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March 24, 2010

Modeling Surface Diagrams

Posted by Simon Willerton

Note This is reposted from the temporary site so that comments can continue here.

I’m currently at MSRI at a knot homology meeting. There are lots of people here with pictures of surfaces, some of these even being categorical, so I thought I would return to the subject of computer manipulation of these. In particular I thought I would make use of our (hopefully) brief sojourn at WordPress and take the opportunity to embed some videos (but then I was told how I could do it at our usual place).

In my paper A diagrammatic approach to Hopf monads I had lots and lots of surface diagrams – these are the one-dimension higher version of string diagrams which can be useful for notating morphisms in monoidal 2-categories or 3-categories. So I had to draw two hundred or so pictures like this one.

A surface diagram

[I hope to write something more about Hopf monads at the café at some point very soon.]

I drew all of these pictures in xfig – a standard 2-dimensional drawing package in which you have to do all of the 3-d stuff by eye and can’t do any fancy shading or anything. It would be really nice if I could actually create these as 3-d objects and then just export some projection of them, possibly suitably ray traced. I had an attempt at creating some categorical surfaces using blender, which is a splendid piece of open source software which you can use to make movies like Shrek or Monsters Inc. Here are two of my efforts.

The first represents the associator natural transformation in a monoidal category (see the Hopf monads paper linked above for the details).

You need the Flash Player installed.

The second is the so-called swallowtail relation (see Figure 25 on page 40 of HDA4).

You need the Flash Player installed.

As well making movies with such models you can also embed the 3d model in to pdf files and, provided you’re using acrobat(!), your readers can rotate the surface for themselves. Here’s an example from meshlab.

Whilst this is all well and good, there are at least two things I would like to be able to do that I currently can’t.

  • Draw on the surfaces. I want to be able to draw what are essentially morphisms, like in the Hopf monad picture above, and also to be able to label them.
  • Integrate with mathematics software such as sage or maple, or possibly even with some scripting language such a python. Then I wouldn’t have to input the surfaces with a mouse incredibly slowly in blender, but would be able to somehow code up the surfaces.

One thing to note is that these “surfaces” can be singular, as in the associator example above, so are not so easy to handle in standard packages with tend to expect non-singular things.

So, has anyone else tried using 3d software for modeling surface diagrams?

Posted at March 24, 2010 5:01 PM UTC

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48 Comments & 1 Trackback

Re: Modeling Surface Diagrams

You have always been able to embed YouTube videos. You just have to do it right.

Posted by: Jacques Distler on March 24, 2010 5:55 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Thanks; I’ve got it sorted now.

Posted by: Simon Willerton on March 25, 2010 7:54 AM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Not quite sure why but I found the static associator surface in the paper easier to comprehend than the video although I had to intuit the hidden surface in the static image. How do I get the manipulable version of the video?
maybe if I can stop it in the position of the static I’ll see it better.

Posted by: jim stasheff on March 25, 2010 2:35 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Well I guess one reason you can’t understand the video so well is that I was just showing off what fancy things I could do and not trying to use it to explain the category theory! You make a good point. The surface diagram has a well-defined up, a well-defined back and a well-defined left, so by spinning it round randomly in space I’m not helping you understand that.

The blender file is available here.

That only helps you if you’ve got blender installed of course! I was trying to export it to some more standard format that can be viewed in a browser such as x3d, but I’m having problems making the resulting exported surface be two-sided – each part of the surface looks transparent when viewed from the ‘wrong’ side. I’ve tried various things but can’t seem to figure it out. If anyone has any success then let me know.

Posted by: Simon Willerton on March 29, 2010 6:45 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

I don’t know this is helpful or not, but one tool that you might use is MATLAB. One of the (in this case) advantages of MATLAB is that it has a relatively flexible visualisation framework which assumes that inputs are actually matrices defining a surface mesh (which it’ll interpolate in a reasonable non-problem specific way) which you generate separately (ie, to do parametric plotting you create the discrete mesh of samples yourself then get it displayed). This is an advantage because you can explicitly control what gets done in singular areas rather than relying on a parametric plotting programs built-in heuristics. The downside is that you’ll either have to come up with programmatic ways of generating the “sheets” of your surface or else import the mesh from some point and click tool. The visualisation part can do everything you want (lighting, captioning, animation, etc) except “automatically drawing on the surfaces”, but since the input is being provided as a mesh if you can figure out a some way of picking those mesh vertices through which you want your curves to go through, or which “mesh face” you want symbols on, you can draw them as separate objects so that they “appear” to be drawn on the surface.

A MATLAB licence is however probably very expensive just for drawing some diagrams. There’s GNU Octave which is an attempt at a FOSS implementation (using GNUPLOT as the visualisation backend) but I don’t know how much of the drawing facilities they’ve actually implemented. (I hold no brief for MATLAB: I think it’s horribly overused outside it’s “optimal domain” but this problem seems to be one where it’s a reasonably good fit.)

Posted by: bane on March 24, 2010 5:58 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Thanks for the suggestion. I’m not sure I want to learn MATLAB for this though. Learning python and sage are currently high on my list, and I think there’s probably a lot of good 3d plotting tools in sage. I’m starting to switch over to sage as I’m getting more and more fed up with the bloated front-end of maple (which I’ve used for years). I was also irritated by the way maple didn’t have much in the way of 3d plot export formats, but that seems to have improved in maple 13 and I think there’s some sort of ray tracing there as well.

One significant plus point for sage is that it’s open source and free. It’s probably worth mentioning that there are supposed to be good free clones of MATLAB like octave and scilab, but I don’t know whether they have the graphical capabilities you mentioned.

Posted by: Simon Willerton on March 25, 2010 8:05 AM | Permalink | Reply to this

Re: Modeling Surface Diagrams

You seem to be more proficient with blender than I am, so I hesitate to say this, but it looks like maybe retopo might be the sort of thing you’re looking for re: drawing on the surfaces?

I’m a little surprised at your second question; while I haven’t actually tried myself, I think I would find it easier to draw some complicated surface (like the ones above — which look awesome by the way!) directly in blender’s 3D editing mode, than to try to figure out an equation whose graph would be that surface. This is the case in 2D, certainly. For instance, when I write calculus exams with problems like “tell me things about this function given only a picture of its graph,” it’s generally much easier for me to draw the graphs with bezier curves in Inkscape than it would be to try to figure out some equation whose graph looks like I want it to and graph it using some mathematical software.

Posted by: Mike Shulman on March 24, 2010 6:03 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Posted by: Mike Shulman on March 24, 2010 6:04 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Thanks for the retopo and shrinkwrap suggestions. I haven’t managed to get them to work satisfactorily, but will have another go. I thought that UV mapping might also be another option. That’s one of the problems with blender — there’s so much functionality that it’s often not clear what the easiest or best way to do something is. You need experience, I guess.

My point about coding was not so much about having equations for surfaces and curves, but more about having something like macros. If I’ve got to do dozens and dozens of surfaces for some calculation in a paper, then I’ve often got to do the same thing over and over, and then at some point I realise that I should have done something a bit different, so I’ve got to go through all of them and change them by hand. That’s the kind of thing I had in mind.

Posted by: Simon Willerton on March 25, 2010 7:50 AM | Permalink | Reply to this

Re: Modeling Surface Diagrams

I think blender has the ability to “share” meshes between objects appearing at different places and even in different files, so that if you want to change something you can just change it in the “source” place and it will change automatically everywhere. Might that be more like what you’re looking for? It’s also scriptable in Python, I think. Just throwing out ideas; I’ve never actually tried any of this myself, but I’ve been reading the user manual.

Posted by: Mike Shulman on March 25, 2010 3:58 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Ah. I’d never thought of reading the manual, well, at least not from cover to cover: the pdf version seems to be 1,400 pages long.

Anyway, do you mean using linked objects? That seems pretty useful thanks. Where did you read about doing this with python? I wish I had the time to really learn this properly. I guess it’s something that you do over long period of time.

Posted by: Simon Willerton on March 29, 2010 6:58 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

do you mean using linked objects?

Yes, I think that’s what I was thinking of.

Where did you read about doing this with python?

I didn’t mean anything specific, just the general fact that “blender is scriptable in python,” so perhaps one could write such a script to function as the sort of “macro” you were describing. I know zero about what such scripting is capable of.

Posted by: Mike Shulman on March 30, 2010 5:36 AM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Here’s a monoidal functor drawn with the shrinkwrap modifier:

a monoidal functor

And here’s the .blend file. You can see that it looks kind of ugly at the junction in the back, but that’s hidden in this view I chose. Putting a box around the junction would also hide this. Perhaps a better approach would be to use an image texture.

Posted by: Mike Shulman on April 2, 2010 11:54 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Why does the swallowtail cobordism look sort of crinkly, as if it had been left out on the dashboard of your car on a hot day?

It almost seems as if the wrinkles move as the surface rotates, too!

I love the swallowtail cobordism so these things matter to me. My student Mike Stay is writing about compact monoidal bicategories, where the swallowtail plays a key role.

Posted by: John Baez on March 24, 2010 6:06 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

It looks crinkly because it was one of the first things that I did and I didn’t do it very well. It still took me several days though! There are many, many ways that you could make such a thing – you can form a bent sheet and then “sculpt” out the scar, or you can form the horizontal cross-sections and sort of lay a sheet on top of them. I can’t remember which method I used, but I would certainly do it faster and better now.

I don’t think the crinkles move – I think it’s just the reflections and transparency that’s having an effect. I would certainly reduce all that fanciness now.

Of course I was thinking about monoidal bicategories as well: I want such pictures for the paper on two-traces that I’m writing. Maybe we should talk about the stuff you and Mike are doing…

Posted by: Simon Willerton on March 25, 2010 7:43 AM | Permalink | Reply to this

Re: Modeling Surface Diagrams

The master is Kenny Baker. Somehow the learning curve on these things still remains steep. It would be nice to embed such foams in 3 and 4 dimensions, project and rotate at will.

Posted by: Scott Carter on March 24, 2010 6:08 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Very nice! Does he use the free version or the Pro version?

I know that there’s a script for taking blender models through scratch to tikz/pgf; that might be useful for people preparing papers with 3d models in them.

Posted by: Mike Stay on March 25, 2010 1:47 AM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Do you have a link for the script Mike?

Posted by: Simon Willerton on March 29, 2010 7:00 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Very impressive. I don’t know how you guys do it. By the way, my personal vote for most amazing static (i.e. not a 3d movie) swallowtail image I’ve seen so far is I think Scott Carter’s diagram on page 3 from the Sphere Eversion write-up (apparently done in Illustrator with nothing more than a mouse??!!). Would like to put a screen capture here of relevant image but having some technical issues with my laptop.

Posted by: Bruce Bartlett on March 25, 2010 12:21 AM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Thanks Bruce. I haven’t been watching this page in a while. I will put individual pdfs of all of the movie moves up on my page. I hope to complete this next week. I’ll let the cafe know when it is done.

I am reediting the text of the SE and getting it ready for a publisher.

Posted by: Scott Carter on March 26, 2010 7:17 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams-meshlab

The meshlab link gave me only the surrounding text - no graphic.

Posted by: jim stasheff on March 25, 2010 2:26 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams-meshlab

You need acrobat version 7, 8 or 9, I think.

Posted by: Simon Willerton on March 29, 2010 6:59 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

What is the formal status of 3D surface diagrams? I was talking to Robin Cockett about this today, and he said there are two ways to make 2D string diagrams precise: you can interpret them as actually being curves in a plane, in which case you have to do all sorts of work in geometry (this is what Joyal and Street did), or you can interpret them as a convenient way of communicating a deduction in a term calculus (this is what he did with Blute, Seely, and Trimble). But in the 3D case, the geometry becomes much harder (so that it makes more sense to view the information as flowing from the algebra to the geometry, rather than vice versa), but an appropriate term calculus doesn’t seem to exist yet either. Is that true? Do surface diagram pictures actually have a generally defined formal meaning, or do we just take it on faith that every reader could translate any particular one we write down into a pasting diagram as necessary?

Posted by: Mike Shulman on March 26, 2010 12:16 AM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Mike Shulman said:

What is the formal status of 3D surface diagrams?

That’s a very good question. I’m kinda hoping Todd will chip in at this point. Is there anything in Trimble-McIntyre on this type of question? I use it as a notation/calculus that I can translate to other notation such as the linear symbolic language of Fausk, Hu and May or Brugières and Virelizier. Is it not true that it’s just Poincaré dual to pasting diagrams, so is translatable to and fro?

Todd: you mentioned a while back that you were thinking of putting some of your surface diagram stuff on your private space at the nlab. Is that still a possibility?

Posted by: Simon Willerton on March 29, 2010 7:14 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Is it not true that it’s just Poincaré dual to pasting diagrams, so is translatable to and fro?

Well, AFAIK pasting diagrams aren’t actually a geometric thing either, but also just a convenient shorthand for the underlying algebra. At the moment, I’m not sure why we tend to regard them as better-justified. I suppose there is the theory of computads which gives some formalization to them, although it is still not completely “geometric.”

There is certainly a coherence question to be addressed before using either surface diagrams or pasting diagrams. For instance, when we draw pasting diagrams in a bicategory, there is a theorem we are implicitly invoking saying that the result of the pasting is uniquely determined once we choose a bracketing of the source and target.

Posted by: Mike Shulman on March 30, 2010 5:40 AM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Simon (and others) – I’ve been away from the math blogosphere for about four or five weeks, so all this is only now coming to my attention.

Anyway, the purpose of that unpublished paper was indeed to formalize a scope of 3D diagrams, as stratified subsets of 3D cubes, so that suitable deformation classes of diagrams would correspond to 3-cells in Gray categories freely generated from Gray computads. This is a somewhat circumscribed goal; my real interest was to describe “surface diagrams” in all dimensions and in a way which would lead to a satisfactory theory of higher-dimensional duals. Continuing in this extremely ambitious way, a more far-off goal would be to understand or rewrite general stratified Morse theory from the point of view of higher-dimensional algebra.

However, none of this is particularly easy, as Mike surmises. Perhaps I was going about it all wrong, but one technical question I was grappling with is getting the right “category” (topological, PL, smooth, or what) in which to work – there is plenty of opportunity for geometric pathology to come in uninvited, unless one is careful. At the point where I left it, I was working with a “tame topology” point of view as advocated by Grothendieck in his Esquisse d’un Programme, as formalized particularly by o-minimal structures used by model theorists, but also with an eye to other possible formalizations of tame topology, as for example these “X-structures” of Shiota and others.

It’s been a long time since I’ve thought about this seriously (and so I may not be able to respond speedily to technical questions), but as is being discussed in emails, yes, I am very interested in doing “lab work” on this, especially as there is interest out there.

Posted by: Todd Trimble on March 31, 2010 4:42 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

I should add that what I actually do in my Hopf monad paper is to use the diagrams to represent the composition of various 2-morphisms in CAT (with its monoidal structure), and apply various rewrite rules, such as a cup and cap representing the unit and counit of an adjunction can cancel out to give a straight line. I never make any claims about the allowable manipulations being the same as arbitrary topological manipulations, but clearly this is the hope.

Posted by: Simon Willerton on March 29, 2010 7:31 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Mike wrote:

What is the formal status of 3D surface diagrams?

Simon wrote:

That’s a very good question. I’m kinda hoping Todd will chip in at this point.

Todd has a paper which is supposed to justify the use of 3d surface diagrams. I accepted it subject to small revisions back when I was an editor of Advances in Mathematics. Unfortunately he never made those revisions so the paper remains unpublished. To add to the problem, Todd lost some work in a hard disk crash, and I cleared out the directory where I kept papers submitted to Adv. Math. So I don’t know if this paper still exists.

I have a paper with Laurel Langford where with massive help from Carter, Rieger and Saito we showed that unoriented 2d surfaces in 4d are 2-morphisms in the ‘free semistrict braided monoidal 2-category on one unframed self-dual object’. The same kind of result would be easier to prove in 3d: unoriented 2d surfaces in 3d should be 2-morphisms in the ‘free semistrict monoidal 2-category on one unframed self-dual object’. But as far as I know, nobody has sat down and proved this.

This would be a big step in the right direction, but one really wants a more refined version that allows certain 2d cell complexes, not just surfaces.

So, we really need some topologists and category theorists to get cracking here.

Posted by: John Baez on March 29, 2010 7:50 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

So I don’t know if this paper still exists.

That’s a real shame!! Is there anyone else who might have kept a copy of it? How about the referee?

Posted by: Mike Shulman on March 30, 2010 5:41 AM | Permalink | Reply to this

Re: Modeling Surface Diagrams

If you are talking about The Geometry of Gray-categories, I have a copy of it, dated August 5, 1999. It even has editorial comments in the margins – John’s, I guess. I’m not quite sure why it’s in my hands.

Posted by: David Corfield on March 30, 2010 9:07 AM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Would the parties involved object to the paper being scanned and put on the nLab? Or such a scan with editorial comments removed?

Posted by: David Roberts on April 1, 2010 1:03 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

David, please write me at my home email address if you have it (if you don’t, I can also be reached at topological period musings at gmail period com). I’ll explain what’s going on with that.

Posted by: Todd Trimble on April 1, 2010 1:34 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

I am not sure what else needs to be done. There will be minor issues with non-orientable surfaces that are mapped into 3-space. These can have branch points or they can be immersed —like the standard Klein bottle or Boy’s surface . If you look at surfaces generically mapped into 3-space, that looks like the free symmetric monoidal category on one object generator. I would guess that if the object generator is self-dual, then cups and caps are oriented arcs, and saddles should preserve orientations.

Surfaces with branch circles are trickier to handle. Even in spin-foam TQFT as developed by Khovanov and Vaz the discussion does not include branched curves in the underlying TQFT. I am having trouble describing what I mean — if objects are points and morphisms are trivalent graphs, then 2-morphisms are branched surfaces. But there is some mighty tricky stuff that has to be dealt with in the Morse theory.

Posted by: Scott Carter on April 1, 2010 1:16 AM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Scott wrote:

I am not sure what else needs to be done.

The simplest project is proving that unframed unoriented 2-tangles in 3d space form the free semistrict monoidal 2-category (= Gray-category) with duals on one unframed unoriented object. For this, all the category theorists need from the topologists is a list of movie moves — like the one you guys came up with in the 4d case.

But I hope you see that the 3d case is vastly simpler, since now the frames of the movie depict tangles that can be drawn in the plane without any crossings. The braiding does not play any role in this game!

I’d always been embarrassed to ask you to work on the 3d case, since it’s less exciting than the 4d case, yet still beyond my technical powers. I mean, I’m sure I could guess the movie moves, but proving I’ve got all of them requires some singularity theory that I don’t understand.

I’d been planning to tackle this issue when I wrote my ‘definitive multi-volume introduction to nn-categories’. I’d been planning to pester you when the time came around. But it looks now like I’ll never write that.

So, this is an open question that’s waiting for someone to tackle it. It would actually be quite manageable in the right hands, and it would begin to provide the foundations for the 3d diagrammatic calculations Simon Willerton is doing.

I guess anyone wanting to work this should bear in mind the existence of Todd’s paper The geometry of Gray-categories, and his continued desire to ponder these issues.

There will be minor issues with non-orientable surfaces that are mapped into 3-space.

This issue is a more fancy than anything I’m discussing here. A 2-tangle in 3 dimensions is (for starters) a 2d surface embedded in a 3d cube.

It would be very nice to get purely 2-categorical descriptions of surfaces generically immersed in 3d space, but the embedded case is the fundamental starting point for all such more sophisticated variants.

Surfaces with branch circles are trickier to handle.

Whoa, whoa! You’re shooting way ahead of us now.

Though admittedly, Simon is using some pretty funky 2d CW complexes that aren’t even manifolds, here.

Posted by: John Baez on April 1, 2010 4:08 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

While you’re in this ball park, what was the conclusion about Lurie’s claims on the free braided and symmetric monoidal categories on one dualizable generator?

Posted by: David Corfield on April 1, 2010 5:27 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

I am not sure what you mean by semi-strict in the above answer. It worries me a bit. When I played with associativity, things got a little funny — surfaces tended to have waves in them when objects were re-associated.

Smoothly embedded surfaces in 3-space can get funny. For example, they can bound cubes with knotted holes, and knotted worm holes can be untied by drilling unknotted holes.

I will think about this some more …

Posted by: Scott Carter on April 1, 2010 11:20 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Looking forward to the graphics you describe

Posted by: jim stasheff on April 2, 2010 12:51 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

I put them on a teeny-tiny flash drive this morning which I forgot to put into my pocket. Meanwhile, look at these slides . The first draft of the Sphere eversion is on this page. But be forewarned, the file of the book is enormous!

None of this addresses Simon’s question on presenting things in space and drawing upon them.

Posted by: Scott Carter on April 2, 2010 3:45 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Fantastic set of slides! You should send some of them to Bridges or other Math in Art

Posted by: jim stasheff on April 3, 2010 2:56 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Simon, do you know how the Sports Engineering Research Group at Sheffield made the picture for the Bending it like Bernoulli poster from AMS “Mathematical Moments” series? I just love the way those streamlines around the soccer ball are plotted like “strings”; I’d love to see some electric and magnetic fields plotted this way.

Posted by: Bruce Bartlett on March 27, 2010 8:58 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

They seem to use the computational fluid dynamics software called Fluent. When I was using the Sheffield’s ‘iceberg’ high performance computing cluster to do lots of magnitude calculations (which basically involved row reducing matrices with 200,000 x 200,000 entries), most of the jobs running on the machine were by people using Fluent.

Posted by: Simon Willerton on March 29, 2010 7:37 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

row reducing matrices with 200,000 x 200,000 entries

That’s pretty much what we get our undergraduates to do for homework. By hand. Said homework then has to be marked. It has sometimes been argued that this is not the best use of anyone’s time.

Posted by: Tom Leinster on March 29, 2010 10:27 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

I just discovered that by using “Shadeless” materials and “Edge rendering” it is possible to get a much more “cartoony” look.

cartoony extranatural transformation

I think I like this version better, actually; these pictures are supposed to be “schematic” rather than “realistic” anyway. Here is the .blend file.

Here is a quick tutorial on cartoon-like rendering in blender, from which I learned these and other tricks.

Posted by: Mike Shulman on April 5, 2010 2:40 AM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Thanks Mike. My blendering is definitely improving and I’ve got the shrinkwrap modifier working. There’s a rather trivial example in my forthcoming paper…

Posted by: Simon Willerton on April 9, 2010 8:26 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

I think this is the nicest version of this diagram. Moreover, it’s well-adapted to being printed on a black and white printer, which is what I guess most of us use day to day.

Posted by: Tom Leinster on April 10, 2010 8:07 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Just add one minor blender comment. I found that using a graphics tablet made it so much easier to work with blender than when using a mouse.

I’d also add that I found it lots of fun!

Posted by: Andrew Stacey on April 10, 2010 8:10 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

People still use mice? I haven’t used a mouse since I got a thinkpad. I like the touchpoint so much better for everything, including blender. (-:

I bet a trackball would also be better than a mouse, though probably not as good as a touchpoint or a tablet.

Posted by: Mike Shulman on April 10, 2010 9:33 PM | Permalink | Reply to this

Re: Modeling Surface Diagrams

Actually, here’s a Top Tip for anyone with small kids (or thinking of getting one or two): the slightly older style Apple mice are fantastic for small kids to get the hang of. I mean the ones where the whole mouse was the button and to click it you just press down with your whole hand.

Posted by: Andrew Stacey on April 10, 2010 9:37 PM | Permalink | Reply to this
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