edge weights. The same points may have more than one minimum spanning tree. For instance, for the vertices of a regular polygon, removing any edge of the polygon produces a minimum spanning tree. Publications on the Euclidean minimum spanning tree commonly abbreviate it as "EMST". They may also be called "geometric minimum spanning trees", but that term may be used more generally for geometric spaces with non-Euclidean distances, such as spaces. When the context of Euclidean point sets is clear, they may be called simply "minimum spanning trees". Several other standard geometric networks are closely related to the Euclidean minimum spanning tree: The Steiner tree problem again seeks a system of line segments connecting all given points, but without requiring the segments to start and end only at given points. In this problem, additional points may be added as segment endpoints, allowing the Steiner tree to be shorter than the minimum spanning tree. In the Euclidean traveling salesperson path problem, the connecting line segments must start and end at the given points, like the spanning tree and unlike the Steiner tree; additionally, each point can touch at most two line segments, so the result forms a polygonal chain. Because of this restriction, the optimal path may be longer than the Euclidean minimum spanning tree, but is at most twice as long. Geometric spanners are low-weight networks that, like the minimum spanning tree, connect all of the points. Unlike the minimum spanning tree, all of these connecting paths are required to be short, having length proportional to the distance between the points they connect. To achieve this property, these networks generally have cycles and so are not trees. Properties Angles and vertex degrees Whenever two edges of a Euclidean minimum spanning tree meet at a vertex, they must form an angle of 60° or more, with equality only when they form two sides of an equilateral triangle. This is because, for two edges forming any sharper angle, one of the two edges could be replaced by the third, shorter edge of the triangle they form, forming a tree with smaller total length. In comparison, the Steiner tree problem has a stronger angle bound: an optimal Steiner tree has all angles at least 120°. The same 60° angle bound also occurs in the kissing number problem, of finding the maximum number of unit spheres in Euclidean space that can be tangent to