Post AtWbASlXsB7qvilkuG by tncowart@assemblag.es
(DIR) More posts by tncowart@assemblag.es
(DIR) Post #AtWDd8rAZnBEgFXgB6 by futurebird@sauropods.win
2025-04-27T14:11:45Z
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Is there a tessellation of the plane where, given a snapshot of a region of the plane (taken at a certain resolution) it would be possible to know the coordinates of that location up to the information density of the image?(This couldn't be infinite could it?)
(DIR) Post #AtWDyDtr1KvMP3qmJs by wtrmt@mastodon.social
2025-04-27T14:15:30Z
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@futurebird without coffee?!
(DIR) Post #AtWE5OQ8IsYr3kMpH6 by dx@social.ridetrans.it
2025-04-27T14:16:48Z
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@futurebird Not sure, but if you have infinite resolution snapshot (but still fixed area) I have a way to get one coord: imagine a row of squares, and above it another row of squares shifted infinitesimally to the right, the row above it shifted once again, etc. then you’d be able to identify the row, provided you could measure the difference in the shift
(DIR) Post #AtWElMXtfEpnPK0lcm by zenkat@sfba.social
2025-04-27T14:24:23Z
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@futurebird Penrose tilings are infinite and aperiodic. I wonder if you could backcalculate position relative to an arbitrary "center"?
(DIR) Post #AtWEmNVbZLR0gSQKIa by dx@social.ridetrans.it
2025-04-27T14:24:33Z
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@futurebird To have this you would need every patch of a tiling to be unique, right? I think then if you had this you would be guaranteed to be an aperiodic tiling: since no two patches are alike, there’s no way to translate it. Conversely, my gut says that aperiodic tilings have this property at *some level*.Also, if you allow for tesselations where each tile is unique, there’s probably something cool that could be done with fractal edges to encode position
(DIR) Post #AtWF4hj18z9eb1R4vA by caragraph@f.cz
2025-04-27T14:27:47Z
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@futurebird I think @ai6yrm.ai6yr.org does this?
(DIR) Post #AtWFIJ6ZTftKIBwIAi by hosford42@techhub.social
2025-04-27T14:30:22Z
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@futurebird If the snapshot is always at the same resolution (by which I'm assuming you mean the same area), even an aperiodic tessellation will have a limited number of different configurations that can fit into the image. So the answer would be no.
(DIR) Post #AtWFT1WlAfNQxhIaB6 by Scmbradley@mathstodon.xyz
2025-04-27T14:32:18Z
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@futurebird any regular grid is out, right, because many locations would look the same? So you'd be looking at aperiodic tilings. I think even these have some symmetries. But let's say you know which way is up (so rotational symmetries are out) and you know what the tiling looks like at the origin, say, then maybe there's a way?
(DIR) Post #AtWFf0Sswmkt8qCTnE by hosford42@techhub.social
2025-04-27T14:31:08Z
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@futurebird Not completely sure I get what you mean when you say, up to the information density, though.
(DIR) Post #AtWFf1XsvgLgUdXzE0 by futurebird@sauropods.win
2025-04-27T14:34:29Z
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@hosford42 The size of the snap shot in relation to the size of the map where it can give you the location?Density is probably the wrong word. Or unhelpful.
(DIR) Post #AtWG8aWrwLflOIAYcq by hosford42@techhub.social
2025-04-27T14:39:49Z
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@futurebird I think you would have to allow the snapshot to grow arbitrarily large as the area you want to choose a location from grows.
(DIR) Post #AtWGOKYj5H52UgXp1U by rrmutt@mastodon.social
2025-04-27T14:42:39Z
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@futurebird There is a 2-dimensional barcode (actually dots) called Anoto that does this, kind of a tessellation if you connect the dots? https://en.wikipedia.org/wiki/Digital_paper
(DIR) Post #AtWGkP6ir17EyGyHwG by ai6yr@m.ai6yr.org
2025-04-27T14:45:04Z
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@caragraph @futurebird I'm missing some context here...But here's photos of the Earth taken from the ISS geolocated to their coordinates. You can do that with any photo from an airplane, manually.https://eol.jsc.nasa.gov/ExplorePhotos/?illum=day
(DIR) Post #AtWGkQPY0Vlh0wmpv6 by futurebird@sauropods.win
2025-04-27T14:46:39Z
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@ai6yr @caragraph But the earth isn't a tessellated surface.
(DIR) Post #AtWQ8CxOa10RpC2BRw by llewelly@sauropods.win
2025-04-27T16:31:47Z
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@futurebird my intuition is that equilateral triangles and a pair of orthonrrmal basis vectors 60 degrees for the co-ordinates are the key, since equilateral triangles can be broken down into smaller triangles, but I guess you already got that far. I don't remember enough about how to prove stuff to prove this intuition right or wrong, unfortunately.
(DIR) Post #AtWXyXkXDa6V36vBBo by IngaLovinde@embracing.space
2025-04-27T17:59:36Z
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@futurebird imagine rows of 1x1 squares. Row in nth position (for positive n) is shifted relative to the previous row by 1/(n+1); for negative n, by 1/(n-1).Then, from any 3x3 snapshot you'd be able to determine its y coordinate precisely. For a tesselation with one kind of piece only (1x1 square)However, combining this with a similar approach for columns will produce a tesselation with infinitely many kinds of pieces, and I assume that you want it to be finite... :blobcatthinking:
(DIR) Post #AtWbASlXsB7qvilkuG by tncowart@assemblag.es
2025-04-27T18:34:08Z
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@futurebird I think that for this question to “make sense” the tesselation has to have a well defined origin (and unit scale), but I think any origin and unit scale is basically an arbitrary decision. I could definitely be wrong though!