A AB A AB ABC D DE DEF AB ABC DE DEF GH GHI
If you look at all the ways these can be layed on a square grid, than the following 86 patterns can occur on the squares made up of two by two unit squares:
AA AA AA AA AB AB AB AB AB AB AD AD AD AG AG BA BA BA BA BC BC BC BD CD DA DD AA AB BA BC CA DE AB DA DG AB DA AD CD EA ED AA AB BA BC BC BD BD BD BG BG CA CA CD CD CG DA DA DA DA DA DD DE DE DE DE CA EF AB EA EG AB EA FA FD FA FG FA AB BD CD GA GD GA AA AB BA BC DE DE DG DG EA EA EA EA EA ED EF EF EF EF EF EG EG FA FA FA FA FA CA GH AB GA AB AD CD HA HD HA AA AB BA CA HI AB HA AB AD BD IA ID FD FG FG GH GH GH GH GH HA HA HA HD HI HI HI HI IA IA IA ID IA AB IA AA AB BA BC CA AB AD CD AB AA AB BA CA AB AD BD AB
These are not exactly Wang tiles, because Wang tiles are labled on their edges, not their corners. But there are 28 different combinations of letters that occur on the edges, thus we could create an equivalent of 86 Wang tiles with 28 different 'colours' for the edges. This is a rather large number, and I thus thought about a way to create Wang tiles with less 'colours'. If one makes Wang tiles around the points where four unit squares meet, and use letters to indicate properties about the edges, I came up with the following set of 76 Wang tiles using nine 'colours' (0 and the letters 'a' to 'h'), where the order in which the 'colours' are give are top, left, right, and bottom side. Note that for every wxyz there is also a xwzy.
0000 0ada 0ae0 0bda 0bf0 0cda 0cg0 0dga 0dh0 0ega 0fga 0fj0 a00e a0ad ab0b ac0b ad0b af0b b00f b0ad bab0 bb0c bbc0 bc0c bd0c bdb0 bec0 bf0c bgb0 bhc0 c0ad cdb0 cec0 cgb0 chc0 d0ag dab0 db0b dbc0 dc0b dd0b ddb0 dec0 df0b dg0b dh0b e0ag eb0c ec0c ed0c ef0c eg0c eh0c f0ag fdb0 fec0 fgb0 fhc0 gab0 gbc0 gd0b gdb0 gec0 gf0b gg0b gh0b hab0 hbc0 hd0c hdb0 hec0 hf0c hg0c hgb0 hh0c hhc0
I did search for algorithms or programs for analyzing Wang tiles, but I failed to find any. Last week, I came across a copy of the book Shadows of the Mind in which there is some reference to tiling the plane with copies from a set of polyominos.
AA AA AA AB AB AB AB AB AD AD AD BA BA BA BA BC BD BD BD CA CA CD BD DA DD AA AB BA BC DE AB DA DG AD CD EA ED EF AB EA EG FA FD FA CD DA DA DE DE DE DE DE DG EA EA EA EA EA ED EF EF EF EF EF EG EG FG AB BD AA AB BA BC GH AB AB AD CD HA HD HA AA AB BA CA HI AB HA FA FA FA FA FA FD FG FG GH GH GH GH HA HA HA HD HI HI HI HI IA IA AB AD BD IA ID IA AB IA AA AB BA BC AB AD CD AB AA AB BA CA AB AD IA ID BD AB
Next, I wanted to know if it would be possible to construct an algorith that would fill one quadrant of the plane with a 'random' pattern, by completing a diagonal at the time. It would be nice to know that if one has finished a diagonal, it is always possible to fill the next, without having to back-track to the current. We look at filling the fourth quadrant. We define a diagonal sequences of colours that can occur on a path that alternating goes up and right, starting with an up movement from the first colour to the second and ending with a right movement between the last two colours. This means that the sequences are always of odd length. I wanted to find out which sequences are not possible. For the sequences of length three this is rather simple to derive from the above Wang tiles. For example the sequence III is not possible because there is no Wang tile in the above set that has I in the left bottom, left top, and right top corner. The program found the following sequences of length five and seven that are also not possible (excluding sequences that contain a shorter sequences that is already not possible. There are no sequences of length nine that are not possible (presuming that they do not contain impossible sequences of length three, five, and seven. This gives some hope that it is indeed possible to implement a back-tracking algorithm that fills the quandrant diagonal by diagonal, not having to back-track to a previous diagonal if the impossible sequences are avoided when filling a diagonal. The sequences are:
BAABA BAABD BAAEA BAAEF BAAEG BAAFA BAAFG BAAHA BAAHD BAAHI BAAIA BAAID DAABA DAABD DAAEA DAAEF DAAEG DAAFA DAAFG DAAHA DAAHD DAAHI DAAIA DAAID ABABA ABABD ABAEA ABAEF ABAEG ABAFA ABAFG ABAHA ABAHD ABAHI ABAIA ABAID CBABA CBABD CBAEA CBAEF CBAEG CBAFA CBAFG CBAHA CBAHD CBAHI CBAIA CBAID EBABA EBABD EBAEA EBAEF EBAEG EBAFA EBAFG EBAHA EBAHD EBAHI EBAIA EBAID EBCBA EBCEA EBCEF EBCHA EBCHI FCAFA FCAFG FCAIA FCAID ADAAB ADAAD ADADA ADADE ADADG BDAAB BDAAD BDADA BDADE BDADG ADGDE AEAAB AEAAD AEADA AEADE AEADG CEAAB CEAAD CEADA CEADE CEADG HEAAB HEAAD HEADA HEADE HEADG AEGDE HEGDE AFAAB AFAAD AFADA AFADE AFADG BFAAB BFAAD BFADA BFADE BFADG IFAAB IFAAD IFADA IFADE IFADG AFGDE IFGDE AHAAB AHAAD AHADA AHADE AHADG AHAGH CHAAB CHAAD CHADA CHADE CHADG CHAGH AIAAB AIAAD AIADA AIADE AIADG AIAGH BIAAB BIAAD BIADA BIADE BIADG BIAGH AABABAA AABABAB AABABDA AABABDE AABABEF AABABGH AABDEBA AABDEBC AABDEBD AABEFCA AABEFCD AABGHEA AABGHED AABGHEF AABGHEG AABHIFA AABHIFD AABHIFG BABABAA BABABAB BABABDA BABABDE BABABEF BABABGH BABDEBA BABDEBC BABDEBD BABEFCA BABEFCD BABGHEA BABGHED BABGHEF BABGHEG BABHIFA BABHIFD BABHIFG DABABAA DABABAB DABABDA DABABDE DABABEF DABABGH DABDEBA DABDEBC DABDEBD DABEFCA DABEFCD DABGHEA DABGHED DABGHEF DABGHEG DABHIFA DABHIFD DABHIFG AADABAA AADABAB AADABDA AADABDE AADABFA AADABGH AADABIA AADADAA AADADAB AADADAD DADABAA DADABAB DADABDA DADABDE DADABFA DADABGH DADABIA DADADAA DADADAB DADADAD ABDABAA ABDABAB ABDABDA ABDABDE ABDABFA ABDABGH ABDABIA ABDADAA ABDADAB ABDADAD EBDABAA EBDABAB EBDABDA EBDABDE EBDABFA EBDABGH EBDABIA EBDADAA EBDADAB EBDADAD FCDABAA FCDABAB FCDABDA FCDABDE FCDABFA FCDABGH FCDABIA FCDADAA FCDADAB FCDADAD ADEBDAA ADEBDAB ADEBDAD BDEBDAA BDEBDAB BDEBDAD GDEBDAA GDEBDAB GDEBDAD HEDABAA HEDABAB HEDABDA HEDABDE HEDABFA HEDABGH HEDABIA HEDADAA HEDADAB HEDADAD AEFCDAA AEFCDAB AEFCDAD BEFCDAA BEFCDAB BEFCDAD CEFCDAA CEFCDAB CEFCDAD HEFCDAA HEFCDAB HEFCDAD IFDABAA IFDABAB IFDABDA IFDABDE IFDABGH IFDADAA IFDADAB IFDADAD AGHEDAA AGHEDAB AGHEDAD AGHEFCA AGHEFCD BGHEDAA BGHEDAB BGHEDAD BGHEFCA BGHEFCD AHDABAA AHDABAB AHDABDA AHDABDE AHDABEF AHDABGH AHIFDAA AHIFDAB AHIFDAD BHIFDAA BHIFDAB BHIFDAD CHIFDAA CHIFDAB CHIFDAD AIDABAA AIDABAB AIDABDA AIDABDE AIDABEF AIDABGH
These results are incorrect! read follow-up story.
Next I saw the Thomas Gainsborough exhibition. I was not very impressed. The work I liked most is The daughters of the artist from around 1758.
From 12:01 to 12:17, I walked around The Rhythm Painter by Jaap Drupsteen. (impressions on vimeo.) After I looked around the rest of the museum, I came back and walked around from 12:38 to be another twenty or so minutes. In the rest of the museum, I liked:
At 13:07:20, I bought the book De Nieuwe Smaak. De kunst van het verzamelen in de 21ste eeuw edited by Josien Beltman written in Dutch published by Rijksmuseum Twenthe in 2016, ISBN:9789072250421, from the museum for € 2.00.
Next, while on my way to TETEM art space, I decided to visit Kunstenlandschap (Art and landscape). At Tetem, I saw the following works I liked:
The GPS-track of how I biked and walked in KML file for Google Earth or in Google Maps. Along the route I liked some works by: Nanon Muskee, Henk Slomp, Mark van Loon, and Imke Beek.
AA AA AB AB AB AB AD AD AD BA BA BA BA BC BD BD BD CA CA CD CD DA BD DA AA BA BC DE AB DA DG AD CD EA ED EF AB EA EG FA FD FA FG AB DA DE DE DE DE DE DG EA EA EA EA EA ED EF EF EF EF EF EG EG FA FA BD AA AB BA BC GH AB AB AD CD HA HD HA AA AB BA CA HI AB HA AB AD FA FA FA FD FG FG GH GH GH GH HA HA HA HD HI HI HI HI IA IA IA ID BD IA ID IA AB IA AA AB BA BC AB AD CD AB AA AB BA CA AB AD BD AB
But this did not effect the algorithm for finding all possible diagonal sequences. Next, I noticed that one sequence of length five being marked as impossible, was actually possible. This uncovered a huge bug in the algorithm for calculating impossible sequences. Now it no longer found impossible sequences with length five and the algorithm enumerating possible diagonal sequences returned many results. But I did find some diagonal sequences for which there was no next diagonal sequence. I changed the algorithm for finding impossible sequences such that it would not stop with length five. I made it search up to sequences of length 23. It found nine impossible sequences that all contain the subsequence "AHIFA". There are possible diagonal sequences that contain the subsequence "AHIFA". It is not possible to claim that these are all the impossible sequences, but there is reason to believe that this is the case. The sequences are:
DAAHIFAAB DAAHIFADEBA DAAHIFADEBD ADEBAHIFAAB BDEBAHIFAAB ADEBAHIFADEBA ADEBAHIFADEBD BDEBAHIFADEBA BDEBAHIFADEBDOn could capture all these sequences with the regular expression ([AB]DEB|DA)AHIFA(AB|DEB[AD]). Below a table given results with respect to the number of diagonal sequences. The first column gives the length of the sequences, the second column the number of diagonal sequences, and the third column the number of patterns that exist till and including this diagonal. The rest columns explain the relationship between the number of sequences and number of columns. For the columns with the number n in the header, the number in the row tells how many sequences are related to n patterns with the sequences. The numbers in the row add up to the number of sequences, while the numbers multiplied with the number in the header of the columns add up to the number of patterns.
l #dia #pat 1 2 3 4 5 6 7 8 9 ------------------------------------------------------------------ 3 3 3 3 5 6 6 6 7 12 12 12 9 30 30 30 11 68 69 67 1 13 178 210 150 24 4 15 483 636 354 105 24 17 1706 2684 1010 478 182 8 28 19 7452 13160 3582 2480 1086 160 144 21 35764 74566 13102 12242 7166 1320 1666 64 162 24 18
May 30: These results are incorrect! Follow-up
But there was also some news on the scientific front. The group at the Academic Hospital Maastricht has now been recognized as an expert center for Kabuki Syndrome here in the Netherlands and there is a process of them also becoming partner of an European group of expert centers for the syndrome. Copies of the paper Body proportions in children with Kabuki syndrome were handed out. If I remember correcly, pictures of Andy were taken in 2013, and I assume that he is one of the 11 children with Kabuki Syndrome included in the study.
The most interesting news were the results from the growth hormone study. The results of the study are very positive. They found a one standard deviation improvement in the length and also the other body parameters (methabolic rate and non-fat tissue) improved. They are now going to request growth hormone therapy to be available for children with Kabuki Syndrome here in the Netherlands as only half a standard deviation improvement is sufficient. They did find that the normal test for the various growth hormone factors gave very conflicting results, not in accordance to normal measurements, but that non the less, all children in the study (even those that otherwise would not have been qualified for growth hormone therapy based on the factors) reacted positively to the therapy. Before it was reported that growth hormone therapy seems to have a positive effect on muscle tone, concentration, and intellectual abilities. In some cases it also seems to stabilize blood sugar levels. But this was not studied in this study and thus also not affirmed.
The GPS-track of how we drove in KML file for Google Earth or in Google Maps.