Usually, card games are one-dimensional. I don’t mean that they’re shallow and boring, but I mean it in a quite literal sense. There is one dimension, and it is usually the vertical one, or, perhaps more accurately, the temporal one. The cards are drawn from stacks and placed in piles. Of course, most games are not strictly one dimensional—the “dimensions” of hands and other common mechanisms seem hard to pin down—but at least predominantly so. These games center around the_order_ in which cards are drawn and played.
It was a delight recently to play card-based games with additional, non-temporal, spatial dimensions. For instance, Innovation,1 in which players compete to build civilisations across the history of technological innovation, has each player develop stacks of cards based on type. It is mainly the top card of each stack which matters, but under certain circumstances a stack may be splayed, or spread out to reveal and activate parts of the cards below. This mechanism is directional: splaying right vs. left vs. diagonally reveals different parts of each card. This flavour of spatial dimensionality is not uncommon: Arboretum2 and Forest Shuffle3 likewise exploit the rectangular geometry of cards. In the former, tree cards are arranged in a grid to score points; in the latter, animal cards are situated on the four sides of the tree cards. In Wingspan,4 bird cards are arranged in habitat rows, affecting the order of effect activation, and certain birds can move between rows.
Each of these games might be rightly called tableau-builders, that is, games in which players develop and manipulate an array, or tableau, of components, often tiles. From this perspective, what’s happening becomes obvious: cards are playing a double role. They are cards, as in the traditional sense, to be drawn and held and discarded; but they are also tiles, to be placed and rearranged. The rectangular card is a familiar, canonical form, and they readily serve as rectangular tiles. But, in principle, tiles may be of hexagonal, circular, or other non-rectangular shapes. Bird cards could be bird-shaped. Perhaps this raises certain mathematical questions of interest to few if any. If we strip away all accidental features, what is the fundamental, algebraic structure of each of these games? Can these informal intuitions about “dimensionality” be precisely characterised? Can informal categories of “game mechanics” be “verified” by mathematical structure? Or we should just stop ruining fun.