Princess Kaguya is a heroine of a famous folk tale, as every Japanese knows. She was assumed to be confined in a bamboo cavity with cylindrical shape, and then fortuitously discovered by an elderly man in the forest. Here, we pose a question as to how long she could have survived in an enclosed space such as the bamboo chamber, which had no external oxygen supply at all. We demonstrate that the survival time should be determined by three geometric attributes regarding her body and the bamboo chamber. We also emphasize that this geometric problem shed light on an interesting scaling relation between biological quantities for living organisms.

This paper deals with the nearest neighbor distance in three-dimensional space. The distribution of the nearest neighbor distance is derived for grid and random point patterns. The distance is measured as the Euclidean and rectilinear distances. An application of the nearest neighbor distance can be found in facility location problems. The nearest neighbor distance represents the service level of facility location. The distribution shows how the distance to the nearest facility is distributed in a study region, and is useful for facility location problems in three-dimensional space. The distribution of the kth nearest neighbor distance is also derived for the random pattern.

A new coding rule for periodic orbits in unimodal one-dimensional maps is derived. The best-known example of a family of unimodal maps is the logistic map. The band merging is observed in the bifurcation diagram of the logistic map. Let a_m^k(k ≥ 1) be the critical value at which 2^k-band merges into 2^{k - 1}-band. At a > a_m⁰, the diverging orbit appears and thus 1-band disappears. The relations a_m^{k + 1} < a_m^k for k ≥ 0 hold. Let s_q be the code for periodic orbit of period q in the parameter interval (a_m¹, a_m⁰]. Assume that the code s_q represented by symbols 0 and 1 is known. In the interval (a_m^{k + 1}, a_m^k], there exists the periodic orbit of period 2^k × q (k ≥ 1). Let its code be s_2^k×q. Let 𝒟 be the doubling operator defined by the substitution rules as 0 ⇒ 11 and 1 ⇒ 01. The following coding rule is derived. Operating k times of 𝒟 to s_q, the code s_2^k×q is determined.