The shape of a system

Shapes. Lesson 1.2 asked whether one set of variables is free. Asking it of every set at once gives a picture, the system's shape:

• each free variable is a point;
• each free pair is an edge between two points;
• each free triple is a filled triangle;
• each free set of four is a filled tetrahedron, and so on.

Simplicial complexes and holes. Every part of a free set is free, since each combination of values on the part extends to a combination on the whole set, and that one occurs. So every side of a filled triangle is drawn too. A collection of sets that contains every part of each of its sets is called a simplicial complex, and a system's shape is one. Topology studies such shapes, and in particular their holes: a loop of free sets with nothing free filling it, or in higher dimensions a hollow sphere of them.

Three examples. On three variables:

• with no rule at all, every set is free, and the shape is a filled triangle;
• under the atom, the three pairs are free but the triple is not, so the shape is a hollow triangle, three edges around a missing face. That is a circle, a hole of dimension one;
• under A ≠ B, with c unused, the edge between A and B is missing, so the shape is two edges meeting at c, with no hole.

Flags and joins. Two terms from topology appear in the next lesson. A shape is a flag shape when every set whose pairs are all edges is filled in, so that its pairs decide all of it. Two systems on separate variables, set side by side, have the join of their shapes as their shape: a set is free when its part in each system is free. The join of two circles is a three-dimensional sphere (Classical).