Conversation (17)
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Danpiker
@Danpiker@mathstodon.xyz ·Mar 25
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Craig S. Kaplan
@csk@mathstodon.xyz ·Mar 25
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@Danpiker@mathstodon.xyz I've seen still images (and animations?) of these sorts of arrangements of squares for a while now, and I still don't believe they're possible.
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Grégoire Locqueville
@glocq@mathstodon.xyz ·Mar 25
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@csk@mathstodon.xyz
The following toot helped me figure it out partially:
https://mathstodon.xyz/@Danpiker/116289884555776002
You need a tiling of the plane by "anchor" quadrilaterals,
where an anchor quadrilateral is a quadrilateral Q such that there is a continuum of squares whose each side passes through a vertex of Q.
Now I'm not sure yet what characterizes anchor squares...
@Danpiker@mathstodon.xyz
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Danpiker
@Danpiker@mathstodon.xyz ·Mar 26
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@glocq@mathstodon.xyz @csk@mathstodon.xyz these quadrilaterals need to be both equidiagonal and orthodiagonal.
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Bathsheba Grossman
@bathsheba@mathstodon.xyz ·Mar 27
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I feel like this is related to
https://www.nga.gov/artworks/61279-fish-and-scales
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kubi
@bikubi@mastodon.social ·Mar 25
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@Danpiker@mathstodon.xyz i just absentmindedly stared into the center of this one (where my cursor was sitting), and when i stopped the animation by removing the cursor I perceived an unusually intense "reverse motion optical illusion"... even the text box i'm typing this in seems to spiral :)
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myrmepropagandist
@futurebird@sauropods.win ·Mar 26
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@Danpiker@mathstodon.xyz
Did you see the 3 blue 1 brown breakdown about the Esher painting?
This isn't the same but it feels similar.
Can this get as small as we like?
https://www.youtube.com/watch?v=ldxFjLJ3rVY
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Danpiker
@Danpiker@mathstodon.xyz ·Mar 26
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@futurebird@sauropods.win Yes - it's a very nice video. There is indeed a link - complex analysis. One way of generating these square tilings is via discrete harmonic functions.
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alg0w
@alg0w@social.vivaldi.net ·Mar 25
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@Danpiker@mathstodon.xyz Unbelievably smooth transitions for squares.
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robinhouston
@robinhouston@mathstodon.xyz ·Mar 26
@Danpiker@mathstodon.xyz I like these very much! Have you explained how you made them anywhere?
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Danpiker
@Danpiker@mathstodon.xyz ·Mar 26
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@robinhouston@mathstodon.xyz Thanks! I'll try and do a little write-up soon.
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RobJLow
@RobJLow@mathstodon.xyz ·Mar 26
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@Danpiker@mathstodon.xyz @robinhouston@mathstodon.xyz Ooh, I'd definitely like to see that. I've only worked out how to do a baby version of this kind of thing that's 'flat' (so to speak), where the centres of all the squares are fixed on a square grid.
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Owen Maresh
@graveolensa@mathstodon.xyz ·Mar 27
@Danpiker@mathstodon.xyz would you be willing to share how you make these? I'm either interested in manufacturing theta series over the centers of the squares weighted their areas or the vertices already in this image, and seeing how those theta series change as these are transformed.