While the golden ratio (1:φ) has been well known since the ancient Greece age and has already been studied in depth by many researchers, the square-root-of-2 ratio (1:√2) and other ratios of this group have gone largely unnoticed until now, despite the fact that the rhombic dodecahedron, one of the most important polyhedra, is based on the square-root-of-2 ratio. According to a popular theory, these ratios are independent of one another. Against it, however, the author found out that there exists close relationship between the ratios of the golden ratio group and those of the square-root-of-2 ratio group through careful study on three kinds of rhombic polyhedron.

We consider the standard map. The stable and unstable manifolds of the saddle fixed point are proved to intersect transversely at the primary homoclinic point u for any parameter value. For the proof, we use the particular objects called the dominant axis (DA) and subdominant axis (SD), and symmetric periodic orbits that have orbital points on these axes. The periodic orbit named 1/q-BE has the orbital point z_k at the intersection point of DA and SD. Let ξ_k be the slope of SD at z_k. Take a sequence of z_k accumulating at u as k → ∞. We prove that the slope ξ_k monotonically decreases to the slope ξ_u(u) of the unstable manifold at u (the monotone inclination property). Using Ushiki's theorem, the hyperbolic region (HR) is constructed. It is proved that the orbital point z_k in HR is a saddle point with reflection. Using the monotone inclination property and the properties of z_k in HR, the transversality at u for any value of a (> 0) is proved.

Many elementary students have a well-known misconception that when the shape of an object changes, its weight also changes. To overcome this misconception, students need to rethink critically on their sensory judgments about shape and weight. In this viewpoint, examples of the visual illusions were introduced to let students be interested in their restriction of visual recognition. In this study, we hypothesized that discussions on the visual illusion are effective for elementary students to understand the volume and weight of objects correctly. Pre- and post-class tests showed that the students in the class who experienced the visual illusions and discussed them made significant improvements in overcoming misconception about the weight of the object.

The Collatz tree, or the directed graph of the 3x+1 problem, was demonstrated to be decomposed into three different subgraphs. An arbitrary positive integer is assigned uniquely to a specific position of the nodes of either of the three subgraphs. The manner of connecting a specific subgraph to its neighboring subgraphs is explicitly given. A comparison with complete chaos synchronization was made.

In 1968, Moser reported a new bifurcation through which the periodic orbits with period-3 or -4 appear. At present, this bifurcation is called the anomalous rotation bifurcation (ARB). The examples of ARB have been already known. Why the anomalous period-doubling bifurcation (APDB) of the elliptic fixed point does not happen in the area-preserving maps? In order to answer this question, we introduce the area preserving map T defined by Cⁿ (n ≥ 1) mapping function and derive the conditions that APDB happens.