RT @KangarooPhysics: 1.2.3.4.

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Similar to what @alytile showed for the H/T/P/F metatiles ( these are the 2 fractal tiles for the substitution system in figure 2.11 of the new aperiodic monotile paper (.

Also - it's not that there aren't any symmetric arrangements possible for 17. It's just that they're not as compact as the one on the right.

I should add - this packing for 17 squares is not rigid - 3 of them can slide, and one even has room to wiggle, but it's not enough to allow any of the other squares to move inward, so it doesn't change the size of the containing square.

The arrangement on the left is a more efficient packing of 49 circles into a square than the one on the right.

That little gap is an essential feature of this proven optimal packing. We can say for certain that no more efficient arrangement exists. Also, even for many numbers where a 'nice' symmetrical arrangement is possible, there's a funky looking one which beats it:

I saw replies suggesting there might be a 'nicer' arrangement. While it's true that there isn't yet a proof of optimality for 17 squares, there are packing problems with similarly awkward solutions which are proven optimal. For instance 10 circles in a square (look lower right):

I picked the solution for 17 because that one looked particularly funny to me, but it's not just this number with an awkward looking solution - the arrangements for most numbers are asymmetrical, the neat ones like 16 are the exception.

(I saw a copy my old post was getting picked up again, and thought people might like a higher resolution image of this beauty.) Generated from the data in

The optimal known packing of 17 equal squares into a larger square - i.e. the arrangement which minimises the size of the large square.

RT @KangarooPhysics: Mastodon: Discord: Dan Piker#9142.

Mastodon: Discord: Dan Piker#9142.

Orbiting quadcopters

context:.

I saw those clips of the tangled plug topology went viral again the other day. Lots of people in the comments were convinced it was fake and just editing tricks. It can be hard to follow in videos with hands getting in the way, so I made an animation to hopefully make it clearer.

RT @MartinSchwab9: . and the Lego Version of this fascinating toy ."Flapping Windows"😀.#lego #kinetic toy

(it can also help to put a small weight such as a clip on this outer one to give it more momentum). You can also do something similar with 7 squares (in 2 different ways) and remarkably each still only has 1 degree of freedom. More on that in the future though!.