Editing 2671: Rotation

{{comic
| number    = 2671
| date      = September 12, 2022
| title     = Rotation
| image     = rotation 2x.png
| imagesize = 380x756px
| noexpand  = true
| titletext = It's okay, we can just feed the one-pixel image into an AI upscaler and recover the original image, or at least one that's just as cool.
}}

==Explanation==
This is another one of [[Randall|Randall's]] [[:Category:Tips|Tips]], this time a phone tip. This tip claims that rotating a phone and taking a screenshot too many times will cause an image to disappear into nothingness and warns the user against doing so. The camera and the display both have limited resolutions, so the detail of the original screenshot at the center of the image will be reduced as it approaches the range of a few pixels, hence the original image will be lost before it reaches the sub-pixel range. This is funny because the default resolution of contemporary camera phones can be too large to meet size requirements for e.g. mobile phone {{w|Multimedia Messaging Service}}, web file uploads, or email attachments, so one or two steps of this awkward procedure are sometimes necessary. Other comics such as [[878: Model Rail]] also use recursion as limits.

[[Image:World lines and world sheet.svg|thumb|200px|{{w|String theory}} describes the {{w|worldline}}s of point-like particles as {{w|worldsheet}}s of "closed strings," forming a topological  foam.]]

For a fuller explanation of the concepts involved, including {{w|Planck units}}, often associated with the topological {{w|quantum foam}} of {{w|string theory}}, please see [https://www.youtube.com/watch?v=pUF5esTscZI this CGP Grey video.] For an explanation of topological string theory, see [[2658: Coffee Cup Holes]]. Please see also [[1683: Digital Data]] for an analogous image processing concept.

The title text refers to producing photographically likely higher resolution images from lower resolutions, an active area of current research.[https://openaccess.thecvf.com/content/ICCV2021/papers/Liang_Hierarchical_Conditional_Flow_A_Unified_Framework_for_Image_Super-Resolution_and_ICCV_2021_paper.pdf] Because reducing the resolution of an image is a lossy process, results obtained through such processes will not be able to perfectly recreate the original. Machine learning can be used to calculate how images of known photographic subjects (or e.g. anime-style art, in the case of {{w|waifu2x}}) behave under certain types of noise or reduction in size, so that images ''of those kinds'' can be upscaled in a way that, if not perfectly recreating the original, at least is a faithful representation, but when the image is scaled all the way down to one pixel, everything except a small amount of data about the image's overall color is lost, making reconstructing the original image impossible. Randall disclaims that, because the AI upscaling is based on ingesting a large corpus of human-made art (with subjects that we find 'interesting' or at least meaningful being predominantly represented), the AI will produce an image that is at least as cool as the original image was, and in fact some image generation AIs actually work on a similar principle — for example, "reverse diffusion" AIs are trained by teaching them to reconstruct images from noise, at which they can produce entirely new images by being fed ''actual'' noise.  He could also be making a pun on {{w|color temperature}}, which the upscaler will be able to match to the original image. The "{{tvtropes|EnhanceButton|enhance button}}" for upscaling images is a common trope in movies and television, especially in crime and science fiction stories.

=== Mathematical corner ===

The scale reduction caused by a rotation can be approximated. If ''a'' is the width of the picture and ''b'' its height, the reduction is ''x=a/b'', the aspect ratio of the picture rectangle. As can be seen in the comic, the first rotation leaves two gray areas on each side of the picture that are roughly square. The width of the reduced picture is ''x.a'' = ''a''2/''b''. Each gray area is ''a'' (high) by (''b-x.a'')/2 (wide). This is roughly square, but will not be exactly square unless
: ''b'' = 2''a'' + ''x.a'' and since ''x=a/b'', dividing by ''b'' we obtain 1 = 2''x'' + x².
This is a quadratic equation, whose only positive solution is √2-1 ≈ 0.414

Returning to the general problem: the reduction is geometric, so that after nine rotations, the picture will be reduced by a factor of ''x''⁹. Since this is "smaller than a pixel", the original screen resolution is fewer than (1/''x'')⁹ pixels. It is not stated whether it is the width, height, or area of the original picture that have been reduced to "smaller than a pixel".

25 rotations reduces a lot further and logarithms are useful to compute that. Let ''L'' be log(''a''/''b''), a negative number since ''a''/''b'' is less than 1. If the original screen is 10cm wide, its reduced picture will be ''x''^25 times smaller in width. The comic tells us that the picture is now "smaller than an atom" (typically 10^-10m). If referring to the width, then 25''L'' is less than about -9.0 using base-10 logarithms.

After 101 rotations, the reduction will be ''x''^101, and the picture is now "smaller than the [http://en.wikipedia.org/wiki/Planck_length Planck length]". The log of the Planck length is about -34.8, so 101''L'' is less than -33.8.

Significantly we know that 100 rotations was ''not'' enough, so 100''L'' is greater than -33.8. If we split the difference and say that 100.5''L'' is equal to -33.8, we get an aspect ratio ''a''/''b'' just about 0.461. Multiple popular phone sizes are within the range, including the iPhone X or XS both with an aspect ratio of 1125/2436 ~ 0.4618.

==Transcript==
:[A phone in portrait orientation shows an image of Cueball standing. It is then rotated, showing the image smaller with bars in landscape orientation, then the next phone is in portrait showing the entire screen of the previous rotated sideways, shrinking it every time. An arrow points from each phone to the phone with the next smaller image, until the last one. The labels, at the 9th, 25th, and 101st rotation, show the decreasing size of the original image as it goes through successive rotations.]

:[Labels:]
:9 rotations: original image is smaller than a pixel.

:25 rotations: original image is smaller than an atom.

:101 rotations: original image is smaller than the Planck length, at which the concept of distance may break down.

:[Bottom caption:]
:Phone tip: don't rotate and screenshot an image too many times or it will become lost in the quantum foam of the universe.

{{comic discussion}}
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[[Category:Comics featuring Cueball]]
[[Category:Physics]]