Rather surprisingly, I note that the sequence 1,4,26,412,11072,... does not appear to be in OEIS. I think that's the first time I've ever looked up a sequence in it that turned out not to be there! Perhaps you should send it in. (After first computing another five terms, I expect, by a cunning dodge I haven't thought of and/or the application of more raw computation than I can conveniently bring to bear or both.)
I'm wondering if I'm missing some other obvious symmetry which would bring the numbers down a bit and make the series different; I too only got to n=6, but was running in Python for minutes and could probably manage to run in C++ on eight cores for a week if anyone were interested in the answer.
Initially I was counting designs of this sort on a 3x3x3 cube, which by choice of axis always begin either 0125 or 014, but there are fewer symmetries of the square.
412 is quite a nice sort of number - comparable to the syllable-count of English, I think, which could lead to some spectacularly cryptic inscriptions slightly too hard to decipher by hand to be fun. The thing I liked about these squiggles is that any two of them do look reasonably different, in the way that two similar Chinese characters look different; they're encoding eight and a half bits while not looking (to me - I have no frustrated nine-year-old Chinese kid to ask) too complicated for humans to work with.
Someone suggested carving them on stone, which I suspect would work rather nicely with the right CAD package and a round-tipped router bit; but I don't need more reasons to buy a CNC router, I just need somewhere to put it. I assume the monumental mason along the road could tell me where to get an A4 slate slab.
There are 45 squiggles which end in the top left-hand corner and so could be written cursively, but I imagine writing cursively would look incredibly confusing; and these will look sillier punctuated than even Chinese in one of those fonts where 002C 'COMMA' is the same size as 9F4E 'TO PRESENT RESPECTFULLY USING BOTH HANDS' or 齉 'SNUFFLE'.
I suppose shorthand is the way to go to look for an extra nineteen characters when writing English; I've naively merged N-gram frequency tables up to N=4, but the recommended polygrams overlap a lot: THE, AND, ING, HIS are clearly useful, DTH is more common than I'd have thought ... at this point I should probably ask hilarityallen about what sort of characters were used by medieval scribes.
The most common five-letter words (as opposed to five-letter sequences) are OTHER WHICH ABOUT WOULD WRITE THESE THING COULD SOUND WATER; single-character nouns would seem too specialist, maybe four conjunctions and twelve prepositions would be a way to go.
If 4x4 is 26 patterns and 5x5 is 412, what is 4x5? When a) the line the line begins by moving in the 4 direction, b) the line begins by moving in the 5 direction?
What would it look like if presented as rectangles 4 wide by 5 high, with the lines starting in the lower right corner with a visible descender, except in those cases where the line happens to end at the top pointing up, where there is a visible ascender instead?
How many patterns in a 4x4 grid if the line need not start in a corner?
5x4 (5 wide, 4 across), beginning by moving in the 5 direction gets 73 patterns; beginning by moving in the 4 direction gets 87. I left the software to do the drawings on the other computer, will try to get them rendered before I flee the country.
Descenders and ascenders sound like an idea; there's also serif potential but I wonder if that might not just look messy.
4x4 starting in arbitrary places gets me 552, but I suspect a lot of those look identical (since the line has no direction to it), and a lot more are rotations or reflections of one another. Basically you can start at 0,0 (52, but they come in pairs depending whether they go to [1,0] or [0,1] first); or at 0,1 (25); or at 1,1 (36)
412 is quite a nice sort of number - comparable to the syllable-count of English, I think
And the number of such glyphs on a 4×4 grid is exactly 26, which suggests a much simpler encoding of English text. Perhaps while the adults are puzzling out your 5×5 syllabic script, the children can be working out a simple substitution cipher based on the 4×4 version :-)
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Initially I was counting designs of this sort on a 3x3x3 cube, which by choice of axis always begin either 0125 or 014, but there are fewer symmetries of the square.
412 is quite a nice sort of number - comparable to the syllable-count of English, I think, which could lead to some spectacularly cryptic inscriptions slightly too hard to decipher by hand to be fun. The thing I liked about these squiggles is that any two of them do look reasonably different, in the way that two similar Chinese characters look different; they're encoding eight and a half bits while not looking (to me - I have no frustrated nine-year-old Chinese kid to ask) too complicated for humans to work with.
Someone suggested carving them on stone, which I suspect would work rather nicely with the right CAD package and a round-tipped router bit; but I don't need more reasons to buy a CNC router, I just need somewhere to put it. I assume the monumental mason along the road could tell me where to get an A4 slate slab.
There are 45 squiggles which end in the top left-hand corner and so could be written cursively, but I imagine writing cursively would look incredibly confusing; and these will look sillier punctuated than even Chinese in one of those fonts where 002C 'COMMA' is the same size as 9F4E 'TO PRESENT RESPECTFULLY USING BOTH HANDS' or 齉 'SNUFFLE'.
I suppose shorthand is the way to go to look for an extra nineteen characters when writing English; I've naively merged N-gram frequency tables up to N=4, but the recommended polygrams overlap a lot: THE, AND, ING, HIS are clearly useful, DTH is more common than I'd have thought ... at this point I should probably ask
The most common five-letter words (as opposed to five-letter sequences) are OTHER WHICH ABOUT WOULD WRITE THESE THING COULD SOUND WATER; single-character nouns would seem too specialist, maybe four conjunctions and twelve prepositions would be a way to go.
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What would it look like if presented as rectangles 4 wide by 5 high, with the lines starting in the lower right corner with a visible descender, except in those cases where the line happens to end at the top pointing up, where there is a visible ascender instead?
How many patterns in a 4x4 grid if the line need not start in a corner?
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Descenders and ascenders sound like an idea; there's also serif potential but I wonder if that might not just look messy.
4x4 starting in arbitrary places gets me 552, but I suspect a lot of those look identical (since the line has no direction to it), and a lot more are rotations or reflections of one another. Basically you can start at 0,0 (52, but they come in pairs depending whether they go to [1,0] or [0,1] first); or at 0,1 (25); or at 1,1 (36)
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And the number of such glyphs on a 4×4 grid is exactly 26, which suggests a much simpler encoding of English text. Perhaps while the adults are puzzling out your 5×5 syllabic script, the children can be working out a simple substitution cipher based on the 4×4 version :-)
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