Escher-Like Spiral Tilings | Hacker News Reader

Escher-Like Spiral Tilings

isohedral.ca

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These maps turn out to be awfully useful for interesting visual effects.

The exponential map shows up in Escher's "print gallery" too, which I tried to explain here:

http://roy.red/droste-.html

And you can use it to make various interesting things:

http://roy.red/fractal-droste-images-.html

I like this paper that extends the Mercator projection of the earth down a few inches so you can see what it does down right next to the South Pole:

http://archive.bridgesmathart.org/2013/bridges2013-217.pdf

I find mercator maps for Mandelbrot to be fascinating, I hope we can use them better than we do right now. Maybe to pre compute values before making a render of a Mandelbrot zoom?

I would love to see the "the complex logarithm function" version of the Powers Of Ten film as a high-resolution image. Or print, even.

Same with various other fractals beside Mandelbrot

You can see a logarithmic spiral ("Complex Exponential" in the article) in my video : https://www.youtube.com/watch?v=CMMrEDIFPZY Obama deformed by holomorphic complex functions (conformal map)

But I forgot to make it twist like the pictures in this article. That would make a nice addition for a new video

Ah very cool! In a couple of those, when his chin is stretched wide, Obama really looks just like Dwayne “The Rock” Johnson!

Did you do anything tricky to blend between functions, or is this simple interpolation?

Something pretty similar I made on ShaderToy: a live Mandelbrot mapping using the camera: https://www.shadertoy.com/view/XsKfDc It sadly doesn’t seem to work on all browsers, so you can swap the iChannel0 texture for one of the built-in videos if the webcam doesn’t show up.

Yes simple linear interpolation eased with a cosine ! Sometimes I make a param of the function vary though

The article mentions a lot of interesting properties of the complex exponential map. I find it especially neat that horizontal and vertical translation map to uniform scaling and rotation. A mapping that shuffles transformations around like that can be used in creative ways.

I'd like to add another interesting property, that naturally stems from the above: the mapping transforms regularly repeating shapes into self-similar shapes. Any geometry that repeats along the x axis (with a given period P) will be mapped to a geometry that is self-similar under scaling by the same value P. You can create mappings in more dimensions that have the same properties, for example in 3 dimensions: https://www.osar.fr/notes/logspherical/

Aha! I knew such a thing (a spherical log transform) had to be possible, but my SDF skills were not up to snuff. Very happy to see it done!

Very good blog article thanks for sharing. It's not really the 3D equivalent, though? Only adapted from 2D

The section titled "Log-polar Mapping in 3D" is, as you say, simply adapting the same map to 3D. The section "Log-spherical Mapping" is about the proper 3D equivalent.

However, while maps in 2D space can be neatly represented with complex numbers, there is no equivalent for 3D space, so things can't be as neat. But the geometric properties are there: using the log-spherical map, translation along one axis maps to uniform scaling (in all 3 axes).

Jos Leys' generated some similar looking images, as briefly mentioned in the article [0]. He uses a different mathematical approach, Doyle spirals [1,2]. Malin Christersson added a Möbius transformation creating even more curious animations [3].

[0] http://www.josleys.com/show_gallery.php?galid=265

[1] http://www.josleys.com/articles/HexCirclePackings.pdf

[2] https://mathematica.stackexchange.com/questions/146377/gener...

[3] http://www.malinc.se/m/MobiusDoyle.php

This would make a great visualization for a bassy music video. Link variables defining the polygon shape with amplitudes for bass frequency ranges and the color to the vocal tone.

the demoscene wants its tunnels back

Why downvote this? It’s (at least could be) a clever callback to the demoscene.

[I didn't downvote it.] It breaks some of the unofficial rules of HN:

No jokes, no kittens, no oneliners.

I recommend to write a longer version of the comment that adds some info. Something like:

fake> This is/was very popular in the demoscene [n years ago]. My favorites examples are [youtube link] and [youtube link]. In particular, there is a post from [famous demoescener] that explain how he/she used this effect [link] in [optional youtube link]

Stationary satellite, "stationary" hurricane;

https://www.star.nesdis.noaa.gov/GOES/floater.php?stormid=AL...

Look at the default GeoColor, then let it load Channel 14 or 16.

The online tool (http://isohedral.ca/other/Spirals/) is really fun to play with, and can be used to create wonderful coloring pages for kids!

Beautiful! This is what the internet is for.

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