
Escher-Like Spiral Tilings - archagon
http://isohedral.ca/escher-like-spiral-tilings/
======
roywiggins
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](http://roy.red/droste-.html)

And you can use it to make various interesting things:

[http://roy.red/fractal-droste-images-.html](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](http://archive.bridgesmathart.org/2013/bridges2013-217.pdf)

~~~
ttoinou
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?

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ttoinou
You can see a logarithmic spiral ("Complex Exponential" in the article) in my
video :
[https://www.youtube.com/watch?v=CMMrEDIFPZY](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

~~~
dahart
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](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.

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

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pierrec
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/](https://www.osar.fr/notes/logspherical/)

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

~~~
pierrec
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).

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jhncls
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](http://www.josleys.com/show_gallery.php?galid=265)

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

[2]
[https://mathematica.stackexchange.com/questions/146377/gener...](https://mathematica.stackexchange.com/questions/146377/generating-
doyle-spiral-painting)

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

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rovyko
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.

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posterboy
the demoscene wants its tunnels back

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

~~~
gus_massa
[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] _

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fuzzfactor
Stationary satellite, "stationary" hurricane;

[https://www.star.nesdis.noaa.gov/GOES/floater.php?stormid=AL...](https://www.star.nesdis.noaa.gov/GOES/floater.php?stormid=AL052019)

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

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

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chrisweekly
Beautiful! This is what the internet is for.

