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Roice Nelson: Hey +Arnaud Chéritat, the rendering works great on my phone too, but I can't adjust the sliders there. Thought I'd mention in case you were motivated to try to make those work on phones.
Arnaud Chéritat: Thanks for the comment, +Roice Nelson. I cannot really test that since my smartphone is rather old. So up to now I have not developed with phones in mind. In your case the slider appears but refuses to slide, or does it slide but without an effect?
Roice Nelson: Hi +Arnaud Chéritat, they show up fine, but when I touch them they highlight as if they are getting selected, and I'm unable to click a new position or drag the slider handles. (btw, the checkboxes do work when clicked.) I wonder if there are some special event handlers for touch that need to be added, maybe something like the "touchmove" Webkit handling described here:
https://css-tricks.com/the-javascript-behind-touch-friendly-sliders/
Roice Nelson: I forgot to mention that touch-dragging the 3D view (zooming and rotating around) does work well!
Refurio Anachro: Cool!
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Vladimir Bulatov: Wow! it is really impressive!
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Roice Nelson: Beautiful :) I like how you made the overlays transparent.
Xah Lee: what's the browser requirement? am on linux, both chrome and firefox just shows a red circle.
Arnaud Chéritat: +Xah Lee , I was able to have it work with FF on Mac, Linux and Windows. With Chrome on Mac and Linux (I have not installed it on Windows). With Safari on Mac, the image works but not the mouse interaction.
Craig Kaplan: Thanks for the comments! Eventually, I'll find some time to play with this code...
Henry Segerman: With a little modification, this would be great as a front page for a website with 24 subpages.
Tim Hutton: +Arnaud Chéritat What are the other tilings invariant under this group?
Arnaud Chéritat: +Tim Hutton There are several groups in the play. Above, I meant the group H of deck transformations of the (universal) cover disk->Klein's quartic* S (*: quartic seen as a compact Riemann surface or as an hyperbolic manifold, not as an algebraic variety). The main group after this one is the symmetry group of S, which has 168 elements (twice this number if we allow for reflections). This symmetry group lifts to a sup-group G of the deck transformation group, and the index [G:H]=168. This said, I have implemented three tilings invariant by G if you ignore the colors: the 24 heptagons, the 56 triangles, and the black and white overlay (invariant also by reflections); the other ones are invariant by smaller groups, between G and H. If you consider colors, I think they are all invariant only under H. I'm not sure this was the questions anyway.
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Craig Kaplan: Very nice! Do you have an explanation of the way it works? I have long wanted to experiment with drawing the Poincaré disc using a fragment shader.
Arnaud Chéritat: Yes Craig (and you can also look at the shader by viewing the source). To compute a pixel's color I use the reflection group of which one the small triangles are fundamental domains. I look at the number of reflections mod 2 used to come back to a fdtal dom, this tells me if the point was in a light or dark triangle. For the black line, I use a distance estimator that consists in keeping tract of the derivative when I reflect the point, and then seeing if I am not too far from a specific line.
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Roice Nelson: Cool!
Btw, is the recent flurry of activity you gearing up for the Illustrating Mathematics workshop? I noticed you listed as a speaker :)
Arnaud Chéritat: No, it is related to other things. (In the workshop I plan to speak of the sphere eversion). Part of my recent applets (Stickman ones for instance) are about a sort of geometry (locally flat connections in 2D) that helped me solve a problem on holomorphic maps. Forgetting about holomorphic maps, this geometry is quite fun. The mirror room is a particular case.
I suppose you are coming to the conference, I'll be happy to meet you.
Roice Nelson: Yes, I'll be there and look forward to meeting you as well!
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Arnaud Chéritat: Check the variant :
http://www.math.univ-toulouse.fr/~cheritat/AppletsDivers/Escher/index-2
paul cheritat: super
Arnaud Chéritat: :)
Jean-Marc Schlenker: Magnifique !
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Roice Nelson: looks cool. For some reason I can't get the dragging to work for me though (on Chrome).
Arnaud Chéritat: Roice: thanks for signalling the bug. It was tested only on Firefox. I'm downloading Chrome to see.
Arnaud Chéritat: Problem solved, now works on FF && Chrome :)
Roice Nelson: Thanks +Arnaud Chéritat, working now!
Sometimes the stickman I'm dragging disappears, but I definitely can get a feel for it now. It's a really neat applet.
Arnaud Chéritat: True, it haddn't happened during my quick test. I added debugging info and it seems Chrome is firing mousemove events with abnormal values of movementX and movementY, like (170, 189). Is that a bug? I could ask the applet to bound the movement values to [-10,10] but somehow it feels wrong.
Arnaud Chéritat: OK clearly a bug in Chrome, got a minimal example. Sorry Roice.
Roice Nelson: no worries, thanks for looking at the Chrome issues!
Roice Nelson: Really well done page!
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Arnaud Chéritat: +Abdelaziz Nait Merzouk : Merci. Oui c'est le bord de l'enveloppe convexe de l'ensemble limite (le coeur convexe), vu de son intérieur. Le groupe contient une translation les 2 plans sont parallèles à cette translation. Leur position est donnée par les points les plus extrêmes de l'ensemble limite (estimés si je me souviens bien en faisant un dessin de l'ensemble limite et en demandant à l'ordinateur de repérer les pixels les plus extrêmes).
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Assembled
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Just received and unpacked
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Work in progress
David Moore: That's fantastic!
Refurio Anachro: That's a nice job you have there +Arnaud Chéritat! Can't wait until you post close ups.
Refurio Anachro: Cool.
Arnaud Chéritat: based on http://arxiv.org/abs/1410.4417
David Moore: I can't help but think after reading your paper: could this be implemented in manipulatable VR? (Thinking: https://www.youtube.com/watch?v=fgo4CMParcw )
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Laurent Bonetto: So all is left now is the 3D printing? That's exciting!
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Arnaud Chéritat: More fun fact: there are two solutions to this puzzle*. Their color schemes are mirror images of each other.
Arnaud Chéritat: There are 4 (now 3) photos here, but Google+ automatically chooses the one appearing in this post without giving me control: choosing "album cover" is ineffective; changing photo order in album is also ineffective. I'll try to arrange this later (maybe merge all 4 photos in a single one).
Laurent Bonetto: You make me curious. What is this ongoing project?
Arnaud Chéritat: It is a new way to perform a sphere eversion. There is a very nice movie about this :
https://www.youtube.com/watch?v=wO61D9x6lNY
I have found another way to do it and I am trying to make a movie too, on an humbler scale. I will also 3D-print a handful of key frames from my animation, to help people (and me) understand what is going on.
The piece above, is one of the important shapes appearing in the process.
Laurent Bonetto: Wow... this makes me dizzy. It's fascinating but I will have to watch that video again a few times before I digest it all!
Julien Chéritat: Et quel est le nom de cette merveille mathématique ?
Arnaud Chéritat: C'est une projection dans l'espace à 3 dimensions du 120-cell, un des 6 polytopes réguliers en 4 dimensions.
Arnaud Chéritat: Il faut voir 120-cell même si on n'y comprend rien, c'est magnifique.
Ilies Zidane: Superbe vidéo ! Un peux mieux que "Dimensions". Mais ça ne vaut pas le "vrai" :)
Arnaud Chéritat: J'ai raconté la génèse de cet objet sur : http://images.math.cnrs.fr/Le-120.html
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Arnaud Chéritat: > Is the scaling hyperbolic?
No.
1. you can tile the hyperbolic 3 space with infinitely* many dodecahedra (put 4 per edge)
2. but you can also tile the 3-sphere with 120 dodecahedra (put 3 per edge)
The picture I made is a projection in 3D of a -Euclidean- 4D polytope whose vertices are exactly the same as the 600 vertices of the tiling of the 3-sphere embedded in R^4, but whose 120 faces are straight instead of being curved (similarly : the dodecahedron in R^3 has its 20 vertices on a 2-sphere but its faces are flat instead of being curved. Those 20 vertices are exactly the vertices of a tiling of the sphere by 12 pentagons.)
*: there is also a hyperbolic 3 manifold called the Poincaré space that can be tiled by finitely many dodecahedra (4 per edge) by starting from the one in the hyperbolic space and identifying polyhedra.
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Arnaud Chéritat: Up to now there's been only one me :)
I'm mostly a researcher in math, but I like programming and computer generated pictures.
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Roice Nelson: Beautiful!