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Consider the following question: In a circular cone, with bus line having length 1, the inscribed sphere is to be maximal. How much is the radius of the base circle? It is easy to see that the answer is (√-1)/2, which is interesting because this is the reciprocal of the golden section. In this paper, we generalize the question to the case that the base circle is generalized to regular polygons., 紀要論文

Consider the following question: In a circular cone, with the sum of the radius of the base circle and the length of the bus line being 1, the inscribed sphere is to be maximal. How much is the radius of the base circle? It is easy to see that the answer is 1/3, which is geometrically interpreted as follows: Consider the section of a cone by a plane which contains the apex and is perpendicular to the base circle. Then the answer corresponds to the case that the section is an equilateral triangle. In this paper, we generalize the question to the case that the base circle is generalized to regular polygons., 紀要論文

Let M_n be the moduli space of equilateral polygon linkages with n vertices, each edge having length 1 in the Euclidean plane R^2 modulo orientation preserving isometry group of R^2. For a given diffeomorphism φ:M_n→M_n, we give a method to determine whether φ preserves the orientation or not. We also determine H_*(M_n/φ;Q) when φ is a certain involution., 紀要論文

Let p: X→Y be a regular 2-cover. We assume that the group H_*(X;Z) are torsion free. It is easy to show that H_* (X;Z) are odd-torsion free. It is also well-known that the torsion subgroup of H_*(X;Z) has exponent 2. Several applications are also known. Since the results are scattered and somehow hidden in the literature, a survey of these researches may be useful. This is the purpose of this note. Our applications include the study of the homology of planar polygon space., 紀要論文

Let M_n be the moduli space of equilateral polygon linkages with n vertices, each edge having length 1 in the Euclidean plane R^2 modulo SO(2)⋉R^2, the orientation preserving isometry group of R^2. The space M_n admits the well-known involution, that is, the reflection about the x-axis. In this paper, we consider the reflection about the y-axis. We determine the rational homology groups of the orbit space., 紀要論文

Let P(n, r) be the moduli space of n-gons in R^2 whose edge lengths are 1, ... , 1 and r. There is an inclusion P(n, r) → (R^2)^n. Restricting the usual Riemannian metric on (R^2)^n, we regard P(n, r) as a Riemannian manifold. In this paper, with the aid of a computer, we compute the volume of P(n, r) for n = 4, 5 and 6., 紀要論文

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