# Isomorphism

Have you ever considered all of the empty space on a Checkers board? Half of the squares are unused. You might legitimately ask why they’re even there.

That, of course, raises the question of whether there’s any way for them not to be there, and it turns out that there is. Imagine removing all of the light spaces from the board, then rotate every dark space 45°. There’s some empty space now, so just sort of slide all the squares together. You’ll end up with something like this:

This is an example of what is called an isomorphism. Loosely put, an isomorphism is a translation between two things that preserves a certain set of their properties. In this case, the playable squares on the original board, with adjacency defined as diagonal, is isomorphic to the new board under orthogonal adjacency. In game design, isomorphism can be used to transform a game into a cosmetically different game, without altering the structure of its gameplay (a well-known example is that Tic Tac Toe is isomorphic to a game of picking numbers from a magic square).

I consider it a feature that the squares that were diagonally adjacent before are now orthogonally adjacent. Orthogonal moves are much more typical in most games, after all. This also seems like a good illustration of just how small the state space in Checkers really is. Of course, I’m talking about Anglo-American Checkers, or Draughts; most international variations are bigger.

Speaking of International Checkers, this leads us to something this variation might be good for. That game is played on a 10×10 board, is reputedly a better game, but is not readily available in the United States. I’ve just started reading R. Wayne Schmittberger’s New Rules for Classic Games, where he discusses both regional variants, and how to make your own equipment. If you were going to do that, you could use a lot less material (and have a more compact board for the same size of checkers) by throwing out the unused squares. Whether you’d like the un-Checkerslike appearance of the board is up to you.

But, as long as we’re going with a different look, let’s go all the way. The checkers themselves don’t occupy all of the space within their squares, do they? You could squeeze things down a bit more by knocking the corners off. To preserve the common edge between adjacent spaces, let’s go with octagons. This actually restores the unplayable squares to the board, but in a form that can’t be mistaken for the playing spaces. And the pattern you end up with (shown here with the 10×10 international board) is, I think, very attractive.

It turns out, by the way, that this is only 75% of the width and height of the same thing as a square grid. This could be cramped to play, but is probably easier to print. At least it fits on an 11″×17″ sheet. In case you want to try, I’ve posted the same image at 10″×10″, 300 dpi.

This is the kind of thing my mind sometimes does when I’m falling asleep at night. I spent some time this morning coding up the pattern in nanDECK; maybe later I’ll put together a proper board.

# Click the Camera

I touched yesterday on one of the problems I wanted to solve with this dice deck, which is the limited number of games that use exactly two six-sided dice. Nothing against The Settlers of Catan or Backgammon, but let’s see what we can do to broaden the scope of the project just a bit.

First, we can include many other games that use just one die at a time, and even a few that use two distinct dice for different purposes. So, we’ll make the dice distinguishable, one white and one shaded. While we’re at it, by varying the color of the shaded die we can produce random colors (and yes, I can name games where that’s useful).

Now, how to produce the needed images? I have little skill as an artist, but I can program, and I can take non-embarrassing photos (the latter thanks in part to the excellent discussions of board game photography on BoardGameGeek.com, particularly the advice of BGG user EndersGame). I start with photos. But this means I need to be able to take very consistent photos of thirty-six different sets of dice.

The solution is an inexpensive tripod and a foam core jig, with marks for the positions of tripod and dice. Natural light is best for this sort of thing in the absence of real lighting equipment (thanks again, BGG), so I set up out on the back deck.

All of this was actually done some days ago, and I’m reasonably happy with the results. I’m not 100% sold on the grey background of the foam core, but I want something that’s not going to argue with the shades of any of the dice. If I were a better photographer I’d probably be able to polish it up more in post. I did edit out a number of blemishes to the pips on my actual dice.

So I printed up some prototype cards using these images. And, sure enough, the prototype is serving its purpose, which is in part to tell me what I did wrong. I didn’t think, at first, that I would need indices on the cards. Indices are the notations found in the corner, so you can hold the cards fanned. Those aren’t necessary if you’re just going to flip cards off the deck to give die results. But once you have the cards, you open up a lot of possibilities other than that one. You can easily imagine more variants, such as drafting or auctioning the cards. Or, yes, playing a hand of them. Plus, I’ve got another cunning scheme for this deck that would absolutely necessitate hands. So, indices it is. Well, the photos aren’t going to provide those.

I can think of few things I’d rather do less than editing little corner symbols onto thirty-odd cards, and then having to redo them all again consistently if I want to make changes. So, the solution for this part is nanDECK, a neat if quirky little card-rendering programming language by Andrea “Nand” Nini. Yes, there really is a programming language for everything.

That’s what I’ve mainly been doing this morning. You can see the results as they currently stand. I’m thinking I may want to further annotate the index for the color-blind, now that I’ve gone this far.

Tomorrow: Aren’t these cards starting to look a little bit like dominoes?