Kiyoko Matsunaga

Reversible Nets of Polyhedra

An example of reversible (or hinge inside-out transformable) figures is the Dudeney’s Haberdasher’s puzzle in which an equilateral triangle is dissected into four pieces, then hinged like a chain, and then is transformed into a square by rotating the hinged pieces. Furthermore, the entire boundary of each figure goes into the inside of the other figure and becomes the dissection lines of the other figure. Many intriguing results on reversibilities of figures have been found in prior research, but most of them are results on polygons. This paper generalizes those results to a wider range of general connected figures. It is shown that two nets obtained by cutting the surface of an arbitrary convex polyhedron along non-intersecting dissection trees are reversible. Moreover, a condition for two nets of an isotetrahedron to be both reversible and tessellative is given.

Generalization of Haberdasher's Puzzle

In this paper, we scrutinize the Haberdasher’s puzzle by Dudeney to produce equi-rotational pairs of figures systematically. We also generalize the puzzle by considering the tessellability condition (strong tessellability) for a pair of figures. As a result of it, it is shown that all pairs of strong tessellative and equi-rotational figures satisfy Conway criterion.

Reversibility and foldability of Conway tiles

Abstract In this paper, we proved that an arbitrary Conway tile is reversible to another Conway tile. We also determine all reversible pairs of figures, both of which tile the plane. Then we prove that the set of all nets of an isotetrahedron is closed under some reversible operation. Finally, we prove that a regular Conway tile is foldable into an isotetrahedron.

Universal Measuring Boxes

Since the old days of Japan, there were many ingenious convenient tools that applied mathematical ideas such as the right-angled scale (kanejaku), and the measuring box (masu). The kanejaku is a right-angled scale that was used to find the center and diameter of a circle (Fig. 11.1.1 (a)). On the other hand, the masu is a simple wooden box with no markings whose volume is, in this example, 6 dL. Using this box we can measure 1 dL, 2 dL, 3 dL, and so on in units of 1 dL up to 6 dL [1, 4].

Nets of Small Solids with Minimum Perimeter Lengths

In our city, the waste treatment center from each ward office sends a garbage truck every other day to the houses in the area to collect garbage. On those days, we sort out garbage into recyclable, non-burnable, burnable, etc.; and some of it, like papers and plastic bottles, are recycled. Also, we help the garbage collection ward by minimizing the volume of garbage containers. For example, if we put empty boxes into a garbage bag, we should flatten them to decrease the total volume. So, I now have a common question that we should consider on a daily basis, especially like for garbage. For a given paper polyhedron P, what is the most efficient way to make it flat? That is to say, how can we minimize the total length d(P) (or simply d) of segments along which the surface of P was cut to make a net of P? A net obtained in this manner is called a net with minimum perimeter length (NMPL), or a minimum perimeter net, for short. If we represent the perimeter length of P by l(P), then l(P) = 2d(P) holds.

Reversible Pairs of Figures

I have here two figures: one is a shrimp and the other is a bream (Fig. 4.1.1). In Japan, there is a well-known saying, “throw a shrimp to catch a bream” which has same meaning as “throw a sprat to catch a mackerel” in English.