@joshmillard@mastodon.social

Josh "cortex" Millard @joshmillard@mastodon.social · Jul 29, 2026

Replying to @joshmillard@mastodon.social

wait shit maybe it was Pauline in DK

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Josh "cortex" Millard @joshmillard@mastodon.social · Jul 28, 2026

Replying to @AceOfWaggery@mstdn.social

@AceOfWaggery@mstdn.social SOMEbody has never gotten to the "kill screen"

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Josh "cortex" Millard @joshmillard@mastodon.social · Jul 28, 2026

With its gripping narrative of a man's struggles against mythological threats (barrels) and the whims of the gods (additional barrels) before finally ousting intruders (a large monkey) from his home and rejoining his beloved Penelope (Penelope), Donkey Kong is the 20th century's boldest and yet most popularly accessible treatment of Homer's Odyssey

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Josh "cortex" Millard @joshmillard@mastodon.social · Jul 10, 2026

Replying to @sodiboo@gaysex.cloud

@sodiboo@gaysex.cloud @N33R@fops.cloud really getting into environmental impact play

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Josh "cortex" Millard @joshmillard@mastodon.social · Jun 01, 2026

Replying to @redsad@ohai.social

@redsad@ohai.social the rare VTOL nyan cat

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Josh "cortex" Millard @joshmillard@mastodon.social · May 11, 2026

never skip vast and trunkless legs day

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Josh "cortex" Millard @joshmillard@mastodon.social · May 06, 2026

Replying to @ax11@mastodon.social

@ax11 @hydropsyche folks this is a Wendys

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Josh "cortex" Millard @joshmillard@mastodon.social · May 06, 2026

Replying to @doctormo@floss.social

@doctormo @Somarasu not much, what's up with you

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Josh "cortex" Millard @joshmillard@mastodon.social · May 05, 2026

Replying to @Kiloku@burnthis.town

@Kiloku i consider micros macro

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Josh "cortex" Millard @joshmillard@mastodon.social · May 05, 2026

mitosis is just a scam by Big Biology to cell more

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Josh "cortex" Millard @joshmillard@mastodon.social · May 02, 2026

Replying to @joshmillard@mastodon.social

@gknauss@mastodon.social "how'd he die?" got zapped by consumer microwave leakage "no, that's a myth, in typical rate of use--" well it wasn't exactly a typical rate of use

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Josh "cortex" Millard @joshmillard@mastodon.social · May 02, 2026

Replying to @joshmillard@mastodon.social

@gknauss@mastodon.social You know what the internet needs? I single-purpose website that just documents the drive train ratios of every known make and model of microwave.

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Josh "cortex" Millard @joshmillard@mastodon.social · May 02, 2026

Replying to @gknauss@mastodon.social

@gknauss@mastodon.social You know, I never once thought about this and now I'm going to every time.

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Josh "cortex" Millard @joshmillard@mastodon.social · Apr 28, 2026

Replying to @mhoye@cosocial.ca

@mhoye threat tween pageants, threat helicopter mothers

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Josh "cortex" Millard @joshmillard@mastodon.social · Apr 21, 2026

Replying to @jessamyn@glammr.us

@jessamyn@glammr.us you could make a button up variant and it'd be a cardigan catalog

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Josh "cortex" Millard @joshmillard@mastodon.social · Apr 18, 2026

RE: @joshmillard@mastodon.social

nerd-sniped, Pizza Hut stained glass lamp edition

mastodon.social

Josh "cortex" Millard: "@jmock@social.lol And because my brain is going d…" - Mastodon

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Josh "cortex" Millard @joshmillard@mastodon.social · Apr 16, 2026

you tellin me there’s an entire faire dedicated to dressing up like a strung-out cartoon chihuahua

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Josh "cortex" Millard @joshmillard@mastodon.social · Apr 16, 2026

Replying to @overholt@glammr.us

@overholt@glammr.us @sheepfilms@mastodon.social hell yeah brother

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Josh "cortex" Millard @joshmillard@mastodon.social · Apr 16, 2026

Replying to @sheepfilms@mastodon.social

@sheepfilms@mastodon.social limitless flavor, zero calories

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Josh "cortex" Millard @joshmillard@mastodon.social · Apr 10, 2026

Replying to @clive@saturation.social

@clive@saturation.social feeling bad for the people on the half of the planet that no longer exists, RIP

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 31, 2026

Replying to @aparrish@friend.camp

@aparrish quis cuslopiet ipsos cuslopes

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 12, 2026

Replying to @joshmillard@mastodon.social

I think that's as far as my brain is going with it at this point. My one lingering thought to play with is whether it'd bear fruit to treat L and J pieces as distinct after all in laying out specimens -- I might be prematurely simplifying things treating them as indistinct and keeping myself from noticing some other interesting patterns.

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 12, 2026

Replying to @joshmillard@mastodon.social

And once we're talking concatenation it starts to feel interesting and tingly to remember that this all started with an observation about the Fibonacci sequence, which is all ABOUT concatenating smaller values. People capable of crunchy analysis can tackle the actual formal combinatorics here, I know better than to try and put myself and others through that. But it's neat!

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 12, 2026

Replying to @joshmillard@mastodon.social

The implication of the uniqueness of this one specimen is that *every other solution* for n >=3 is simply the concatenation of two or more previous solutions for smaller values of n. Which makes sense: other than this one configuration that constantly overlaps its layers of I shapes, every other solution ends in a nice flat wall on both sides. They can be mashed together arbitrarily to create a longer solution made up of distinct unrelated parts.

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 12, 2026

Replying to @joshmillard@mastodon.social

And that just keeps going. For every n >= 3, there is a new specimen that follows this pattern of 2L + (n-2)I, just sticking in a new I piece each time. You could think of it as flip-flopping a bit for each n, or as extending two related odd-n and even-n configurations in a more straightforward "add to I's" way.

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 12, 2026

Replying to @joshmillard@mastodon.social

The other interesting thing is where totally new specimens show up -- that is, solutions that aren't a combination of two smaller solutions. The answer seems to be "literally only in the I+L row", and only and exactly one for each successive value of n >= 3. I've highlighted all brand-new specimens in green. n=1 introduces the O as a 2x2 solution n=2 has two I's and two L's respectively as new 2x4 solutions n=3 has a new L+L+I 2x6 solution n=4 has a new L+L+I+I n=5 has a new L+L+I+I+I...

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 12, 2026

Replying to @joshmillard@mastodon.social

Coming back to this to look at some patterns and make a couple more notes. First off: for odd n, there's never going to be a solution that uses ONLY L or ONLY I shapes. Which makes sense: for a 2x2n rectangle for odd n, you'd need to arrange n-1 I's or L's into most of the space (and the only way we know how to do that is 2x4 blocks) and then fit one extra one into...a 2x2 space. Oops! So for odd n, those specimens just will never exist.

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 12, 2026

Replying to @AmyZenunim@unstable.systems

@AmyZenunim@unstable.systems @pronoiac@mefi.social without a profit motive, the shittiest dudes you ever met wouldn’t do anything. Shitty dude culture would collapse.

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 10, 2026

Replying to @joshmillard@mastodon.social

(as always, I manage to goof when doing this stuff on the fly; there are in fact [at least] eleven specimens for n = 4, missed a I + L variant in my haste!)

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 10, 2026

Replying to @joshmillard@mastodon.social

Once we hit n = 4, we're in full swing: there's finally specimens for all seven combinations of pieces, and ten total unique solutions controlling for symmetry. I have to stop and pay attention to work for a bit, but one of the things I'm excited to fiddle with more is looking at where genuinely new non-decomposable patterns show up (like the sixth one down on the left in this image) where it's not just a concatenation of previously attested smaller specimens.

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 10, 2026

(If we don't control for symmetry on the n=3 specimens, we get the nine total that we see in Mark's set, the results of flipping some of these horizontally and or vertically.)

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 10, 2026

Replying to @joshmillard@mastodon.social

Does this mean we don't ever have to care about the mirroring of L pieces, tho? Nope! We just had to yet. That changes immediately with... n = 3 For which we have four solutions, controlling for symmetry (several more that I've ignored if you don't, my set will keep diverging more and more from Mark's because of this). The bottom-most of which is a case where we have BOTH versions of L/J in play -- there are times when that WILL matter! There's also no single-piece solutions except 3 O's.

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 10, 2026

Replying to @joshmillard@mastodon.social

The first couple of values of n are very trivial: n = 1 (so 2x2n = 2x2) the only solution is a single O. n = 2 (so 2x4) has four solutions: a pair of O's, a pair of I's, and two different orientations of a pair of L's. But me being me, I want to avoid repetitions across mirror/rotational symmetry, so I'm going to decide that second pair-of-L's isn't interestingly different from the first one. I'm only going to count that once. So really we have three n=2 solutions, controlling for symmetry.

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 10, 2026

Replying to @joshmillard@mastodon.social

Now, L/J can be thought of as the same piece mirrored across one axis; we can't actually just treat it as the same at all times (more on that) but for organizational simplicity let's call it L. O, I, L are our three piece types. From there we can say that every possible tiling of a 2x2n rectangle is going to be made up of one of these seven subsets of O, I, and L: {O}, {I}, {L}, {O,I}, {O,L}, {I,L}, {O,I,L} So that might be as good a way as any to organize our specimens.

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 10, 2026

Replying to @joshmillard@mastodon.social

Boy this is a nice potent nerdsnipe. So, scratching around with it, a couple observations so far:

This is done with Tetris pieces (aka square tetrominoes), but it's not done with all the Tetris pieces -- some are never gonna fit in any finite 2xn tiling (let alone the 2x2n here): T and S/Z pieces are out of play. That leaves us with only O, I, and L/J pieces to work with.

(We're also omitting the tall/skinny rotations of I and L/J because we only have two blocks of height.)

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 10, 2026

RE: @mjd@mathstodon.xyz

that's the stuff baby

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 09, 2026

TIRED: femme fatale
WIRED: homina homina hominous

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Josh "cortex" Millard @joshmillard@mastodon.social · Mar 07, 2026

[finally picking up where I left off training steer to be assassins] guess who’s back on their bulls hit

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Josh "cortex" Millard @joshmillard@mastodon.social · Jan 22, 2026

Replying to @middleclasstool@phire.place

@middleclasstool@phire.place you are not counting wrong, I was tired last night when I drew that and didn't double check!

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Josh "cortex" Millard @joshmillard@mastodon.social · Jan 22, 2026

Replying to @joshmillard@mastodon.social

And then show that for each of those four examples for an nxn square transformation, you can do a similar n+4xn+4 transformation with the exact same frame plus a couple more 2x4 rectangles popped on at the edges of the frame. So this is a family of four frame transformations for n, n+1, n+2, and n+3, and each of them also trivially works to add for any multiple of 4 larger value of n. Neat!

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Josh "cortex" Millard @joshmillard@mastodon.social · Jan 22, 2026

Replying to @joshmillard@mastodon.social

And, in fact, we can think of this as an infinite family of transformations: it's easy enough to rearrange and extend those last two examples so that we have four specimens for 6x6, 7x7, 8x8, and 9x9 squares each doing a somewhat similar framing move...

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Josh "cortex" Millard @joshmillard@mastodon.social · Jan 22, 2026

Replying to @joshmillard@mastodon.social

More specifically, that transformation works if we treat the source square as 6x6 if we want the new rectangles to have nice integer values for their side lengths. But it's all just proportional. We can do a similar idea with more rectangles that have a different proportion: for this example consider the square 7x7, and we see a collection of new 1x2 and 2x4 rectangles creating a DIFFERENT frame.

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Josh "cortex" Millard @joshmillard@mastodon.social · Jan 22, 2026

Replying to @joshmillard@mastodon.social

What about this little move? Take an nxn square, and add some 2x4 and 1x2 rectangles to it to get a new n+2xn+2 square! I'm using an abstract square here rather than any other solution; we can do this with *any* known full solution of n, we're just looking at what gets added: a frame around two sides, which add five new rectangles.

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Josh "cortex" Millard @joshmillard@mastodon.social · Jan 22, 2026

Replying to @joshmillard@mastodon.social

I think this by itself is NOT enough to find all specimens -- there are solutions that don't proceed solely from subdividing existing rectangles -- but it feels like a fun space to explore. There's other kinds of transformations we can document as well; one is the basis of all this, transforming a square into a set of rectangles, and all our solutions for n are new members of that family. But there's other things to look at to, like...

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Josh "cortex" Millard @joshmillard@mastodon.social · Jan 22, 2026

Replying to @joshmillard@mastodon.social

So maybe part of what we're doing is building up a vocabulary of valid transformations from a starting rectangle to distinct sets of multiple sub-rectangles. And then we could find some of the solutions for any given n by looking at various kinds of transformations that add some number m rectangles to a smaller solution of size n-m. i.e. if we know an n=2 solution, and we know there's a +3 transformation, that gives us an n=5 specimen. (Which is sort of where we started!)

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Josh "cortex" Millard @joshmillard@mastodon.social · Jan 22, 2026

Replying to @joshmillard@mastodon.social

So here's one thing on my mind: these solutions can be thought of in part as a transition from a single rectangle to a set of sub-rectangles. Here's a couple that have shown up in the specimens so far:

a four-way split of a single 2x4 rectangle into four 1x2 rectangles, like we did to get the original n=5 solution from the n=2 solution
a five-way split of a 4x8 into a 3x6, a 2x4, and three 1x2, like in the bottom of one of the n=6 solutions

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Josh "cortex" Millard @joshmillard@mastodon.social · Jan 22, 2026

Replying to @joshmillard@mastodon.social

So how many ways ARE there to solve n=6 with distinct sets of rectangles? Are these two it, or are there more? How about 7? 8? And how can we prove that we've exhausted all the possibilities? I cannot emphasize how much I do not know the answer to any of those questions. We are done with the "saying things I've already satisfied myself about" part of the discussion, now it's just mucking around and seeing what we notice. So!

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Josh "cortex" Millard @joshmillard@mastodon.social · Jan 22, 2026

Replying to @joshmillard@mastodon.social

Reiterating: the actual numeric values don't matter -- I am descrbing those two n=6 solutions above as filling a 6x6 and an 8x8 square respectively, but that's just for notational convenience: I'd rather describe the smallest rectangle as 1x2 in both cases than as 1/6x1/3 in one case and 1/8x1/4 in the other. It's all just proportions and we're breaking toward readability here.

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Josh "cortex" Millard @joshmillard@mastodon.social · Jan 22, 2026

Replying to @joshmillard@mastodon.social

This, for example, is what I have in mind: two solutions for n=6 which use different sets of rectangles. (Someone already noted btw that I accidentally showed this right example upthread as an n=7 specimen. Whoops! Should have double-checked my late-night diagramming before posting.) One uses two 1x2s and four 2x4s to make a 6x6 square; the other uses three 1x2s, one 2x4, one 3x6, and one 4x8 to make an 8x8 square. Qualitatively different species!

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Josh "cortex" Millard @joshmillard@mastodon.social · Jan 22, 2026

Replying to @joshmillard@mastodon.social

Similarly, using the exact same set of rectangles in a different layout doesn't feel like it is necessarily that interesting. Is there anything very distinct about these two setups, where it's a pair of 1x2s and a quartet of 2x4s and they're just rearranged a bit? Which, honestly, I'm not quite ready to declare that there's zero distinction here. But for now let's just say: what I'm most interested in is solutions with different *sets* of rectangles entirely.

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