Abstract. We have obtained the best constant of discrete Sobolev inequality on theregular polyhedra including double bond. Let N be the number of vertices. We introduce the discrete Laplacian A which is N × N real symmetric matrix. A has an eigenvalue 0 whose corresponding eigenspace is 1 dimension. If we introduce the pseudo Green matrix G∗, then G∗ is reproducing kernel by setting appropriate vector space and inner product. The maximum of the diagonal values of G∗ is the best constant of this inequality.

Abstract. Polyhedrons by Nojima and by Tachi-Miura, which both are two symmetrical origami structures, can be folded in the axial or radial direction, and it is convenient if they can be applied to aluminum cans. In this paper, we investigate whether both structures are rigid folding or not, and consider the influence of this on the energy absorbing characteristics. Moreover, we studied the crushing and spring back characteristics of both structures by simulation, and explore their possibility.

Abstract. The geometric aspect of origami has been attracting attention for a long time, and has been the subject of research. In recent years, studies using computers have been actively carried out, and the term "computational origami" appeared. Applications of origami related technology to the field of engineering are also actively discussed. In this paper, we summarize the researches on flat-foldability discussed so far in origami mathematics. In addition, we will introduce researches on computational complexity on origami, and origami design methods. We also outline the fields of non-flat origami and rigid origami that are important in application of origami mathematics.