Revisiting examples of computer assisted mathematics

I'm in the middle of a finishing off some last things for a brand new course we're teaching at +Cardiff University starting in October. I plan on using my first lecture to explain to our new students how important computing/programming/coding is for modern mathematicians.

I will certainly be talking about the 4 colour theorem which states that any map can be coloured using 4 colours. I'll probably demo a bit of what +Sage Mathematical Software System can do. Here's a sage cell that will demo some of this (click evaluate and you should see the output, feel free to then play around with the code).

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a =[[0,1,1,1,1,1,0,1,1],[1,0,0,0,0,1,1,1,0],[1,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,1],[1,0,0,0,0,1,0,1,1],[1,1,1,0,1,0,1,1,0],[0,1,0,0,0,1,0,0,0],[1,1,0,1,1,1,0,0,1],[1,0,0,1,1,0,0,1,0]]

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g = Graph(matrix(a))

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print g.is_planar()  # Check that graph is planar

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g.show()  # Showing the 9 vertex planar graph with the above adjacency matrix

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p = g.coloring()

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g.show(partition=p)  # Showing the same graph with the coloring.

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print g.chromatic_number()  #Print the chromatic number of the graph

Language: Sage Gap GP HTML Macaulay2 Maxima Octave Python R Singular

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There I've written some very basic code to show a colouring of a graph on 9 vertices.

I expect that Students might find that interesting in particular if I show colourings for non planar graphs. For example here's another cell showing the procedure on complete graph on 9 vertices:

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g = graphs.CompleteGraph(9)

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print g.is_planar()  # Check that graph is not planar

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g.show()  # Showing the 9 vertex planar graph with the above adjacency matrix

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p = g.coloring()

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g.show(partition=p)  # Showing the same graph with the coloring.

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print g.chromatic_number()  #Print the chromatic number of the graph

Language: Sage Gap GP HTML Macaulay2 Maxima Octave Python R Singular

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That is just a couple of 'qwerky' things that don't really go near the complexities of the proof of the 4 colour theorem.

I took to social media in the hope of asking for more examples of cool things that mathematicians use computers for. Here's a link to the blog post but without a doubt the most responses I got was on this G+ post.

You can see all the responses on that post but I thought I'd try to compile a list and quick description of some of the suggestions that caught my eye:

• This wiki was pointed out by +Kevin Clift which contains a large amount of great animations (like the one below) made by 'Keiff':

• Kevin also pointed out that +Luis Guzman had shared a link to this great article that does not only mention Hale's 'proof' of the Kepler conjecture but also discusses some of the negative sides of computer assisted proofs.
  
• On a similar note +Artem Kaznatcheev pointed out a great blog post of his. In this post Artem discuss the 4 color theorem and also points out 'the curse of computing':

'This is the curse of computing: giving up understanding for an easy verification.'

• +Joerg Fliege mentioned chess endgame tablebases which I think would be a cool thing to point out to students.
  
• +David Ketcheson, +Dima Pasechnik and others pointed out how computer algebra systems are more or less an everyday tool for mathematicians nowadays. When I'm finding my way through a new project I normally always have a sage terminal open to try out various algebraic relationships as I go...

There are a couple of other things that I'm not listing above (mainly because I don't know enough about them to be able to comment), but interestingly enough +Timothy Gowers posted the other day a link to a paper that he has co-authored entitled: 'A fully automatic problem solver with human-style output'. In that paper a new program is described able to produce human style proofs of theorems. A blog post that +Timothy Gowers put up a while back was actually an experiment for this paper.

I'm obviously missing a large amount of other stuff so please do let me know :)