Proof. By Example 19 and Prop. 20.

## 4 Constraint Reasoning With The Dipole Calculus 4.1 Consistency

We now consider the question of whether algebraic closure decides consistency.

We call the set of constraints between all dipoles at hand a *constraint network*. If no constraint between two dipoles is given, we agree that they are in the universal relation. By *scenario*, we denote a constraint network in which all constraints are base-relations22. We construct constraint-networks which are geometrically unrealizable but still algebraically closed. We do this by constructing constraint networks that are consistent and algebraically closed, and then we will change a relation in them in such a way that they remain algebraically closed but become inconsistent. We follow the approach of [58] in using a simple geometric shape for which scenarios exist, where algebraic closure fails to decide consistency. In our case, the basic shape is a convex hexagon, similar to a screw head.

Consider a convex hexagon consisting of the dipoles A, B, C, D, E and F.

Such an object is described as
(A errs B)(B errs C)(C errs D)(D errs E)(E errs F)(F errs A)
where the components r of the relations ensure convexity, since they enforce an angle between 0 and π between the respective first and second dipole, i.e., the endpoint of consecutive dipoles always lies to the right of the preceding dipole.

Such an object is given in Fig. 26 To this scenario we add a seventh dipole G
with the relations
(G rrll A)(G lrll F)(G llrr D)(G rlrr C)
We have the overall constraint network:
(A errs B)(B errs C)(C errs D)(D errs E)(E errs F)(F errs A)
(G rrll A)(G lrll F)(G llrr D)(G rlrr C)
Because of the relations (G lrll F) and (G rlrr C), line lG intersects line lF
as well as line lC . Because of the first two components of the relations, dipoles F and C are oriented into qualitatively antipodal directions. This network is consistent and is of course algebraically closed.

22In this case, a base-relation between every pair of distinct dipoles has to be provided