Gisaku Nakamura

Dudeney Dissection of Polygons

Given an equilateral triangle A and A Square B of the same area, Henry E. Dudeney introduced A partition of A into parts that tan be reassembled in some way, without turning over the surfaces, to form B. An examination of Dudeney’s method of partition motivates us to introduce the notion of Dudeney dissections of various polygons to other polygons.

A solution of Dudeney's round table problem for an even number of people

Abstract Dudeneys round table problem asks for a set of Hamiltonian cycles in the complete graph Kn with the property that every 2-path lies on exactly one of the cycles. In this paper, we construct a solution of the problem when n is any even number.

Enumeration of Polyominoes, Polyiamonds and Polyhexes for Isohedral Tilings with Rotational Symmetry

We describe computer algorithms that can enumerate and display, for a given n > 0 (in theory, of any size), all n -ominoes, n -iamonds, and n -hexes that can tile the plane using only rotations; these sets necessarily contain all such tiles that are fundamental domains for p4, p3, and p6 isohedral tilings. We display the outputs for small values of n . This expands on earlier work [3].

Radial Perfect Partitions of Convex Sets in the Plane

In this paper we study the following problem: how to divide a cake among the children attending a birthday party such that all the children get the same amount of cake and the same amount of icing. This leads us to the study of the following. A perfect k-partitioning of a convex set S is a partitioning of S into k convex pieces such that each piece has the same area and \(\frac{1}{k}\) of the perimeter of S . We show that for any k, any convex set admits a perfect k-partitioning. Perfect partitionings with additional constraints are also studied.

Dudeney's round table problem

Abstract A set of Hamilton cycles in the complete graph on n vertices is called a Dudeney set, and denoted D ( n ), if every path of length two lies on exactly one of the cycles. In this paper it is shown that: 1. (a) There is a Dudeney set D ( p + 2) if p is prime and 2 is a generator of the multiplicative subgroup of GF( p ). (b) If there is a Dudeney set D ( n + 1), then there is a Dudeney set D (2 n ). (c) For n ⩽ 50, the only n for which the existence of a Dudeney set D ( n ) remains in doubt are n ϵ {27, 29, 35, 37, 41, 47}.

Convex developments of a regular tetrahedron

The best-known developments of a regular tetrahedron are an equilateral triangle and a parallelogram. Are there any other convex developments of a regular tetrahedron? In this paper we will show that there are convex developments of a regular tetrahedron having the following shapes: an equilateral triangle, an isosceles triangle, a right-angled triangle, scalene triangles, rectangles, parallelograms, trapezoids, quadrilaterals which are not trapezoids, pentagons and hexagons, and furthermore these cases exhaust all the possibilities of convex developments with sides n= =7. Here, we mean by a development of a polyhedron a connected plane figure, from which one can construct the polyhedron by folding it without getting overlap or gap. In so folding we do not require that the sides of the development should end up as the edges of the polyhedron.

Congruent Dudeney Dissections of Triangles and Convex Quadrilaterals – All Hinge Points Interior to the Sides of the Polygons

Let α and β be polygons with the same area. A Dudeney dissectionofα toβ is a partition of α into parts which can be reassembled to produceβ in the following way. Hinge the parts ofα like a chain along the perimeter of α, then fix one of the parts to formβ with the perimeter of α going into its interior and with its perimeter consisting of the dissection lines in the interior of α, without turning the pieces over. In this paper we discuss a special type of Dudeney dissection of triangles and convex quadrilaterals in which α is congruent toβ and call it acongruent Dudeney dissection. In particular, we consider the case where all hinge points are interior to the sides of the polygonα and β. For this case, we determine all triangles and convex quadrilaterals which have congruent Dudeney dissections.

Exact coverings of 2-paths by Hamilton cycles

We construct a C(2m, 2m, 2) design which is a family of Hamilton cycles in K2m so that each 2-path of K2m lies in exactly two of the cycles.

Foldings of regular polygons to convex polyhedra i: equilateral triangles

To fold a regular n-gon into a convex polyhedron is to form the polyhedron by gluing portions of the perimeter of the n-gon together, i.e. the n-gon is a net of the polyhedron. In this paper we identify all convex polyhedra which are foldable from an equilateral triangle.

Resolvable coverings of 2-paths by 4-cycles

Abstract A C(n, k, λ) design is a family of k-cycles in Kn in which each 2-path of Kn occurs exactly λ times. We construct a resolvable C(n, 4, 1) design when n ≡ 0 (mod 4) and a near-resolvable C(n, 4, 1) design when n ≡ 2 (mod 4).