The most efficient origami torus.

Area of Science:

• Mathematics
• Computational Geometry
• Topology

Background:

• Origami tori are three-dimensional shapes constructed from triangles meeting at vertices.
• The sum of angles around each vertex in an origami torus must equal 2π.
• Previous research explored the existence and properties of origami tori with varying vertex counts.

Purpose of the Study:

• To determine the existence of origami tori with specific numbers of vertices.
• To settle the question of the most vertex-efficient origami torus construction.

Main Methods:

• Utilized geometric and topological principles to analyze origami torus structures.
• Investigated the constraints imposed by angle sums at vertices.
• Employed proof-based methods to establish existence and non-existence results.

Main Results:

• Demonstrated that an origami torus cannot be constructed with exactly 7 vertices.
• Proved the existence of an origami torus with 8 vertices.
• Established a boundary for the minimum number of vertices required for origami torus construction.

Conclusions:

• The non-existence of a 7-vertex origami torus and the existence of an 8-vertex one are confirmed.
• These results provide definitive answers regarding the most efficient origami torus construction in terms of vertex count.