All done using R, of course (showtext and igraph packages).
I’ve been having some fun with Six Degrees of Wikipedia, which finds shortest paths inside the network of links between Wikipedia pages. Here are some connections between ASC teams (of various lengths):
Teams connect to locations too:
And my personal favourites:
The New Zealand game of mū tōrere is illustrated above with a beautiful handmade wooden board. The game seems to have been developed by the Māori people in response to the European game of draughts (checkers). Play is quite different from draughts, however. The game starts as shown above, with Black to move first. Legal moves involve moving a piece to an adjacent empty space:
Game play continues forever until a draw is called (by mutual consent) or a player loses by being unable to move. Neither player can force a win, in general, so a loss is always the result of a mistake. For each player there is one “big trap” and four “small traps.” This is the “big trap” (Black wins in 5 moves):
Here is one of the four “small traps” for White. The obvious move by Black results in White losing (but avoiding this does not require looking quite so far ahead as with the “big trap”):
Here (click to zoom) is the complete network of 86 game states for mū tōrere (40 board positions which can occur in both a “Black to move” and a “White to move” form, plus 6 other “lost” board positions). Light-coloured circles indicate White to move, and dark-coloured circles Black to move, with the start position in blue at the top right. Red and pink circles are a guaranteed win for Black, while green circles are a guaranteed win for White. Arrows indicate moves, with coloured arrows being forced moves. The diagram (produced in R) does not fully indicate the symmetry of the network. Many of the cycles are clearly visible, however:
I have previously mentioned my strong interest in science / technology / engineering / mathematics education and in networks and in board games. This has prompted me to start designing educational games, such as the World Solar Challenge game. Joining the collection is my new Chemical Compounds game, which looks like this:
The online game store (faciliated by the wonderful people at The Game Crafter) has a free download link for the rules, should anyone wish to take a look. I also have a few other educational games there.
“About once every hundred years some wiseacre gets up and tries to banish the fairy tale,” C.S. Lewis wrote in 1952. The wiseacre of our time seems to be Richard Dawkins who, two years ago, told the world that fairy tales could be harmful because they “inculcate a view of the world which includes supernaturalism” (he had said similar things in 2008). In a later clarification, he added that fairy tales could “be wonderful” and that they “are part of childhood, they are stretching the imagination of children” – provided some helpful adult emphasises that “Do frogs turn into princes? No they don’t.”
But many scientists grew up with, and were inspired by, fantasy literature. For example, Jane Goodall tells of growing up with the novel The Story of Doctor Dolittle (as I did!). In fact, many science students and professional scientists avidly read fantasy literature even as adults (as they should). The booksthatmakeyoudumb website lists, among the top 10 novels read at CalTech and MIT, Harry Potter, Dune, and The Lord of the Rings. And Alice in Wonderland was written by a mathematician.
This is a science blog, so I have a strong emphasis on scientific truth, which tells us many important ecological and physiological facts about, for example, frogs. Without science, we’d all still be struggling subsistence farmers. But there is actually more than scientific truth out there.
There is also mathematical truth. Are the links in this frog network all equivalent? Yes, they are – but that is decided by mathematical proof, not by scientific experiment. It is in fact a purely abstract mathematical question – the background picture of the frog is actually irrelevant.
And there is ethical truth. Is it OK to eat frog’s legs? Science does not give us the answer to this (although logic can help us decide if our answer is consistent with our other beliefs), but fantasy literature often helps us to explore such ethical questions. Tolkien’s The Lord of the Rings is one superb example. Would you “snare an orc with a falsehood”? Would you attempt to take the One Ring and “go forth to victory”?
There is metaphorical truth. A frog may, in spite of what Dawkins says, be a handsome prince – there’s more to the universe than can be seen at first glance. Or, as Antoine de Saint-Exupéry put it, “What is essential is invisible to the eye.” Children often learn this important fact from fairy tales.
And there is even religious and philosophical truth. Does the frog-goddess Heqet exist, for example? Does the universe exist? Is there a spoon? The methods of philosophy are different from the methods of science, and some amateur philosophers simply state their beliefs without actually justifying them, but philosophy is actually very important. Science itself is based on certain philosophical beliefs about reality.
I’m excited at the publication of a joint paper on network ecology, with a focus on the Australian dingo: “Trophic cascades in 3D: Network analysis reveals how apex predators structure ecosystems” (by Arian D. Wallach, Anthony H. Dekker, Miguel Lurgi, Jose M. Montoya, Damien A. Fordham & Euan G. Ritchie, and appearing in Methods in Ecology and Evolution).
Associated with this publication is an animation I put together for the paper showing how the ecological network changes if the role of the dingo as apex predator is weakened. I’m grateful to my ecologist co-authors at the opportunity to contribute my mathematical skills to such an interesting project.
Network colouring is an fascinating branch of mathematics, originally motivated by the four colour map theorem (first conjectured in 1852, but proved only in 1976). Network colouring has applications to register allocation in computers.
For each network there is a chromatic polynomial which gives the number of ways in which the network can be coloured with x colours (subject to the restriction that directly linked nodes have different colours). For example, this linear network can be coloured in two ways using x = 2 colours:
The corresponding chromatic polynomial is x (x − 1)3, which is plotted below. Zeros at x = 0 and x = 1 indicate that at least 2 colours are required.
For the Petersen network below, the chromatic polynomial is:
x (x − 1) (x − 2) (x7 − 12 x6 + 67 x5 − 230 x4 + 529 x3 − 814 x2 + 775 x − 352)
This polynomial has zeros at 0, 1, 2, and 2.2051, and is plotted below:
Chromatic polynomials provide an interesting link between elementary and advanced mathematics, as well as an interesting case study of network algorithms.