Geometry
A chain of tetrahedra saves more
In the plane, junctions never seem to shorten a network of roads by more than the equilateral triangle's 13.4%. In space the regular tetrahedron allows 18.7%, and the obvious guess — that it is the worst case, as the triangle is in the plane — is wrong. Points spaced along a chain of tetrahedra glued face to face do better with every point added: 20.5% at twelve, 21.4% at sixty, and about 21.6% in the limit. Random points come nowhere near, and a search that only goes downhill does not find the chain.
