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I write, therefore I am

With this variation on a famous statement by the philosopher Descartes, I would like to express that the act of writing about what happens in my life is important to me.

Monday, July 13, 2026

I was mistaken

The formulea that I presented yesterday for the determinant (related to convex shapes of hexagons) is incorrect. The correct formulea, written slightly different, is: 
    -28(a² + c²) + 8ac + 8n(a + c) + 4n² - 32l
Taking out a factor of four, we get: 
    -7(a² + c²) + 2ac + 2n(a + c) + n² - 8l
So, how do we find integer values for a and c for given n and l such that the above formulea returns a square? Even if l is a negative number, there are only a limited number of pairs to be evaluated, due to the negative factor with the quadratic terms. The values for n are always positive. It looks like all positive values are within a certain circle or elipse. The function is symmetric around the line a = b, The highest value probably lays on that line. If we replace a and b we get the formulea: 
    -12x² + 4nx + n² - 8l
From the first derivative we know that the highest point is at x = n/6. This might be used to determine the center of the circle. I wrote some code to investigate values for which the formulea returns squares for various values of n and l and discovered that only for some multiples of 3 for n and multiples of -6 for l there are no squares. The others have many squares and it seems that the number increases with smaller values for l.

Sunday, July 12, 2026

Some interesting relationships

I found some interesting relationships with respect to convex shapes of hexagons. I realized that the each hexagon on a triangluar grid can be viewed as the intersection of two triangles. I also realized that there is a simple relationship with respect to the sizes of the hexagon. Let the sizes of the hexagon, be denoted by a, b, c, d, e, and f, then the following equations are true: 
    a + b = d + e
    b + c = e + f
    c + d = f + a
Note that the third equation can be derived from the first two. If n is the sum of all sides, the following equation can be derived: 
	d = n - a - 2b - 2c
	e = -n + 2a + 3b + 2c
	f = n - 2a - 2b - c
Now it is also possible to calculate the number of enclosed triangles from the values of a, b, c, and n with the formulea: 
     n(4(a + c) + 6b) - n² - 4(a² + c²) - 8b² - 10b(a + c) - 6ac
The formulea shows that a and c can be interchanged, which is what one would expect. This formulea can also be used to find generic formuleas on k of the form 6k² + nk + l, where l is equal to the above formulea for enclosed triangles. I wonder if there is an efficient method to discover for giving values for n and l there is a solution for some values of a, b, and c or not. We could for example, turn the formulea in a quadratic equation of b, which results in (if I am not mistaken): 
     -8b² + (6n - 10(a + c))b - n² + 4n(a + c) - 4(a² + c²) - 6ac - l = 0
The determinant of the quadratic equation on b is (if I am not mistaken): 
    -28a² - 28c² + (128n - 120)a + (128n - 120)c + 8ac + 4n² - 32l
For there to be a integer solution for b, the determinant must be a square number.

Remark July 13: I was mistaken with respect to last formulea.

Saturday, July 11, 2026

Cyber Ægg

In the afternoon, I went to TkkrLab to help with the 'sweatshop' for assembling the Cyber Ægg badge for BornHack. I first unpacked some PCBs, next I helped with attaching the E-Ink display, because that was the bottleneck in the whole assembly line. I did the same in 2017 for the SHA2017 badge. I felt it was a little bit more difficutl this time. I did a bit over thirty before I left.

Exhibitions

I went into the city and at photo gallery Objektief, I saw the exhibition Meesterwerken 2026 with students finishing some the photography training. I saw the presentations by:

At Concordia, I saw the exhibition AKI Finals '26 with works by the following graduates:

I also saw the exhibition Expositie Boswinkel with amateur art from the neighborhood with that same name. I was rather surprised by the quality as the neighborhood is not known as one of the best neighborhoods in Enschede.

Tuesday, July 7, 2026

Flowers again

Just like last year and some years before, our magnolia has some flowers again. The three plants that I planted about a month ago are still doing well. They still rather small, but I understand that that is normal as they focus on growing roots first. I regularly give them water with the waste water from rinsing vegatables.

Friday, July 3, 2025

Graduation Show at KABK

Today, I went to the Graduation Show 2026 at KABK. I found the works of the following students noteworthy, which is very subjective and often based on my first impression, in the order I encountered them:

Wildsam Field Guides: Los Angeles

In the train home, I finished reading the booklet Wildsam Field Guides: Los Angeles, which I started reading on March 22 after I bought it on February 18. Initial I read some pages during commercial breaks. In the past days, I finished reading it in the train. Although it has many pages with fun and mundane facts, it also has some serious content giving you a good impression of the city.

Thursday, July 2, 2025

Graduation Show at Rietveld Academie

I traveled to Amsterdam to attend the Graduation Show 2026 at Gerrit Rietveld Academie. I found the works of the following students noteworthy, which is very subjective and often based on the first impression, in the order I encountered them:

Museums

Afterwards, I went to Stedelijk Museum Amsterdam. There I first saw the exhibtion Kho Liang Ie, which contained the following works from his collection: From some of the other exhibtions: I also saw the exhibition Danh Vo - πνεῦμα (Ἔλισσα).

At FOAM, I saw the exhibitions Foam Talent 2026 and Martin Parr - Very Modern and Rather Ugly.

Wednesday, July 1, 2026

More on convex shapes of hexagons

I continue looking into convex shapes of hexagons and looking into the conjecture that the sequence deviates at most one from the sequence A216522. With respect to this sequence and the sequence A135711 the following are true: 

A216522(n) = ceiling(sqrt(12n + 9)) - 3
A135711(n) = 2.ceiling(sqrt(12n - 3))
A135711(n) = 2.A216522(n-1) + 6
A216522(n) = A135711(n+1)/2 - 3
(It is important to note that the sequence A135711 is one based where the sequence A216522 is zero based.) If we take the sequence A216522 as a reference then the perimeter should be defined as the distance between the hexagons on the outside where the distance between the center of two hexagon is defined as one. The table below gives for a given perimeter (for which k is positive) the number of hexagons is can enclose. So for k is one, seven hexagons can be enclosed with a perimeter of six. 

perimeter:  sides:                         #triangles:     #hexagons:
6k          k,   k,   k,   k,   k,   k     6k²             3k² + 3k + 1
6k + 1      k+1, k-1, k+1, k,   k,   k     6k² +  2k - 1   3k² + 4k + 1
6k + 2      k+1, k,   k,   k+1, k,   k     6k² +  4k       3k² + 5k + 2
6k + 3      k+1, k  , k+1, k,   k+1, k     6k² +  6k + 1   3k² + 6k + 3
6k + 4      k+1, k+1, k,   k+1, k+1, k     6k² +  8k + 2   3k² + 7k + 4
6k + 5      k,   k+2, k,   k+1, k+1, k+1   6k² + 10k + 3   3k² + 8k + 5
If we use a zero based variant of sequence A216522, the formulea becomes: ceiling(sqrt(12n - 3)) - 3 for the number of perimeters. Lets denote this number with p, then the equations with this formulea are: 
p + 3 - 1 < sqrt(12n - 3) ≤ p + 3
(p + 2)² < 12n - 3 ≤ (p + 3)²
p² + 4p + 4 < 12n - 3 ≤ p² + 6p + 9
p² + 4p + 7 < 12n ≤ p² + 6p + 12
Now if we fill in the values for the perimers and matching number of hexagons, we get: 
6k:      36k² + 24k +  7 < 36k² + 36k + 12 ≤ 36k² + 36k + 12
6k + 1:  36k² + 36k + 12 < 36k² + 48k + 12 ≤ 36k² + 48k + 19
6k + 2:  36k² + 48k + 19 < 36k² + 60k + 24 ≤ 36k² + 60k + 28
6k + 3:  36k² + 60k + 28 < 36k² + 72k + 36 ≤ 36k² + 72k + 39
6k + 4:  36k² + 72k + 39 < 36k² + 84k + 48 ≤ 36k² + 84k + 52
6k + 5:  36k² + 84k + 52 < 36k² + 96k + 60 ≤ 36k² + 96k + 67
It is quite obvious that for all positive k the above equations are true. This means that for all those minimal perimeter for the given number of hexagons is equal to the given formulea. So, there is not a value, above which the perimeter might be always one larger than given by the formulea. This seems to support the conjecture that it is at most one higher than the value given by the formulea. It should be noted that the number of hexagons for which there is a simple solution for the minimal perimeter appear further and further apart leaving more and more room for complicated cases.

End of heat wave

Today, is the last day of a two week long regional heat wave. In the past two weeks the maximum temperature was at least 25° Celsius. There were eight days on which the temperature was 30° or higher of which one day higher than 35° with a new 37.9° record for the day. There was a total of four days with temperature records for the day of the year. There was also one night that the minimum temperature was 21.6°, which is also rather exceptional. The national heat wave was only eleven days, but nevertheless a record number of days. In the South of the country there was even a 'super heat wave' with four days with temperatures of 35° or higher.

Tuesday, June 30, 2026

Link

Monday, June 29, 2026

AKI Finals

The AKI Finals already opened last Friday, but I did not go due to the high temperatures. I heard that many invited people cancelled and that there was no speech as usual. I went today to see the exhibition. Below the list of exhibitions from the graduate that I found noteworthy (based on my very subjective impression) in the other I encountered them when walking through the building. Some names are mentioned twice, because I encountered them twice.

At 13:39, I bought the catalogue 25/26 edited by Ruben Steeman and written in English for € 25.00.

Saturday, June 27, 2026

Sequence A067628

In the past days, I have been looking into the problem of compact shapes of hexagons again. I worked on developing a faster algorithm to generate more values for the sequence to see if there are maybe cases where is deviates with more than one with the sequence A216522. This afternoon, I realized that maybe the sequence for convex shapes on triangular grid is included in the On-Line Encyclopedia of Integer Sequences. I found the sequence A067628: Minimal perimeter of polyiamond with n triangles. At first it looked the same, but on further inspection, I discovered that it has many differences, the first being the 84 triangles, which with the convex requirement is 26 instead of the 24 in the sequence. The new algorithm starts with a parallelogram on the triangular grid, which is defined by the length of the two sides, and next removes all combinations of triangles on the triangle grid from the two sharp corners. This also shows that each shape can be defined by four parameters. If the lengths of the parallelogram are a and b and the lengths of the sides of the triangles that are removed are c and d, then the length of the perimeter is: 2a+2b-c-d and the number of triangles equal to: 2ab-c²-d². Of course, the values for c and d should both be smaller or equal to both a and b. This can be achieved by enforcing: a≤b≤c≤d. I checked that the algorithm returned the same values as generated with the program convexhexagon.cpp. I calculated values to 10,000 and did not find any value for which the differences is more than one.

33.9° Celsius

The temperature at Twente Airport has gone up to 33.9° Celsius, which breaks the previous record of 32.2°C on this date in 1976.

Friday, June 26, 2026

37.4° Celsius

The temperature at Twente Airport has gone up to 37.4° Celsius, which breaks the previous record of 31.3°C on this date in 1976. Because we knew it was going to be hot today, we left around a quarter past seven o'clock for a walk. When we crossed the bridge over the pond called 'Vijver Smuddeboshoek', we saw a kind of rainbow on the water as shown in the photograph below. It is probably caused by iron-oxidizing bacteria that result in a thin film of iron oxide on the water.

Link

Thursday, June 25, 2026

Stoeptegel 1

Around five o'clock in the afternoon, I bought the artwork Stoeptegel 1 from Jobke Eijsink, which consists of two prints each 30 cm square, the size of regular paving tile. One is a life-size print of a paving tile, and the other has black dots on a white background, where the dots correspond to certain features on the tile, resulting in an intriguing pattern that lies somewhere between totally regular and purely random. I had already seen the work earlier this month, on the sixth, during the Open Studios day, and was immediately fascinated by it.

Wednesday, June 24, 2026

34.1° Celsius

The temperature at Twente Airport has gone up to 34.3° Celsius, which breaks the previous record of 33.1°C on this date in 2005. We are in the middle of a heat wave (according to Dutch definition) with the hotest days still ahead (according to the latest predictions). It started with a maximum temperature of 30.1° on last Thursday. On Friday the maximum temperature of 33.3° broke an earlier record of 32.6° of 2000 on that day. This was followed by maximum temperatures of 31.1°, 28.7°, 27.6°, and 29.3° in the following days. This means that on Monday we already had a regional heat wave.

Monday, June 22, 2026

History of the modern literatue

This morning, I finished reading the book Geschiedenis van de moderne literatuur (Dutch for History of the modern literatue) by Herman Brusselmans, which I started reading on December 2, 2022 after I bought it on November 29, 2022. (I have read some other books in the mean time.) This not an academic non-fiction book one would expect from the title, but considered by the author himself to be a novel. I would not so much call it a novel, but rather a mixture of an autographic rant and a diary. In the first part of the book he critizes Belgian and Dutch authors often interspersed with accounts of personal encounters with them and anecdotal stories about them. But increasingly, he writes about his daily activities, with the occasional bizarre story thrown in. It is rather boring book, unless you enjoy his kind of humor.

Sunday, June 21, 2026

Biking along MUPIs

From 9:20 to 13:24, I biked around the city to visit all the MUPIs that had a poster of a work by this year graduates of the AKI. I mostly followed the route I calculated last Thursdays. I only made one mistake in the order. At one place I took bit different but not really longer route. I found the posters of the following graduate the most interesting: I found the posters of the following graduates interesting: I found the posters of the following graduates noteworthy: Eline de Boer, Tamar den Bok, Manal Al-Harazi, Loes Kingma, Marieta Kritikou, Sofie Reijerman, Borys Glowacki, Floris Gies, Lea Hadadi, Yordan Kirev, Dahyun Lyu, Benedetta Zamboni, Irene van Dijke, Dorsheila Malisama, Dilara Tramm, Emil Ardla, Ece Bakirhan, Havva Malyalizade, and Caroline Schot. It is difficult to judge the quality of a graduate based on one single image not knowing the range of works produced and the story behing the image.

Milan van Zuilen

I saw the exhibition Milan van Zuilen at Rijksmuseum Twenthe with works by Milan van Zuilen. I found the following works noteworthy:

Friday, June 19, 2026

Sign extension in RISC-V instructions

Sign extension is when a signed value represented with a number of bits is converted to a binary number with more bits, such that negative values stay negative. For the two's complement method for representing binary numbers, this is done by assigning higher bits with the value of the highest bit. Hence why it is called sign extension. I am studying the RISC-V instruction set, in particular the RV64I: Base integer instruction set, 64-bit. I am a bit surprised that all immediate instructions working on 12-bit values (encoded in highest 12-bits of the instruction) are signed value, meaning sign extension is applied. I understand the reasoning for doing this for the immediate add instruction, because it removes the need to also have an immediate substract instruction. However, I do not understand why it is also used for all bitwise logic operations. This restricts the use of the twelved bit for logic operations. If you want to do use it, you first have to load an immediate value in another register before performing the operation. The lui instruction can be used to set the bits 13 to 32 with a 20-bits value, which is also sign extended to fill the highest 32-bits of the 64-bits register. If you want to store a certain value in 32-bits, this means that you first use the lui instruction and than follow it by a addi (add immediate) instruction. But you have to take into account that if the 12-bit is set, it actually means a substraction (due to the sign extension) of a constant. In that case you also have to increase the value used in the lui instruction with one to compensate that the sign extension will result in a substraction of one. If the ori (or immediate) would not have used sign extension, one could have used that instead of the addi instruction without having to add one to the value for lui. It should be the hex2.c program that has to deal with this, but I have not found the code that does it. The cc_emit.c generates values prefixed with ~ and !, however, I do not see the corrections in hex2_AArch64.hex1, which I expect to be in the code following StorePointer_rel3 and StorePointer_rel1.

Thursday, June 18, 2026

AKI MUPI route 2026

Yesterday evening, I biked along one of the MUPI advertisement signs to see if they already had put the photographs with works by the graduates of the AKI on the back side, because I had not seen an official announcement yet, only to discover that they had done it. This morning, I found a facebook post announding the locations of the MUPIs with a link to a Google map. When I ran my program ParseMUPIkml2.cpp on the downloaded map, it reported two errors. One was due to one pin being a bit too far from the location of the MUPI and the other was due to a MUPI that I had not yet included in my map of possible biking routes between them. After having added the new MUPI and routes to it, the program returned a route along all the MUPIs with a length of 35,294 meter. A slightly shorter route, not including my home, of almost 35Km can be found in Google Maps. I do not know yet, when I am going to bike the route, because according to the weather and climate plume of today (at midnight) for Twenthe by The Royal Netherlands Meteorological Institute we are in for a heat wave (according their definition) for almost two weeks including today.

Links

Tuesday, June 16, 2026

Link

Sunday, June 14, 2026

Faster live-bootstrapping

Harmen Stoppels has developed a version of live-bootstrapping in the repository shpack, which bootstraps a basic, reproducible Linux environment natively from a few hundred bytes of binary seed to a working GCC 4.7 (C and C++) in 2 minutes and 30 seconds. It makes uses of both the original live-bootstrap project as the tcc_cc compiler that I have developed and can be found in the MES-replacement repository. It also has replaced the standard library that comes with the GNU Mes compiler with the musl libc standard library.

Last week I created the boot_cc repository that is a hard fork of the MES-replacement repository for developments that go beyond the goals of that repository, namely to create an easy reviewable live-bootstrap solution. After having seen the above solution, I also looked into performance improvements. I first looked at the hex2 program, which was taking most of the time during the bootstrapping. I first implemented a hash tree mapping, based on code I developed for IParse before. It did give some improvement. Then I noticed that the time command was showing that a lot of time was spend on executing system calls. I spend some time reducint the number of calls by using buffers for input and output. That gave a hugh performance boost. That probably also proof why the above solution was so much faster than MES-replacement. Today, I spend time implementing buffered file input and output in the standard library that is used for the tcc_cc compiler. It does work well for x86 and x86_64/amd64, but for arm64/AArch64 it looks like the M1 executable becomes a lot slower. I have not looked into it yet why this is the case. I nevertheless commited the results.

Saturday, June 13, 2026

Links

Thursday, June 11, 2026

Weeds and plants

I enjoy the various weeds and plants in our garden in the back of the house. Below a picture of a part of our garden. In the left bottom corner the shadow and on leaf of our magnolia can be seen. Against the fence there are various fruit plants. The tallest is a rubus. There are also two grapes. There are also six mustard greens and three little magnolia plants among all the weeds. In the back there are three Jerusalem artichoke, still rather small.

Monday, June 8, 2026

Links

Sunday, June 7, 2026

Exhibitions

I first went to the exhibition Herman Nijhof with photographs by Herman Nijhof, who was a Dutch garden and landscape architect, but better known in pop music circles as a photographer. I also looked around a bit at some of the studios of the artists in the building where the exhibitions was held. I met with an old friend in the city and together we went or B93 where we went to the opening of the exhibition Van de plank with photographs and works by Jojanneke Dekens, Loeske Bult, and Anja Wisseborn. Afterwards we also visited the studios of Judith Schepers (from whom I bought a postcard sized reproduction of her charcoal drawing Labyrinth from 2024) and Susan Pinkster.

Saturday, June 6, 2026

Open Studios Enschede

This afternoon, I visited all ten locations of the Open Studios Enschede weekend. I visited the following locations: By the end of the afternoon, I felt that my mind could no longer absorb information. I also realized that I had not visited all the studios that were supposed to be open according to the brochure.

Friday, June , 2026

Two more sprouted plants

This afternoon, I cleaned the drain in the alley, because yesterday there was some flooding in the alley due to some heavy rain showers. After I was done, I discovered two more new plants in the seed tray that contained the magnolia seeds from last year. I decided to plant them in the garden, close to where I planted the other plant. These new plants even looked a bit larger than the first.

Thusday, June 4, 2026

Progress on Task 5

I have complete two sub tasks of Task 5 of the GNU Mes replacement project, namely the implementation for x86_64/amd64 and for arm64/AArch64. The pass1.kaem script of tcc-0.9.27 step had no support for the x86_64/amd64 and arm64/AArch64 targets. I had to fix this and also apply some of the patches for tcc-0.9.26 to get it compiled by the tcc-0.9.26 compiler. After updating the documentation and optimizing the code generation of stack_c_arm64.c a bit, I completed these sub tasks. Next I will have to work on the riscv64 target.

Links

Monday, June 1, 2026

The Mandalorian and Grogu

This afternoon, Conny and I went to see the film The Mandalorian and Grogu. The movie was in the style of the earlier Mandalorian series. It was a fun watch, but not really memorable. We watched this in IMAX 2D. I was a little annoyed by the IMAX 'promo' at the start of the film.

Sunday, May 31, 2026

Progress with 64-bits targets

Harmen Stoppels submitted some pull request to fix some problems with getting the Tiny C Compiler compiled for some 64-bits targets using the tools I have been developing to replace the GNU Mes compiler that is used in the first stage of the live-bootstrap project. I understand that he used Claude Code to find fixes. The fixes are for compiling the tcc-0.9.26 sources to x86_64 (also know as amd64) and AArch64 (also known as ARM64). However the tcc-0.9.27 sources cannot be compiled yet with the compiled tcc-0.9.26 giving different kind of error messages for both targets. I also did find a bug (Issue #15) in the compiler with respect to the use of the continue statement inside a switch statement that is placed inside a for, while, or do statement. I had 'hacked' the implementation of the switch statement with the do statement of the Stack-C intermediate language. To execute the programs that are compiled for the AArch64 target under Linux one has to install the Arm System emulator of QEMU. When installed this will automatically call the emulator whenever an AArch64 ELF file is executed. Very handy for testing purposes.

Link

Thursday, May 28, 2026

Link

Wednesday, May 27, 2026

Proof for A216522

I am going to give a proof that the sequence A216522: 'Integers of the form 2*x + 3*y with nonnegative x and y, with repetitions,' can be expressed with ceiling(sqrt(12*n + 9)) - 3 with n starting at 0 for the first element in the sequence. Given a certain number, is quite easy to determine in how many ways it can be written as a sum of 2's and 3's. It all comes down to the fact that 6 equals three 2's and two 3's. This means, that for example 18 there are four ways to write it as a sum of 2's and 3's. This leads to the formulea: f(6*k) = k + 1 with k larger or equal to 0, Because, 2, 3, 4, and 5, can only be written in one way as a sum of 2's and 3's, we also know that f(6*k+2), f(6*k+3), f(6*k+4), and f(6*k+5) result in the same value as f(6*k). Because there is no way to write 1 as the sum of 2's and 3's and there is but one way for 7, we can also establish that f(6*k+1) = k. The function f(n) returns the number of times n will occur in the sequence A216522. We can also deduce from this, that the distance where the number f(6*k+6) from where number f(6*k) occurs for the first time in the sequence is k+6. The same is true for f(6*k+6+j) and f(6*k+j) where j is a number from 1 to and including 5.

Define the function g(n) to return the last (zero base) location for which the value n occurs in the sequence A216522, where g(1) is equal to 0 although 1 does not occur in the sequence. We define this as the sum of all values of f(i) with i ranging from 0 to and including n minus 1. When g(6*k+6) is expressed with g(6*k), we get: 

g(6*k+6) = f(0) + .. + f(6*k) + f(6*k+1) + .. + f(6*k+6) - 1
g(6*k)   = f(0) + .. + f(6*k)                            - 1
g(6*k+6) = g(6*k)             + f(6*k+1) + .. + f(6(k+1))
g(6*k+6) = g(6*k)             + k + k+1 + k+1 + k+1 + k+1 + (k+1)+1
g(6*k+6) = g(6*k) + 6*k + 6
Similarly we can show that for i from 1 to and including 5 that g(6*k+i+6) is equal to g(6*k+i)+6*k+i+6. These equations can be combined to g(n+6) = g(n)+n+6.

We can verify that ceiling(sqrt(12*n + 9)) - 3 returns the correct values for low values, but that still does not proof it is correct for all values. If we equate this expression (without the rounding up with y and replace the n with x, we can derive the following equations: 

    y            = sqrt(12*x + 9) - 3
    y + 3        = sqrt(12*x + 9)
   (y + 3)^2     = 12*x + 9
   (y + 3)^2 - 9 = 12*x
    y^2 + 6*y    = 12*x
    x = y*(y + 6)/12
If we take into account the ceiling function, than for each whole number filled in y this expresses where on or before the value of x occur all values of y in the sequence. For the first six numbers that occur in the A216522 sequence, we get: 
 0:   0
 2:   1 + 4/12
 3:   2 + 3/12
 4:   3 + 4/12
 5:   4 + 7/12
 6:   6
From this we can observe that they do match with the position of the last occurence of the value in the sequence. Now, when we substract value for x for n+6 from n, we get: 
   (n+6)*((n+6) + 6)/12 - (n)*((n) + 6)/12
   (n+6)*( n+6  + 6)    - (n)*( n  + 6)/12
   (n+6)*((n+12)        - (n+6)*((n)/12
   (n+6)*((n+12 - n)/12
   (n+6)*(12)/12
   n+6
From this we can conclude that g(n) = floor(n*(n+6))/12). Which completes the proof.

Tuesday, May 26, 2026

31.2° Celsius

The temperature at Twente Airport has gone up to 31.2° Celsius, which breaks the previous record of 29.7°C on this date in 2005. This also marks the first tropical day of this year.

Link

Monday, May 25, 2026

On Being Elsewhere

In the afternoon, I visited the exhibition On Being Elsewhere at Concordia. It is a group exhibition with works by Tessa Langeveld, Geertje Brandenburg, and Rossella Nisio. In the description it says something about video installations, but I did not see any moving images and/or videos being displayed. Maybe those were switched off.

28.1° Celsius

The temperature at Twente Airport has gone up to 28.1° Celsius, which breaks the previous record of 27.0°C on this date in 1992.

One little plant

This afternoon, I saw a little plant in one of the two boxes I put all the seeds we harvested last year from our magnolia. Some months ago, I found a lot of fungus on one of the boxes. Some weeks ago, I already empted the contents of one box, because it was totally soaked with water. I did not see any seeds. I had totally no hope for the other box. So, I was a little surprise to see one little plant appear. I am not even sure it is indeed a magnolia plant.

Saturday, May 23, 2026

KunstenLandschap 2026

I biked around the route of KunstenLandschap 2026. At 11:06, I arrived at Rijksmuseum Twenthe where I bought the entrance band and got a map of the route and additional information of the artist and performances. There I first viewed the exhibition Abstracties van het Landschap with collages of landscape photographs by Ger Dekkers. I found the following works noteworthy: Next, I saw: A few months ago, it was still unclear whether the festival could go ahead due to the loss of subsidies. Fortunately, enough private individuals donated, but it is clearly 'smaller' than in previous years, with fewer large installations.

Sequences are different

When I wanted to calculate the area of a hexagon placed on a triangle grid, I decide to split it in two halfs and use inner-product method of calculating the area for the lengths a, b, and c and calculate the points starting at the origin. I was a bit surprised that the are was equal to ab+bc+ac triangles. I verified it to be correct. I also modified the program convexhexagon.cpp to calculate the first 1000 values (instead of 100 as before) and see how it matches with the A216522 sequence and I discovered that they are different. The first value where the sequence is one smaller than the most compact arrangement is 121. Some of the following are: 163, 211, 235, 265, 292, 325, 355, 391, 424, 463, and 499. Up to 5000 there are 160 differences of one smaller. So, now there is no need anymore to prove that they are the same. But it still surprising that they are so close. It is also noteworthy that the value for the most compact arrangement for 122 is one lower than the one for 121. After some more searching in The On-line Encyclopedia of Integer Sequences, I found sequence A135711: 'Minimal perimeter of a polyhex with n cells'. It calculates the the perimeter along the outside of the hexagons. It states that the formula for this sequence is: a(n) = 2*ceiling(sqrt(12*n-3)). That means it is monotone increasing sequence, which the sequence for the most compact arrangement with the 'convex' requirement is not. It seems that the sequences A216522 and A135711 are related.

Friday, May 22, 2026

Exhibitions

At the end of the afternoon, I visited two exhibitions. The first was the exhibition Friendly Secrecy System curated by Janne Schimmel at Sickhouse. The list of works on display (with videos) see this post on instagram. I am not really into console games. I found the following two works noteworthy:

The second was the exibition Everything In-between with works by second year students from the AKI at Creative Broedplaats Enschede. Not all the works on display had information with them. I saw the following works:

I hope that more information will be added. I will try to visit the exhibition again on a later date.

Thursday, May 21, 2026

Compact shapes of hexagons (Part 2)

Below a visualization of the compact shapes of hexagons with the minimum perimeter with each number of hexagons. I made it to see if there are some obvious patterns, but I could not see one. A problem is that for some sizes there are multiple shapes with the minimum perimeter for that number of hexagons. I still find the connection with the A216522 sequence very mysterious.

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The Belly of the Beast

In the afternoon, I saw the exhibition The Belly of the Beast with photographs by Anca Axinte at photo gallery Objektief. In many of the photograph the model holds an object that resembles a heart. The artist wrote about this, translated from Dutch: 'A figure consuming her own stone heart.' The object was hanging from the ceiling in the middle of the room. The text with the exhibition starts with (translated from Dutch): 'There is a primal animalistic force within us, built into our instinct, our ego, and our will to survive. We call it the beast. It acts not out of cruelty, but out of fear, and fear is the absence of love. To live, the beast must consume what it lacks.' A very different exhibition from the previous. Quite interesting.

Wednesday, May 20, 2026

Compact shapes of hexagons

I was wondering if there was a more compact shape to arrange 64 hexagons than the one I have been using for place the pieces from The China Labyrinth. I searched The On-line Encyclopedia of Integer Sequences® with some keywords hoping to find the sequence for the number of compact (convex) shapes for a given number of hexagons. I did not find a sequence related to this. I wrote the program convexhexagon.cpp to calculate all the shapes up to 99 hexagons. With the numbers, I did find the sequence A116513: 'Number of distinct hexagons of n points chosen from triangular lattice A_2 with sides parallel to the principal axes of that lattice. Degenerate sides (of length 1) are permitted.' This sounds like it is the same, I was looking for. I also searched the sequence of the most compact shapes, which are the shapes with the shortest perimeter. I found the sequence A216522: 'Integers of the form 2*x + 3*y with nonnegative x and y, with repetitions,' that matches the perimeter when measured around the centers of the hexagon for first 99 numbers. Although it is quite likely that the two sequences are the same (beyond the part they are the same), I have no idea how to prove that they are equal.

Christian Freeling

I read that Christian Freeling was found died yesterday at the age of 79. According to a news report in Dutch he was found dead at the bottom of the stairs by a neighbour, who already suspected something might be wrong because his raccoon dog Snowy was left outside during the night. There are reasons to believe that he already died on Monday. Yesterday, I still visited his website with respect to The China Labyrinth. I met in 1981, when I started studying at the university during the game evenings of Fanaat. He introduced me to some of his abstract board games with complete information. We also had some philosphical discussions and was impressed by his intelligence and his ideas.

Links

Tuesday, May 19, 2026

The China Labyrinth: I Ching

This evening, I did some experimentation with the visualization of compact solutions of the The China Labyrinth to connect it with the I Ching. First I decided to adapt the shape of the pieces suitable for a physical puzzle. I also decided to place the hexagrams on them. The upper three lines reference to the top three sides from left-to-right and the lower to the bottom ones also from left-to-right. The I also decided to have the names in Chinese characters as an alternative to be selected. (Note that the selection will become effective for the next solution to be realized.) I am not really happy with the result.

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Monday, May 18, 2026

Seven Mystic Questions

I was watching a Dune: Part One first reaction video on YouTube, when I realized that the phrase "Who are you?" that is heard when Paul and Jessica enter the room where the reverend mother Gaius Helen Mohiam is sitting is not found in the book and that it might be taken from the Aql (from the Arabic عَقْل) as found in the appendix "Terminology of the Imperium" where it is defined as: "the test of reason. Originally, the 'Seven Mystic Questions' beginning: 'Who is it that thinks?'" I have always wondered what the other six question could be. I used Google and its AI gave the following list:
  1. Who is it that thinks?
  2. Who is it that loves?
  3. What is the beginning of the end?
  4. Where is the path to truth?
  5. How is the soul weighed?
  6. When does the drop become the ocean?
  7. Why does the universe dream?
It also explained that the first question serves as a gateway into the consciousness and inner-awareness traditions of the Zensunni Wanderers and the Bene Gesserit. It explained that it invites the thinker to look past their day-to-day ego, memories, and personality to identify the true, core source of their consciousness. It mirrors real-world mystical practices, like the Atma Vichara (self-inquiry) found in Advaita Vedanta. I found the sixth question quite interesting when you realize that the story of Dune plays on a dessert planet that has not seen an ocean for a very long time. I asked Google's AI for the origin of the questions and it gave an answer with another set of questions.
  1. Who is it that thinks? (The canon prompt: identifying the observer)
  2. From where do these thoughts arise? (Traces the origin of consciousness).
  3. What is the shape of the mind? (Realizing that thought has no physical form).
  4. Who experiences the senses? (Separating biological input from the soul).
  5. What remains when the thought is gone? (Finding the stillness of pure awareness)
  6. Is the observer separate from the observed? (Dissolving the barrier between self and universe).
  7. Who am I? (The ultimate dissolution of ego).
Although this list is more of the line of self-inquiry, I judge it as less mystic compared to the first list.

Sunday, May 17, 2026

Negative space

This afternoon, I visited the exhibition Negative space at Het Robson (Stichting Vierkant), the first solo exhibition of Kashyap Krishna. Although, I was not able to attend the artist talk he gave later this afternoon, as announced, and hear his story behind the works on display, I was quite impressed by the exhibition also through the titles given to the works. The works are given on negative space page on the website of the artist. I saw the works as they were numbered at the exhibition:
  1. Don't Carry the World Upon your Shoulders
  2. Broken
  3. Dissection of the Objectified Female Human
  4. N01 is trapped
  5. Genocide
  6. Life is loneliness
  7. can he dance?
  8. The Exhibitionist & The Voyeur
  9. Stranded (36×)
  10. what's right (in me) what's left (of me)
  11. U R N01
  12. suicide
  13. ಸಮಾಧಿ | Samādhi (a state of meditative consciousness)
  14. ಅಂತ್ಯಕ್ರಿಯೆ | Antyakriye (funeral)
  15. ಸಂಸಾರ | Saṃsāra (ongoing cycle of life, death, and rebirth)
  16. Markers
  17. Alone
  18. ನಮಸ್ಕಾರ | Namaskāra (a customary Hindu manner of respectfully greeting)
I do not know if the artist attached some particular meaning to this order, but it would not surprise me.

Saturday, May 16, 2026

Link

Friday, May 15, 2026

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