Affine spaces over F3

The card game Set is played rather regularly at gatherings such as the MathsJam events held nationwide on a monthly basis. The original game involves trying to identify sets of three cards which form lines in the affine hyperspace . Here is an example of such a set:

Each card can be represented by a vector . Each coordinate represents one of the four attributes (quantity, shape, colour and shading, in that order), and is interpreted modulo 3. The three cards shown above are , and .

This coordinate system is somewhat unsatisfying, due to the presence of an ‘origin’. To circumvent this, it is necessary to prepend another coordinate, , to produce projective coordinates. The leftmost card in the diagram above is now . We can now identify the necessary and sufficient condition that three cards form a ‘set’ (henceforth referred to as a line, since it’s more accurate):

• Line: Three linearly dependent vectors, no two of which are identical.

Equivalently, we want three vectors, , such that some linear combination of them (with coefficients of ±1) equals the zero vector. By considering the fixed coordinate, it is evident that the number of ‘positive’ terms in the linear combination differs from the number of ‘negative’ terms by a multiple of three. In this case, the only admissible equation for a line is this:

• Line: Three vectors a, b, c, such that .

These definitions are more mathematically interesting than the seemingly arbitrary (albeit equivalent) rule of ‘either all the same or all different, for each attribute’. Also, the first one lends itself particularly well to a generalisation (named after Edward Kirkby, who proposed this major modification to the rules of the game):

• Kirkby: Four coplanar points, no three of which are collinear.

An example of a Kirkby

By the same argument, the only possible structure for a Kirkby is a set of four vectors satisfying . This was the standard definition of a Kirkby for quite some time, so it was only recently that Gabriel Gendler realised that lines and Kirkbies were special cases of a (potentially infinite) sequence. His own surname became associated with the next term, defined thusly:

• Gendler: Five cohyperplanar points, no four of which are coplanar.

The desired structure is . If this equation is obeyed and there are no lines, then we can deduce that there are also no Kirkbies and thus we have an actual Gendler.

Restricting ourself to four-dimensional space, there is only one more term. I’m not precisely sure of the exact etymology of this, but I believe that the following name and definition became associated:

• Carlotti: Six points in general linear position.

Since we have six cards, it is necessarily true that any one can be expressed as a linear combination of the other five. In particular, there are two admissible structures for a Carlotti, namely and .

Any affine transformation in this space can be expressed uniquely as a matrix with 20 variable entries (the top row is fixed). Consequently, the group of invertible affine transformations is transitive on sets of five points in general linear position. A corollary is that any two lines are equivalent, any two Kirkbies are equivalent and so are any two Gendlers.

We can generalise to more dimensions. At a Cambridge MathsJam, the clumsy dining abilities of one of the attendees led to the idea that three packs of Set cards, each one of which has been smeared with a different flavour of preserve (strawberry, blackcurrant and apricot, for an explicit example). In this case, Carlotties are no longer in general linear position, and a different type of structure arises.