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Adaptative mesh applied to the Gridcoin logo by zipity

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· @zipity · (edited)
$15.56
Adaptative mesh applied to the Gridcoin logo
Here is a mesh I computed in order to evoke the new Gridcoin logo 
https://steemitimages.com/0x0/http://i.imgur.com/0UPGfqI.png (thanks  @joshoeah).
First, I decomposed the logo into elementary shapes like lines, circles and meeting points.
Then I computed the distance function to the global shape defined as the minimum of all the distances from those elementary shapes.
Then I ran a mesher which adapts to the metrics defined by that function.
Et voilà!

https://upload.wikimedia.org/wikipedia/commons/4/4b/GridcoinMesh.png
By Zipity.grc (Own work) [<a href="https://creativecommons.org/licenses/by-sa/4.0">CC BY-SA 4.0</a>], <a href="https://commons.wikimedia.org/wiki/File%3AGridcoinMesh.png">via Wikimedia Commons</a>

The hardest part (beside writing the mesher) was to tweak half a dozen of coefficients in order to stress the 'active' locations -- the shape itself -- from the backgound by varying triangle density.
The result has T = 7202 triangles, and as you can count there are B = 12 border edges (and 12 border vertices obviously).
Fun fact : this is enough to compute E the total number of edges!
Indeed, as
<center> <a href="https://www.codecogs.com/eqnedit.php?latex=2E-3T=B" target="_blank"><img src="https://latex.codecogs.com/gif.latex?2E-3T=B" title="2E-3T=B" /></a></center>
for any triangular mesh, we have E = 10809 edges. And now, since the Euler characteristic is 1 for a planar mesh
<center> <a href="https://www.codecogs.com/eqnedit.php?latex=V-E+T=1" target="_blank"><img src="https://latex.codecogs.com/gif.latex?V-E+T=1" title="V-E+T=1" /></a></center>
so V = 3608 vertices.

(edit: hints for a proof [here](https://steemit.com/mathematics/@zipity/the-dehn-sommerville-equations))
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@gregan ·
Instead of trying to understand it i'll just gaze in awe.. &#128565;
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@h202 ·
$0.04
As a physicist, I can appreciate this! :D
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@zipity ·
Physics rule! Would you help me run some fluid simulation on this ?
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@h202 ·
Perhaps, but I'm not a fluid dynamicist :P Not sure how much work that would involve. Maybe there are some existing packages out there that can simulate Navier-Stokes on a custom mesh. Not the kind of thing I'd want to code from scratch..!
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@jefpatat ·
I don't understand a bit of it but it sounds scientific ;-)
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@zipity ·
Lol ! Metrics is like the scale on a map. Except it is not written once on the map, it is defined differently for every point on  the map . This kind of mesh is just a series of equilateral triangles, except they are drawn with respect to a special metrics : the metrics (the scale) which zooms according to the distance to the shape of the logo. I hope I could give the general idea :/
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@joshoeah ·
$0.12
Nice work! I have no idea how you achieved this - anything mathematics goes way over my head :) - so this is like some kind of wizardry to me! Is the final result capable of being exported in a  vector format? It might be fun to play around with for interesting artistic applications :)
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@zipity · (edited)
$0.09
Thanks @joshoeah! You can't say your logo has nothing mathematical in it! Lines and circles, that's all there is ;-)
About the format, as a matter of fact I used an SVG version for producing the png.
<a href=http://svgshare.com/i/4uX.svg>Here</a> it is .
By the way, this is @originalworks
Speaking of applications, the inital use of a mesh is to  simulate  fluid dynamics  equations on it. I'm thinking of a gif where the logo moves like a liquid or a fire.
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@grider123 ·
$0.04
Wow fluid dynamics with this image sounds incredible cool! Would love to see this.
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