Public Google+ posts of Arnaud Chéritat

Category: Professional

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14 Sep 2018
Same as before, but with the bifurcation locus mapped on the graph.
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Roice Nelson: Beautiful!

14 Sep 2018
The graph of a function,* rendered so as to look like a snowy mountain.

*: minus the logarithm of the conformal radius of cubic polynomials fixing a point with rotation number the golden mean, as a function of the remaining parameter, if you really want to know...
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03 Sep 2018
Just rendered a graph z=f(x,y) with a fancy style. Can't help seing a volcano.
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31 Oct 2017
One step in the sphere eversion that I propose.
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20 Oct 2017
Upgraded my 4D folding app to all 5 platonic solids.
You can even mix platonic solids provided they have the same faces.
In this case, the folding will occur so that all vertices stay on a 4D 3-sphere (sorry for this limitation but otherwise it seems complicated to compute).
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15 Aug 2017
Try to create your favorite pattern in 3D space for the 600 cell, the 16 cell or the 5-cell, and fold it in 4D with the slider. The program will only show the orthogonal projection (shadow) in 3D space.
Note: you can save your work.
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05 Jul 2017
I converted my torus eversion to an applet.
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09 Apr 2017
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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:

Roice Nelson: I forgot to mention that touch-dragging the 3D view (zooming and rotating around) does work well!

06 Apr 2017
Render of the potential associated to a polynomial in complex numbers, using an unusual representation (the idea is not mine).
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06 Apr 2017
The same procedure, applied to a post-critically finite rational map of non-Lattès type (same rational map as in It has a dense set of bumps but most are too small to see.
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04 Feb 2017
I have received a strange email a few days ago,

From Times Higher Education (THE) inviting me to fill a survey to "take part to the next Times Higher Education World University Rankings" organized in partnership with Elsevier. Then they claim they will give one £ to a charity (UNICEF) for each participant filling the survey, up to a maximum of 10 000.

Have you received this too?

Notwithstanding the fact that the charity thing is ridiculous, I will not fill this form for two reasons.

1. I have been applying the boycott of Elsevier (won't publish,* won't referee, won't do editorial work) since I signed The Cost of Knowledge petition.

2. Rankings are of poor significance and have been misused. There are many people explaining why they should not be used for evaluating departments (or worse, individuals) so I won't repeat their arguments here. This is especially true for pure mathematics where research output cannot be predicted and works on the long term.

Comments welcome.

*one exception: I was co-author for a note in Comptes Rendus, it was written and submitted by one of the author and I did not realize the editor was Elsevier until the paper was ready for production.
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08 Jan 2017
Ester Dalvit just released on YouTube a series of videos about knots in 4D. A realization of high quality and excellent explanations.
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04 Nov 2016
The puzzle now has a transparent shell.
Transparent PLA on a Makerbot 2.
(continuation of
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19 Sep 2016
Another projection of the 120-cell to 3D space, this one has a smaller symmetry group than my first model: the tetrahedron group instead of the dodecahedral group. There are 4 touching gold-coloured pieces in the center.

Pieces: SLS nylon powder bought from Shapeways, then I hand painted them with acrylic and a brush, and varnished them also with a brush.
Stand: FDM with PLA plastic, done in the fablab of my university.
Design: by me with C++ and Blender.

Problems: took a lot of time to paint. Friction between the pieces removes the varnish and paint. Maybe I should have used a primer. The ancient model does not have this problem: it was printed in coloured sandstone and varnished with a spray.
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30 Aug 2016
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Refurio Anachro: Cool!

20 May 2016
Now implements real-time Voronoi cells (Dirichlet domains).
It may be a bit demanding on your graphics card (I did not do optimize the code).
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Vladimir Bulatov: Wow! it is really impressive!

18 May 2016
Updated: more patterns, more controls, and a bit of French touch
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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.

18 May 2016
v7 : Mobius maps are now implemented (this was a request of +Vladimir Bulatov )
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10 May 2016
Navigate in the universal cover of Klein's quartic
I have commented a little bit the shader's code
This code is not optmized.
I plan to include a few other tilings invariant under the same group.
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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.

07 May 2016
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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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04 May 2016
Mirror room
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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!

29 Apr 2016
v5's out
Suggestion: download a map of any planet from the Internet, save it on your computer, then go the the link above, click "filename" and choose the file you saved. The map can be in any of the following formats: cube map, sphere map, equirectangular, Mercator, and a couple more.
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25 Apr 2016
More features added in v3.
Note : you can use it to visualize any map your computer.
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24 Apr 2016
Ready : a first version of my online spherical maps viewer.
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20 Apr 2016
Go to the following link:
Thanks to GPU pixel shader support in browsers via WebGL, you can navigate in real time in the hyperbolic plane tesselated by Escher's Circle Limit IV.
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Arnaud Chéritat: Check the variant :

paul cheritat: super

Arnaud Chéritat: :)

Jean-Marc Schlenker: Magnifique !

19 Feb 2016
Friend's there. Also, warmer colors.
(this is part of a series about a 2D world with a locally flat connection).
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18 Feb 2016
Yet another one. I'll add the friend when I have time.
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13 Feb 2016
I just added a stick-world with two poles:

Basically, you can move the stick figures around in the plane, nothing really bad will happen unless you pass between the two poles.

You may also check a variant with four:
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04 Feb 2016
v2's ready: now Stickman has a friend !

Note: I gave up the pointer lock feature, just drag the mouse to move the characters. Pointer lock works on Firefox, though.
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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!

16 Dec 2015
Made that web page a year ago :
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Roice Nelson: Really well done page!

22 Nov 2015
The sun sets on the vale of a quasifuchsian group.

C++ / Luxrender

For once, this image is not free of use : a variant will be included in a book.
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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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27 Aug 2015
Finished, at last.

I look forward to seeing the public's reaction: this will be first shown at a general public science festival, with many events. In my case, I will be in a big hall with other researchers from different fields, each with a table and their material. The public will wander randomly and be able to do discuss, watch and do some experiments. I have prepared a few 3D puzzles and will include the eversion models.

It took so much time painting them, maybe I will leave the second series white (the ones with wooden stands and plastic sticks).

These models will then be brought to talks I plan to make about the eversion in the coming years. Ultimately, they will be stored at the Institut de Mathématiques de Toulouse, in a permanent display.
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15 Aug 2015
Close-ups of one of the objects, as requested.

I have painted and varnished 6 out of 8 of the spherical objects.
But there are 7 more objects, those with stands.


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09 Aug 2015
The Math Institute of Toulouse accepted to fund my project : they bought the 3D models. This include the set I showed earlier, plus another set I proposed, and the total amounts to approx 1250€, taxes included. I paid for the rest: the paint, wood boards, plastic straws, neodyme magnets, glue and varnish. All the tools are mine. Now I am spending part of my summer carefully painting the models, this is taking forever. I also assembled stands for the second set, this was faster and more pleasant to do.

Just received and unpacked


Work in progress

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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.

12 Jul 2015
I'm looking for 800€ to make this become real. These are steps in a sphere eversion.

Rendered with Blender/Cycles.
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Refurio Anachro: Cool.

Arnaud Chéritat: based on

David Moore: I can't help but think after reading your paper: could this be implemented in manipulatable VR? (Thinking: )

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Laurent Bonetto: So all is left now is the 3D printing? That's exciting!

06 Jul 2015
Here's an ongoing project related to the sphere eversion.
On the pictures, one half is raw, the other has been colored with acrylic paint using a paint brush, and varnished.
Both are equipped with neodyme magnets fixed with epoxy glue.

Desing by me with C++ and Blender
3D-printed by Sculpteo
Sorry for the photo quality: I just took them with my compact camera.
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10 Apr 2015
Shadow of the 120-cell.
This is a bigger version of an object I realized earlier.
Photo by Patrick Dumas.
3D-printed PLA on a Makerbot.
Brush-painted with matte Acrylic,
Sprayed with matte varnish.
Approx 16cm, 700g.

I have to enhance the varnish, because friction between pieces leaves marks of paint.
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23 Feb 2015
A new puzzle protoype I just finished (CC BY SA). I'll try another one with 4 magnets per face instead of one.
Fun fact: rotate each roof by a half turn: colors still match.
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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).

31 Jan 2015
New version: 3d-printed, painted. Stand build out of wood and 2 big nails. Size, stand included : 9cm long, 6cm wide, 6.6cm high.
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27 Jan 2015
A 3D-printed model, painted.
(part of an ongoing project)
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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 :
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!

27 Jan 2015
Image de synthèse.
Je n'explique pas, c'est juste joli.
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27 Jan 2015
Image de synthèse : un modèle da surface de (Werner) Boy, en morceaux simples. J'essayerai d'en faire un puzzle 3D un jour.
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21 Nov 2011
Allez, c'est parti pour mon premier post...

Je me suis récemment offert un objet imprimé en 3D.
Il est inspiré par un objet "classique" en mathématiques et consiste en 45 dodécaèdres plus ou moins applatis qui s'assemblent d'une certaine façon.
J'avais réalisé une représentation 3D de cet objet l'année dernière (première image).

J'ai envoyé le fichier (et une certaine somme d'argent) à la société qui m'a renvoyé l'objet par la poste.
Le voici avant montage (2e image; la coque en papier---c'est moi qui l'ai faite avec une imprimante 2D ;-)---est obligatoire pour que l'assemblage tienne).

Sympa, mais ça reste un peu cher.
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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 :

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Arnaud Chéritat: > Is the scaling hyperbolic?

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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