The n-Category Café

Re: Graphical Category Theory Demonstrations

John, that’s really useful. Yours and the other comments are just the kind of discussion I hoped to provoke. The categories you suggest — finite-dimensional vector spaces over the rationals or a finite field — sound feasible, and as I explain below, it shouldn’t be a problem to tie in software for solving systems of linear equations.

As Aaron Denney says below, my current demonstrations aren’t applets. It might be useful if I explain the difference, and also the various programming methods available for writing such demonstrations. Each has its pros and cons.

Firstly, how do my demonstrations work? When you press a button on my input form, your Web browser sends the name of the button to my Web server. It also sends whichever inputs I ask you to type, such as the two sets wanted by the product and coproduct demonstrations.

When my Web server gets this data, it runs a little program — a “script” — that I wrote in a language called PHP. The script finds the Prolog program I wrote to do my calculations, and runs it. More precisely, it runs one of the predicates in it, such as the one that makes and explains a product cone.

That predicate does the following. It makes a data structure holding the product cone. It adds other categorical stuff lying around the environs, such as a second product and the isomorphism between it and the first. It converts the product cone to input for GraphViz, then runs GraphViz to generate a categorical diagram image file. And finally, it spits out a load of HTML into a text file. That HTML is my results page, containing: my explanatory text; listings of the objects and arrows; and a link to the diagram. The Prolog program then halts, and the script tells my Web server to send, to your browser, a version of my demonstration page similar to the one whose button you pressed, but with a link near the top pointing to the results page.

Having Web pages be generated by programs in this “one-shot” fashion is what Aaron Denney refers to when he talks about “going through a CGI call”: “CGI” stands for Common Gateway Interface, and is a standard protocol for interfacing programs such as my PHP script to a Web server. Its advantage is that everything happens on one Web server, which is completely under my control. So I could indeed call a program for solving linear-equation systems, or anything else, without needing to implement anything new on the user’s computer.

(In fact, I have done something vaguely similar. Colleagues and I once built a Web-based simulation of the UK economy, and for several years thereafter, put a copy on the BBC’s servers each Budget Day, as a game called “Be Your Own Chancellor”. This ran: a microeconomic model; a macroeconomic model; a graph plotter; code for turning Web pages into printable output; scripts for passing data between these; and a package for solving the Treasury equations that defined the macroeconomic model. Probably very few of the thousands of people who ran our simulation knew how many different programs were co-operating inside it.)

The disadvantage of doing things this way is that interaction is coarse-grained. Fill in a form; submit it; get back a page of output. You can’t draw diagrams in your browser and pick up arrows with your mouse and probe objects with them, and so on. Which I suspect is what Urs has in mind when he says below “I’d imagine the more ‘hands on’ such an application is, the more it would serve the intended purpose”.

Such coarse-grained-ness is why programmers got excited when Sun brought out Java in 1995, then demonstrated Java “applets”: Java programs running inside a Web browser, doing graphics that you could interact with, such as these nifty examples by Ken Perlin. By the way, part of the word “applet“‘s meaning is that the Java program must be coded according to certain conventions, so not every Java program is an applet.

Applet interaction is much finer-grained than what my demonstrations currently have; but a disadvantage would be that I’d have to rewrite most of my Prolog program, especially the categorical calculations and the example generator, in Java. Or would I? An applet can be programmed to request data from a Web server. So I could continue to do the calculations on my Web server, but implement the graphics in the applet in the browser.

But this raises another question. Even graphics as simple as drawing an ellipse with nice-looking letters inside it is hard to program using only the primitives Java provides for drawing. My demonstrations use Graphviz suite: I’d have to find or write a substitute for that, and as Aaron Denny says in another comment, that would not be much fun.

Moreover, what level of rendering do I want? Suppose I want to make a really memorable demonstration by having a knight “probe” an object with a map depicted as a lance, or a physicist “probe” an object with maps depicted as particles flung out by a cyclotron. I’d have to find an artist to draw and animate all these: sadly, depicting reality in pictures is much harder than doing so in words.

But perhaps not for much longer. I’m going to devote the next few paragraphs to the Java-based Alice, “an innovative 3D programming environment that makes it easy to create an animation for telling a story, playing an interactive game, or a video to share on the web”. Alice may eventually be extremely useful to those building maths animations, so I’ll explain a bit more. Alice is intended for teaching programming to novices. It contains a stock of “objects” — some of which are people, such as an ice-skater. You program these by dragging-and-dropping tiles representing actions (for the skater, these include spinning on the ice) and events such as mouse clicks. Thus you build “worlds” inhabited by animated objects that react to one another and to a user. Alice includes a 3D rendering engine — so if you program the ice-skater to spin, you’ll get a nicely rendered view, including details such as hair, tutu, and bare legs bending.

Alice has a built-in tutorial, which invited me to extend some supplied animations. One animation features a bunny whose nap is disturbed by a mobile phone; you have to make him jump up and down on the phone and squash it. Another has coloured penguins, and you have to make each one play a note when the mouse clicks on it. It only took five minutes or so to do these, by following the tutorial.

It sounds silly to be writing in a category theory blog about ice-skaters in tutus, bunnies jumping on mobile phones, and musical penguins that play scales when you hit them. But the point is that to make programming fun, and to concretely represent object-oriented “objects”, Alice’s creators are tying up with Electronic Arts’s video game The Sims. The next version of Alice, Alice 3.0, will let users build Sim-style animations using The Sims characters. And, I hope, Sims furniture, houses, etc. And, I also hope, simple shapes with which we can depict (categorical) objects and various types of arrow.

Alice runs in Java, so I’m hoping that those of us who don’t have the artistic and computer-game-programming skills to animate our ideas, will be able to do so with Alice, and make some really nice instructional applets. (And perhaps even a category theory visualisation kit, to help professionals express their ideas more efficiently than with LaTeXed diagrams? I recall reading a story, perhaps by Greg Egan, where mathematicians routinely built such animations to accompany published papers.) Anyway, to quote from Alice Press Release - 03/10/06:

EA [Electronic Arts] has agreed to help underwrite the development of Alice 3.0 — a popular, object-oriented, Java-based computer-programming environment created by Carnegie Mellon researchers — and provide essential arts assets from “The Sims™” — the best selling PC video game of all time.

The Sims content will transform the Alice software from a crude, 3D programming tool into a compelling and user-friendly programming environment. … Experts say that when the transformation is complete, the new programming environment will be in position to become the national standard for teaching software programming.

… “Getting the chance to use the characters and animations from The Sims is like teaching at an art school and having Disney give you Mickey Mouse,” said Computer Science Professor Randy Pausch, director of the Alice Project at Carnegie Mellon.

Alice 3.0 isn’t yet generally available, but anyone who wants to try Alice 2.0 can download it from the home page at www.alice.org: you want the “Alice 2.0” link at the top of the “Downloads” panel on the home page. If you download for Windows (which is what I use), you’ll get a zip file. Unzip this into a clean directory. Then just run the file Alice.exe. This pops up a window, in which you’ll see a tab or button labelled “Tutorial”.

Getting back to methods for delivering demonstrations over the Web, I’ve talked about interactive Web pages like my demonstrations, and about Java applets. There’s also a programming language named JavaScript — no relation to Java, despite the confusing similarity in name — which is implemented to run inside Web browsers. It’s more closely integrated with them than Java is, and can do things like changing the colour of buttons when you click them — all the rather pointless special effects that many commercial Web sites seem to like. The Web browsers available when JavaScript first came out didn’t enable it to do graphics; newer ones can, by using either “SVG” (scaleable vector graphics) or the “CANVAS” element. These are what Jacques Distler refers to in his comment below that “ or Javascript+SVG seem like a much better bet, for this sort of thing, than Java/Flash”. The demo he points to is of Penrose tiling, implemented both ways.

I’m no expert on using these in JavaScript, but a Google search for “SVG Canvas” found me several comparative evaluations. The trouble with both is the same as for Java graphics: you must either code all the higher-level stuff, or find a rendering engine to do it.

Finally, Mike Stay commented that “For all practical purposes, Flash is available everywhere.
Flash 8 is in more than 99% of all browser installations.” The link is to a survey by Adobe, Flash’s makers, about the market “penetration” of Flash. That’s something I’ve no experience with yet.

Re: Graphical Category Theory Demonstrations

I just wanted to clarify that these aren’t Java Applets – they go through a CGI call to a server that does the computation instead of doing it on each user’s computer.

Would it still be possible to maybe have the output more directly related to the input? Maybe it’s just my system, but for me when I hit one of these buttons I am eventually presented with a page that asks me to follow a link, and there finally the output if visisble.

I can’t tell about the technical problems faced here, but I’d imagine the more “hand on” such an application is, the more it would serve the intended purpose.

There are other aspects that don’t quite seem to work for me:

for instance I just hit “colimit” and the output displays a single arrow from one bubble to another. Is that intended?

The output of “terminal object” is similar, but I take it that here it is intended.

Re: Graphical Category Theory Demonstrations

Re: Graphical Category Theory Demonstrations

If you’d like collaborators, the trendy way to get them these days is to open-source your project and put it on GitHub. Does that appeal to you?

Re: Graphical Category Theory Demonstrations