the evening I read through the book skipping some parts. The book is about the constructed language Toki Pona, which only has 120 words. The book contains lessons for the language, but also a kind of hieroglyphs of each word and a sign language. A few years ago, I made some attempt to learn the language. I think I am going to try again. The only negative point that I have about the book is that here and there it contains some reference to the (recently acquired) religious convictions of the authors, which I find are not appropriate for this kind of work.
GasI try keep track of our gas usage and milages. For this purpose, I always fill up the gas tank and note down the distance we travelled. Our car also has a milage indicator but it is not always very reliable. Last week, Annabel got some gas, but she did not fill up the tank, because the price was quite high. Now I don't know how much gas was used during both periods. But I can use the readings from the milage indicator to make a good guess. Annabel filled 20.02 liter of gas and noted down a distance of 586.5 Km with a milage reading of 15.6 Km/l. Yesterday, I filled 35.55 liter with a distance of 309.6 Km and a milage reading of 16.0 Km/l. The total distance traveled is 896.1 Km and the total gas is 55.57 liter, which gives an average milage of about 16.1256 Km/l. The weighted milage according to the indicator is equal to (15.6 * 586.5 + 16.0 * 309.6) / 896.1, which is: 15.7382 Km/l. This means that the reported values of the milage indicator have to be multiplied with 1.02461 to arrive at an estimation of the true usage during the given periode. This results in an estimated usage of 36.69 liter of gas during the first periode and 18.88 liter during the second periode. These add up to 55.57 liter for both periodes. Yesterday, I paid € 61.47. But taking into accound the estimated gas usage during the last periode, it means that actually only € 32.65 was used. The remained has to be added up to the amount of the first periode (€ 35.42) resulting in € 64.24. Now I can use these numbers to calculate how much Annabel has to pay for gas that she used during both periods based on the administration that we keep, assuming that she agrees with the above calculations, of course.
De SlegteToday, it was the last day that books were sold (with a 75% discount) at the location of the former De Slegte bookshop. In the last half hour before they closed at 4 PM, I looked around, seeing if there was any book that I still would like to buy, but I could find non, not even when I was told at the end, that I could take some for free (because I had been such a frequent customer). I think, it was because I already had bought all the books that I did find interesting in the past months. At the moment, I have a whole stack of books that I still need to put in my bookshelves. They have to leave the shop empty by July first, and someone was already busy with taking down the shelves along the walls and not without making some loud noises.
Twitter pic: Historic moment.... The doors of Polare Enschede being closed for the last time. De Slegte now part of Broekhuis. — Broekhuis Enschede (@BoekEnschede) June 29, 2014
Finals AKI 2014I decided to have a look at Tetem and I discovered that the 'Finals AKI 2014' exhibition was held there, and in the AKI ArtEZ building and 21Rozendaal. At Tetem, I saw the works of, among others, Mercedes Marin and Pauli Beutel. In the main building, I:
I also visited 21Rozendaal, the last of three places where the finals exhibition was held. There I watched the movie "Part of the Empire/Plague" (trailer) by Jeffry Spekenbrink consisting of time-laps fragments for which he composed the music. In this sense it reminded me of Koyaanisqatsi. I also looked at the book "Showing my father the world" by Gunter Gruben from a project with the same title.
Seventh leafIn the past week the sixth and seventh leaf of the little magnolia plant have become fully grown. The start of the eight leaf is already visible. The leaves are becoming bigger and bigger, but still only half the length of an average leaf on the mother tree.
75% or moreSince last Thursday, bookshop Broekhuis is giving 75% off on all books at the former Polare / De Slegte location. But often it is even more, because the prices are rounded down. At 17:56:22, I bought the book Jalozie by Catherine Millet, ISBN:9789023462774, for only half a Euro.
Deventer murder trialToday, some interesting developments in the Deventer murder trial were announced. A team of the police has been researching the case again. They found two pieces of evidence suggesting that Mrs. Jaqueline Wittenberg was murdered on Friday, September 24, 1999 and not on the day before. This would rule out Ernest Louwes as the murderer because he has an alibi for that day. (Many years ago, some lay person already had concluded based on the report of the state of the body when it was found, that it was very unlikely that she was murdered on September 23.) Moreover, forensic doctors say that based on all available photos and research material there are indications that the body has been moved 6 to 24 hours after the murder. This would mean that someone has been on the crime scene before the body was found. (It was already known that the body was moved from, most likely, the hall to the living room before the picture of her late husband. Some lay person has concluded that the seven stabbing wounds were made after the body was moved.) There is also mentioning of new perpetrator knowledge was discovered. The details of this are not revealed. (This could point to an other suspect that is being investigated at this moment.)
In addition, the investigators discovered a crucial mistake regarding the phone call that Louwes made to his client, the victim, on the evening of September 23. Louwes declared that he made the phone call 'on the A28, near Harderwijk'. But in the police report this was mistakenly interpreted as a call from 't Harde, which is a difference of at least 8 kilometers. This also invalidates an important statement by expert witnesses with respect to the possibility that the phone call was picked-up from Deventer. Faced with this new information, expert witness J. Rijnderslaan from KPN Security, has retracted a rather incriminating statement against Louwes, and he encourages other experts to do the same.
Fifth leafIn the past week the fifth leaf of the little magnolia plant occured. It is now larger than the other leaves and it also is on a small stalk. The signs for a sixth leaf is already there. It seems it is picking up some speed now.
Trip to China 2010
-- contact -- Frans
My life as a hacker
The Art of Programming
HTML to LaTeX
eXtreme Programming Hamilton cycles