Vertex-Unfoldings of Deltahedra

• Main Authors: Wen-Tsang Chang, 張文蒼
• Other Authors: Keh-Ming Lu
• Format: Others
• Language: zh-TW
• Published: 2010
• Online Access: http://ndltd.ncl.edu.tw/handle/72593303763682429653

碩士 === 亞洲大學 === 資訊工程學系碩士在職專班 === 98 === Abstract The study of origami blocks is an ongoing process of innovation and discovery. Besides being able to be disassembled and reused, the dimensional paper blocks can be cut into a variety of launch map. Through proper unfolding and folding of the different forms of plane blocks, we had the thought of 2D converted to 3D. With the introduction of mathematical combination theory, the development of the paper blocks can become multiple by various methods of creation, games, and thinking. In daily life, we often contact the information products such as mobile phones and notebooks. The shells of these products are mostly sheet metal parts. The 3D parts of which are near the polyhedron. Their surface graphics are the combinations of 2Ds. How to judge whether a convex polyhedron can be cut along the edge of incision and unfolded into a non-overlapping plane is a common question. This research studies the deltahedra composed of the equilateral triangles to see if we can connect the surfaces at vertexes or edges. We used vertex unfolding algorithms to unfold the deltahedra with zero, one, and high genus numbers, and built the non-overlapping but connecting diagram. After cutting the polyhedron along the edges, the diagram is connected as a whole, but there are unconnected areas inside. These triangular planes are connected at the vertexes but not necessarily along the edges. Examples of zero genus deltahedra included 8 convex and 4 non-convex deltahedra. Deltahedra with one genus are Conway toroidal deltahedron, Stewart toroidal deltahedron and a 96-hedra torus. Examples of high genus number are the 4-Bowtie of Napoleom paper building block model, Conway spherical bodies, and Napoleon Bowtie basic one model. The research tried to explore if it is possible to unfold these polyhedra by cutting along the edges of the surfaces into unfolding but connected diagram, and used the vertex unfolding algorithms to complete vertex unfolding. It was found that it would be more persuasive if we provide more high-genus deltahedra as research objectives to certify the vertex unfolding. Similar methods can also be adopted to verify Vertex unfolding of other types of polyhedron.