Increasing insightful thinking in analytic geometry using frequent visualisation and a synthetic approach Theoretical background and practical implementation As of 2014, synthetic geometry will not be part of the Dutch 'VWO – Mathematics B' programme anymore. Instead, the focus will be more on analytic geometry. Mark Timmer explored possibilities to connect the two disciplines in order to have ...

Let P ⊂ R3 be a polyhedron. It was conjectured that if P is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. P can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additionnal assumption of codecomposability. The proof relies on a result of independent interest concerni...

Let P be a (non necessarily convex) embedded polyhedron in R, with its vertices on an ellipsoid. Suppose that the interior of P can be decomposed into convex polytopes without adding any vertex. Then P is infinitesimally rigid. More generally, let P be a polyhedron bounding a domain which is the union of polytopes C1, · · · , Cn with disjoint interiors, whose vertices are the vertices of P . Su...

A tensegrity polygon is a planar cable-strut tensegrity framework in which the cables form a convex polygon containing all vertices. The underlying edgelabeled graph, in which the cable edges form a Hamilton cycle, is an abstract tensegrity polygon. It is said to be robust if every convex realization as a tensegrity polygon has an equilibrium stress which is positive on the cables and negative ...

The synthesis and characterization of triynylbenzenes, 1-iodo-2,4,5-tri(4-alkoxyphenylethynyl)benzenes (1), are reported, and their structure–mesomorphism relationship is investigated. These compounds possess a y-shaped geometry with rigid linear spacers and a stable nematic phase that was achieved by lengthening the alkoxy chains or replacing the iodo group with a smaller atom. The low aspect ...

The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed the case for two classes of polyhedra: those obtained from a convex polyhedron by “denting” at most two edges at a common vertex, and suspensions with a natu...