Peterson Graphs, Semantic Networks, Glass Bead Games

The HipBone Games can be seen as "semantic networks" and their boards as "graphs" -- there's a neat connection here with areas that I'm not personally too familiar with at a crossover between mathematics, computer science and artificial intelligence... This page shows you some of the "game board" resources I have been running across on the web, and talks, too, about the curious connection between the great mathematician George Boole and the Holy Trinity.

Warning: there is a poet trying to understand mathematics loose on these pages!

Peterson Graphs

My friend and colleague Walter Logeman of PsyberNet introduced me to Peterson Graphs, by posting these two illustrations on the web and inviting the participants in his Psyber-L mailing list to take a look at them:

Two boards, right? Each of them has ten "positions" with three links from each, but they also have the curious property that they can be "mapped" onto one another, so that a player using one of these boards could (if the position numbers are carefully correlated) play with an opponent who was using the other -- while the links would still be valid whichever board they were recorded on, the symmetries would obviously be different...

I have formatted these images as the Pentagram and Mercedes HipBone Boards.

In any case, this sent me off in search of more Peterson Graphs, and I was fortunate to come across Dr. Tomaz Pisanski of the Department of Theoretical Computer Science at Institute of Mathematics, Physics and Mechanics in Ljubljana, Slovenia, and found these illustrations on the Institute's web pages.

Here's an example:

Dr Pizanski named this particular graph "the PetersonOrnament above".

It is not actually the most interesting graph from my point of view as a game designer, since a graph with 10 vertices seems optimal in most games, and somewhere around 20 would be the upper limit except on very rare occasions -- though we might use a fullerene molecule or a map of the London Underground on some occasion where there were, eg a larger group of players who could "start" simultaneously from different vertices.

The next graph isn't so pretty, perhaps, but it interests me as a potential board because it contains 14 points or positions, which I see as 3 groups of 4 with a single pair which connects to all of them. I'm not sure that that's in accord with the mathematics of the graph, but it's what the visual image suggests to me.

For my own poetic purposes, this "three fours with a two" closely resembles the form of the Shakespearean sonnet, which rhymes:

abab cdcd efef gg

and I can easily see how one of my games could be structured to mimic sonnet form using this graph.

If you think of the graph as looking like three longish oblongs which cut through each other at the center, then the three oblongs would correspond to the three sonnet quatrains which end in the rhymes "abab", "cdcd" and "efef" respectively, and the two points at which all three oblongs intersect would correspond to the couplet "gg" -- which in any case tends to tie up all the threads of the sonnet.

I'm interested by this graph because it reminds me of the setup in parallel processing, and because from a game point of view, the multiple links between the two central "columns" would be very demanding in play.

And perhaps best of all for my purposes is this graph, which strikes me as very elegant:

There's an eerie spatiality to this graph, as though it's the groundplan for one of those strange architectures of Maurits Escher...

These graphs form part of the Vega Project of the IMPM, Vega being a system for manipulating discrete mathematical structures. Dr. Pizanski kindly directed me to the Vega Graph Gallery index, which contains a few hundred graphs of this sort...

Margaret Masterman, George Boole and the Holy Trinity:

some meta-mathematical meditations on Three...

In the course of my correspondence with Dr. Pisanski, I pointed him towards some of the "early game boards" that I have collected, ranging from the Pythagorean Tetraktys via a mediaeval "Trinity" board, a board from Bartholomaeus Anglicus and the Kabbalistic Tree of Life, to Sir Thomas Browne's elegant Quincunx. Dr. Pisanski expressed interest in these "historical examples" of the use of graphs -- and this in turn made me want to tell him about an extraordinary article linking my mediaeval "Trinity" graph with the work of George Boole...

The article in question is by Margaret Masterman, and forms part of her intriguing work, "Theism as a Scientific Hypothesis". In it, she talks about George Boole's fascination with the idea of "three", and reproduces a variant on the Trinity "graph" shown below, which I tend to think of (only a little facetiously) as a "mediaeval rendition of a HipBone Game played in heaven":

She proposes that this image (versions of which are commonly found in mediaeval churches) can be mapped onto a Boolean lattice / Hasse diagram of eight elements -- and that a "reading" of this diagram in terms of Boolean algebra "translates" into theological propositions concerning the persons of the Trinity which are remarkably coherent with those in the Athanasian Creed...

If you'd like to know more about Masterman's article, you can read my brief intro, or go directly to Masterman's article itself.

Semantic Networks:

All this brings me in turn to the connection between Margaret Masterman and "semantic networks", which were invented at the Cambridge Language Research Unit where she worked in the late 1950's.

I browsed the web for further references to "semantic networks", and came across Casey Reas' page, where I read:

Semantic networks are usually depicted in a graphic format. They consist of concepts represented by nodes and lines showing their relations represented called links.

Again, I was amazed. Casey Reas continues:

The links can take many different forms. The two kinds of hierarchal relationships are type (a bird is type of animal) and part (a beak is a part of a bird). Other links can specify a certain characteristic of an object or if one concept is the result of another. In a type of semantic model called a Spreading Activation Model (see example to the right), the links are different lengths based [on] how closely they are related with another concept...

And Reas' "example to the right" turned out to be yet another elegant board:

Now there's a fascinating suggestion -- that some HipBone Games might be played on boards where linking lines of different length require moves that are "more" or "less closely" related...

Final Thoughts

Mathematically inclined players may also be interested in a "Hamiltonian" Game board, which has another interesting property from the point of view of our games: it would require players to make "directional" links. Many links in current play are metaphors or analogies of one sort or another, but players using this particular board would need to find links which possessed the sort of directionality we associate with (eg) cause and effect or before and after...

That's about as far as I have got at the moment, and I would welcome comments, clarifications, corrections and pointers from any mathematicians, semanticists or other interested parties who visit this page.

Charles Cameron

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