A generating theorem of punctured surface triangulations with inner degree at least 4

Given any punctured surface F2, we present a method for generating all of F2 triangulations with inner vertices of degree ≥ 4 and boundary vertices of degree ≥ 3. The method is based on a set of expansive operations which includes the well-known vertex splitting and octahedron addition. By reversing this method we get a procedure to obtain minimal triangulations by a sequence of intermediate triangulations, all of them within the given family.

Vertex-Splitting on a Diamond Origami Pattern

A diamond origami pattern is a well-known origami pattern consisting of identical six-crease vertices. As each vertex can be modeled as a spherical 6R linkage with three degrees of freedom (DOF), the tessellated pattern with multiple vertices is a multi-DOF system, which makes it difficult to fully control the motion in the desired symmetric manner. Here, two splitting schemes on the diamond vertex are proposed to generate three types of unit patterns to reduce the DOF. This vertex-splitting technique is applied to the multivertex diamond origami pattern to produce several one-DOF basic assemblies, which form a number of one-DOF origami patterns. Two of the one-DOF origami patterns are discussed: one of which is a flat-foldable origami pattern mixed with four- and six-crease vertices and the other is a nonflat-foldable one mixed with four-, five-, and six-crease vertices. In the one-DOF patterns, the symmetrically kinematic property of the original diamond origami pattern is well kept. Such property would significantly facilitate engineering applications comparing to the multi-DOF origami patterns. It also paves a new road to construct one-DOF origami patterns.