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Dairy, July 2014



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Thursday, July 3, 2014

Toki Pona: The language of good

This 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.


Saturday, July 12, 2014

Flowers in magnolia

I 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.

Book about math

At 17:01:48, I bought the book Speeltuin van de wiskunde: Opties, Kansspelen, Esher, pi, Fermat en Meer, with editors Bart de Smith and Jaap Top, ISBN:9789076988207, from bookshop Broekhuis for € 8.95. The book is about some popular math topics at the M.Sc. level.


Tuesday, July 15, 2014

Double tree graphs

Some 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.


Thursday, July 17, 2014

Moleskine notebook

At 20:48, I bought a violet Moleskine Notebook Pocket Plain Hard for €11.95 from bookshop Broekhuis, which I plan to use as next diary


Thursday, July 24, 2014

4-connected

The 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.


Friday, July 25, 2014

The Jordaan

Today, I got the book De Jordaan: 28000 meter gevelwand (The Jordaan: 28000 meter facades), ISBN:9789062740079, which I had bought through the internet for € 15.00. The book is about all of the facades of the Jordaan district in Amsterdam. The book was published in 1978 and the drawings of the facades were made in the five years before. All facades were measured up to the height of 3 meters and a total of 2,300 photographs of the facades were taken. The facades were drawn on 72 sheets, 60 x 150 cm, on a scale of 1:200. For the book the drawings were reduced to a scale of 1:400, each covering two pages. The book is 39.8 cm wide and 26.8 cm high. The book contains introduction in Dutch and English. I bought this book because it is quite unique.


Saturday, July 26, 2014

Another dead-end

Last 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: Yesterday, I found the paper The diameter of associahedra by Lionel Pournin of which the abstract starts with: "It is proven here that the diameter of the d-dimensional associahedron is 2d-4 when d is greater than 9." The associahedron is related to 'diamond switches' in tree or flips in polygon triangulations, which are related to Catalan number. The OEIS sequence A005152 gives the first values. This shows that in many cases it is not possible to convert a tree of size n with n diamond switches (or triangle flips) and thus not show that they have a mutual coloring.


Tuesday, July 29, 2014

Triangulation of regular polygons

The 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 program to count these numbers and 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 columns 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 OEISsequence 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

(Addition on November 7, 2015: The numbers in the first column are OEISsequence A257887.)


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