The ${\theta }_5$-graph is a spanner

Given a set of points in the plane, we show that the $\theta$-graph with 5 cones is a geometric spanner with spanning ratio at most $\sqrt{50 + 22 \sqrt{5}} \approx 9.960$. This is the first constant upper bound on the spanning ratio of this graph. The upper bound uses a constructive argument that gives a (possibly self-intersecting) path between any two vertices, of length at most $\sqrt{50+22\sqrt{5}}$ times the Euclidean distance between the vertices. We also give a lower bound on the spanning ratio of $\frac{1}{2}(11\sqrt{5}-17)\approx 3.798$.