Current project(s) - the mathematical side of doodling

[downwardsfromzero]

I've written a couple more pages of notes today covering a logical consideration of what makes specific glyphs permissible. I'll attach this, along with some illustrations, over the next few days.

Jacubey - thanks for the further pointers about the possibilities with various types of matrix identities and operations. It really feels like progress is being, and will continue to be, made. And I'll happily clarify the dividing line ideas with a bunch of diagrams - the text description was a bit of a mind splurge

Gifs of glyphs by Christmas! :lol:

 

[Hyper Turtle]

Heh, i think i found all the rules for my algorithm now...

At least my generator produces only valid glyphs anymore for the range 0..27 (attached). Up to glyph 9, i checked by hand and they are correct for that number.

Next i'll make it produce glyphs up to higher numbers and start identifying repetitions and symmetries in the sequence. If it turns out that the code really is correct and complete now, i'll share my findings and the code (after cleaning it up a lot).

Just wanted to say i finally broke through the wall that hindered my progress


EDIT:

I can also say definitively that there are vastly less than 6^36 glyphs:
- Each square needs to connect in one of just 4 directions to the next. 4^36 aka 2^72. (Actually there's never more than 3 directions because one of the four was already used to connect from the previous square, but i like powers of 2)
- In the corners plus one square adjacent to each corner, there is always only one legal option. Divide by 4^8 aka 2^16 makes 2^56.
- On the edges, there are never more than 2 options (because each corner must connect to the next, there is no crossing, so if viewing it in a directed way, there is no going back on the edge between the two. There is also no going all the way to one of the other edges). Divide by 2^12 makes 2^44.
- When moving from square to square, most of the time there are only 1 or 2 legal options. These depend on what's already there, so i can't quantify the reduction off the top of my head, but it is massive. I currently suspect that there are no paths that involve more than 15 decisions, with only few having 3 options.

 

[Hyper Turtle]

I talked with my father (who is an actual mathematician) about this today.

It turns out, this is a special case of "Hamiltonian circuits".

I haven't read into them, and not going to, until i either am happy with my code or believe that i won't get there on my own.

Interestingly enough, due to all angles being 90° and all distances being the same length, any solution to this problem is also a solution to the travelling salesman problem! And trivially, all the glyph paths have the same length.

My code still isn't quite there, numbers 30 and 33 produce invalid paths. I hope the solution i found yesterday, adapted to the other 2 corners as well, should solve these. I'm convinced there is a generalisation of it that i still fail to recognise.

 

[downwardsfromzero]

Fantastic! Putting the correct name to it really helps - and that's the value of being able to talk to an actual mathematician. I suppose this really is just a question of a pawn's tour of a shrunken chessboard. Hamiltonian paths, or Hamiltonian cycles, sounds more impressive, what with name-checking the inventor of quaternions.

I'm still working on brute-forcing the rules with pencil and paper by considering valid permutations of partial glyphs. What I'd like to know is whether you've established a clear figure for the total number of glyphs within my stipulated rules of ignoring all symmetry operations.

As it stands, I'm up to 149 glyphs now - and I've identified practically all of the same rules as you (despite being still int the process of writing them all down...) And I agree, the number of possible glyphs is going to be way, way smaller than the frankly over-pessimistic first guess. I'd be very surprised to find more than 67 more. 25, maybe.

It's always a teeny bit exciting to find a new glyph, and finding another way of looking at the problem is also interesting. Having developed a notation method, I now want to see if there's a convenient way of eliminating duplications that occur from working through the permutations of different sizes of partial glyph.

The glyphs that sensibly cut in half - meaning they yield two halves, each with a single contiguous path - have given up most of their secrets to me now. The glyphs that similarly cut into one third and two thirds are not far behind. I just need to examine their permutations in the same way as with the half glyphs.

And I suppose I still ought to provide some further diagrams to make this easier to understand. What will probably happen is that I'll type up all of the notes first, in order to force myself to include diagrams there

One somewhat interesting result that I obtained, in terms of assigning (semi)meaningful unique identifiers to each glyph, is what happens when I use the matrix description I outlined in post[#]. Taking the vertical and horizontal matrices separately, they can be multiplied together to yield a single 5×5 matrix, and I get the feeling that each matrix should be unique to the glyph that produced it. Whether this turns out to be of any value remains to be seen.

I have that same feeling as you that there is a generalisation hiding behind all this, just out of my grasp.

 

[Hyper Turtle]

I'm still working on brute-forcing the rules with pencil and paper by considering valid permutations of partial glyphs. What I'd like to know is whether you've established a clear figure for the total number of glyphs within my stipulated rules of ignoring all symmetry operations.

I wish i had made myself actually sit with pen and paper... it's the best!

I don't know yet how many there are, but i have started to code that aspect as well. I'll produce a bunch of glyphs counting from zero, comparing each one to all the previous ones and their 90°/180°/270° rotations, vertical and horizontal mirrors, and the different combinations thereof.

I agree that the number of 'archetypal' glyphs is probably not beyond the hundreds.

First i have to get the generator correct though. Right now i still have 2 invalid glyphs in the first 50, and 20 in the first 150. When i can generate up to 10k with no invalid ones, i'll check some random samples by hand - it's one thing to never make illegal moves, but it's also necessary to never forget about a legal one. When i'm reasonably sure everything is right, i'll brute-force the inventory - every unique glyph, every transformation, and the (lowest) number that produces each of them.

 

[downwardsfromzero]

Up to 10k? I'm gonna need a new box of pencils!!! :S

The great thing about the pencil and paper approach has been that I've already begun to observe how with each of the sensible subdivisions of the basic glyph the number of possibilities is far smaller than one might at first imagine. Right at the start of all this, conversely, I was rather surprised by the realisation that the number of possible glyphs was beyond single digits.

On the other hand, the pencil and paper approach has so far cost me two months of roughly 15 hours a week as a conservative estimate of the average - so, something like 100 hours plus. Actually, I'm quite pleased about that :lol:

 

[Hyper Turtle]

Almost out of bugs now

I got 1900 unique valid glyphs already (without subtracting any rotated or mirrored ones) from the range 0..65536, plus 159 invalid ones (and 63477 repetitions). I have reason to believe that there will be no new glyphs beyond that number. So the remaining uncertainty is how many of the 159 will be previously unseen once the code is correct. I think.

Actually i'm not sure yet that my 1900 are unique - i just know that after closing the path, there was no remainder of the input number. I'm pretty sure that means that it's different from any other glyph with a lower number.

EDIT: i changed my repetition detection and the numbers are lower. In the range 0..65536, i have 1026 right now that aren't repetitions, 74 of which are still invalid.

 

[Hyper Turtle]

I got these suckers (invalid paths) eliminated!

Now my rules aren't too lax anymore. They could still be too strict i guess, gonna check that soon.

Now i have a total of 917 glyphs i believe to be unique, if we don't consider mirrored or rotated ones to be the same. They can each be generated with a (sometimes bunch of) number(s) from the interval 0..32768. The largest number that actually results in a unique one is 29120.

The distribution is clearly not in a continuous range. Different glyphs have different periodicities, depending on the number of decisions there were and the number of choices in each. This (preliminary) table shows that most of the time, there's just 2 options. No glyph had 3 options more than 3 times along the way. Some glyphs involve up to 15 decisions, others only 5. The vast majority takes 9-13.

Period (occurrences): decisions x options
48     (2):  4x2, 1x3
96    (12):  5x2, 1x3
128    (1):  7x2
144    (2):  4x2, 2x3
192   (27):  6x2, 1x3
216    (2):  3x2, 3x3
256   (22):  8x2
288   (10):  5x2, 2x3
384   (31):  7x2, 1x3
432   (11):  4x2, 3x3
512    (9):  9x2
576   (24):  6x2, 2x3
768   (38):  8x2, 1x3
864    (4):  5x2, 3x3
1024  (48):  10x2
1152 (103):  7x2, 2x3
1536  (68):  9x2, 1x3
1728  (34):  6x2, 3x3
2048  (41):  11x2
2304  (47):  8x2, 2x3
3072 (131):  10x2, 1x3
4096  (30):  12x2
4608  (72):  9x2, 2x3
6144  (33):  11x2, 1x3
8192  (34):  13x2
12288 (53):  12x2, 1x3
16384 (20):  14x2
32768  (8):  15x2

Attached are:
- a list of all the unique numbers
- ASCII renderings of all the glyphs i found - one file has HT format, one has DFZ format

 

[Hyper Turtle]

Here's two more files.

The first has only the archetypical glyphs (they are 147), the other lists the archetype and transformation for all 'unique' glyphs from the previous post as well.

I haven't completed verification of my code, it's possible but not very likely that a few are missing.

'mir' means mirrored horizontally, rotX means rotated by X * 90°, mir+rot2 is equivalent to mirroring vertically.

 

[Hyper Turtle]

I have more ideas. I dislike that my generation method doesn't nicely enumerate the glyphs. Neither does it produce them in one continuous range of numbers, nor do they appear in an order that makes much intuitive sense.

Luckily, i can fix that

Here's a different way to represent the glyphs as strings:
- start in NW corner, make one step E. We're now at (x=1,y=0) and looking E.
- Follow the path. For each step, note 'L' for left turn, 'f' for straight ahead, 'r' for right turn.
- Do this until at (0,0). There's always a right turn at the corners, we don't have to note the final 'r'.
This produces 'ffffrffffrffffrrfffLLfffrrfffLLfffr' for glyph 0.

We now have a string representation that is 35 characters and starts with 'f' or 'r'. ASCII upper case has smaller numbers than lower case, so in ascending alphabetical order, 'L' comes before 'f' comes before 'r'.

If we take the set of different strings produced by all valid paths, and we sort them alphabetically, glyph 0 changes to the one that looks like a spiral (archetype 207 but with mir+rot1 applied). Then come all the other glyphs that move straight along the first 3 edges of the matrix (including the current #0). Then all the glyphs that do a right turn sometime after moving straight for the first two edges, etc. Because 'L' before 'f', a one-square deviation from the edge comes before a 2-square deviation, and so on. I expect this to lead to a natural-feeling order, and it certainly is well defined.

Once they are in order, we can filter them if we please. Reduce them to the archetypical ones, or leave them all in.

Finally, we assign them consecutive numbers starting at 0 or 1.

I'll do this soon and attach the resulting file to this post.

 

[downwardsfromzero]

Thanks for your efforts again, HT.

Using pencil and paper alone, I've established that there are 33 archetypal quarter glyphs which out of necessity includes mirror pairings of the 16 asymmetric ones. By examining how these could all pair up in the 'quasi-valid' way of all open paths aligning only with other open paths, but including invalid glyphs with closed loops smaller than the whole glyph, I was able to set an absolute upper value of 2753 glyphs, 4 of which could be immediately eliminated because they have a dividing line right through the middle. Another 'group' comprising a single glyph was also obviously invalid. So I'd got as far as reducing the remaining 2748 to one quarter of that because of symmetries before having a day off yesterday.

Next on my list would be working out how to deal with this putative figure of "about 687" to see how it relates to the 149 glyphs that I've devised by hand so far. This raises the question - how certain are about the 147 that you've discovered as being the total number of archetypal glyphs? Checking for duplicates by hand has become more and more laborious as the number of glyphs has increased. The systematic notation that I've developed has helped a lot but there is still a group that escapes classification so I'll be going over them one more time at least.

I've had a brief look at your files and it's of course immediately apparent that your glyphs numbered 0 and 21920 are archetypally identical and your methods eliminate this in the 'archetypes only' list. I am wondering why you define some of the mirror-symmetric glyphs as having rotational symmetry (rot2) and others not, e.g. 478 vs. 579. They both have a single mirror plane and no rotational axis. It seems to me that the ones with the mirror plane horizontally orientated are (in my view, correctly) being counted exclusively for mirror symmetry whereas those with a vertically oriented mirror plane are (incorrectly) being counted for 'rot2' symmetry. Is this an artifact of your algorithms?

[Several hours later: Now, since working part way through your archetypal list, I've found one glyph (HT39) of yours which I hadn't found yet. This appears to bring the total number of glyphs up to at least 150, assuming I haven't overlooked any duplicates - although I'll have to keep this statement provisional until I've gone through all your archetypes and matched them off against my list. This will have to wait until tomorrow at the earliest, however.]

[EDIT: I see that you've posted more while I've been working on this. Perhaps your ideas with the ascii code can be ported into the colour schemes for each glyph. That is, once a specific value is assigned, that value can easily be used to control other parameters within the image. The context of the data can then also control factors such as orientation, size and movement.]

 

[Hyper Turtle]

... This raises the question - how certain are about the 147 that you've discovered as being the total number of archetypal glyphs? ...

... I am wondering why you define some of the mirror-symmetric glyphs as having rotational symmetry (rot2) and others not, e.g. 478 vs. 579. They both have a single mirror plane and no rotational axis. It seems to me that the ones with the mirror plane horizontally orientated are (in my view, correctly) being counted exclusively for mirror symmetry whereas those with a vertically oriented mirror plane are (incorrectly) being counted for 'rot2' symmetry. Is this an artifact of your algorithms?

Certain that i was off... In the meantime, i have brute forced the entire space (the new ascii representation helped make it easy and pretty quick), and found 1072 total unique valid glyphs. Of those, 149 are archetypes. See attachment.

So i currently believe that you actually found all archetypes by pen and paper and HT39 is in there, or that you missed HT39 but have a duplicate somewhere. Then again, i wouldn't put it past me to make mistakes even in brute forcing

Yeah that was just unfortunate nomenclature on my part. I made two simple functions, one that mirrors across X axis, one that rotates 90°. "mir+rot2" is one operation and means "mirrorX then rotate twice" - it has the same result as "mirrorY", and i changed it to 'mirY' in the new file. When there is more than one self symmetry, they are separated by comma and space.

 

[downwardsfromzero]

That ASCII solution is rather satisfying.

I just brute forced a comparison by eye through the night and found that, of my 149, four were duplicates - and there were four of yours that I had missed! I also found two of mine that weren't in your original 'archetypes only' set so this means I got the number right just by chance!

I might as well mention, the ones that I missed are numbered 39, 60, 95 and 204 in your original scheme. These correspond to 110, 135, 11 and 104 in the 'bruteforce arch with transforms' scheme.

Attached are the two missing glyphs, in your format but minus your sophisticated numbering method (rather, with my historical numbering method and mildly uninformative classification scheme). While I say my classification scheme is uninformative, it at least had enough precision to help me weed out my duplicates when using your list. (Edit: I see that these two glyphs are now numbered 3 and 7 in your brute-forced scheme.) And just what does the periodicity value refer to?

[Edit2: I'm just a teeny bit worried that the combination ffrfLLfrrffrfLLfrrffrfLLfrrffrfLLfr might be illegal in my country of residence and don't want to get into any trouble. Do you think I should leave it out of any future art exhibitions?]

 

[Hyper Turtle]

Haha nice! Glad that i could help you find them

The periodicity is a thing in my original approach.

It means that if you added a glyphs period to its number arbitrarily often, you would get the identical glyph. That's because the period was the product of the numbers of choices at each step - so adding it to the number would result in a different choice only after the glyph was already complete.

However, since my rule based code was never quite correct, neither were the periods. Some of them might have been, others not.

Lol yeah, it might be better to leave that one out. I mean, it's mathematics, and that can't be illegal. In an art piece that has all or many of them, i don't think it's a problem to include it. But you actually might get in trouble for exhibiting a piece which shows that glyph in a prominent way.

 

[downwardsfromzero]

Here's a wee pic of my quarter glyphs. I'll show how they combine to form half glyphs, and then full glyphs, in the coming posts. You'll get to see how the exhaustive, scattergun approach evolves into systematisation and a notation system (unless my methods take me in a different direction :lol: ) The idea is that through systematisation, a route is developed that links specific numerical values to any other form of data, which in turn provides a link between, sound, form, textue and colour, and, indeed, anything else quantifiable - a bit like the NMR music that was produced recently, for example.

Since an important part of this project is wrecking eyeballs along with minds, I've attached another recent effort.

 

[RhythmSpring]

These kinds of designs have always nourished a deep sense of aesthetics in me, kind of like a lower-chakra alignment or something.

I wonder if it is because these kinds of designs resemble how cells are situated amongst each other--no single block completely isolated or out of place--all reasonably adjacent to some kind of snaking pattern which could represent some kind of vein or flow bringing nourishment or energy.

 

[downwardsfromzero]

These kinds of designs have always nourished a deep sense of aesthetics in me, kind of like a lower-chakra alignment or something.

I wonder if it is because these kinds of designs resemble how cells are situated amongst each other--no single block completely isolated or out of place--all reasonably adjacent to some kind of snaking pattern which could represent some kind of vein or flow bringing nourishment or energy.

There was that special shape discovered not so long ago - I forget the name right now - that allows cells to pack in a particularly efficient way. Perhaps you've heard of that too? It wasn't very rectilinear though - but I have been considering ways to move away from the overly rectilinear approach I seem to end up taking with at least my computer-generated artwork. I'll be adding these thoughts to the background ideas mill and we'll see what may emerge.

Rectilinear spirals and similar designs are a practically universal feature of decorative patterns the world over so I shan't be getting too hung up about this either :d At the fundamental level, pixels are almost inescapably rectilinear.

 

[RhythmSpring]

...but I have been considering ways to move away from the overly rectilinear approach I seem to end up taking with at least my computer-generated artwork. I'll be adding these thoughts to the background ideas mill and we'll see what may emerge.

Rectilinear spirals and similar designs are a practically universal feature of decorative patterns the world over so I shan't be getting too hung up about this either :d At the fundamental level, pixels are almost inescapably rectilinear.

Rectilinear is a great word. :d
I don't know what special shape you're talking about, but I feel like mentioning Shipibo artwork, as it gives me even more of a sense of cellular organization, particularly because it's less rectilinear, although it is suggestive of rectilinearity overall.

 

[downwardsfromzero]

Now that's the kind of thing I'm talking about. I have sketched a few of the glyphs with curvier lines like that. The combination of wiggly curves and stepped, straight lines has a great effect. (And I was entirely thinking of the Shipibo while composing the previous post, so I'm especially grateful that you've come up with an example that really nails it!)

 

[donfoolio]

https://art.kunstmatrix.com/fr/system/files/styles/full/private/art/24514/204_allyson_grey_order-with-chaos-letters-2001-allyson-grey-watermarked_-_alexa_spadafora.jpg?itok=B7QJuMGm

I cannot put the second picture in for whatever reason, but is a picture of the language project from Allison grey. Reminds me of her and the I ching of course, beside the shipibo patterns.