Jin Ichi Itoh

Acute triangulations of the regular dodecahedral surface

In this paper we consider geodesic triangulations of the surface of the regular dodecahedron. We are especially interested in triangulations with angles not larger than @p/2, with as few triangles as possible. The obvious triangulation obtained by taking the centres of all faces consists of 20 acute triangles. We show that there exists a geodesic triangulation with only 10 non-obtuse triangles, and that this is best possible. We also prove the existence of a geodesic triangulation with 14 acute triangles, and the non-existence of such triangulations with less than 12 triangles.

Star Unfolding Convex Polyhedra via Quasigeodesic Loops

We extend the notion of star unfolding to be based on a quasigeodesic loop Q rather than on a point. This gives a new general method to unfold the surface of any convex polyhedron ℘ to a simple (nonoverlapping) planar polygon: cut along one shortest path from each vertex of ℘ to Q, and cut all but one segment of Q.

Acute Triangulations of the Regular Icosahedral Surface

Abstract We prove here that the surface of the regular icosahedron can be triangulated with 8 non-obtuse and with 12 acute triangles. We also show these numbers to be smallest possible.

Thaw: A Tool for Approximating Cut Loci on a Triangulation of a Surface

The cut locus from a point on the surface of a convex polyhedron is a tree containing a line segment beginning at every vertex. In the limit of infinitely small triangles, the cut locus from a point on a triangulation of a smooth surface therefore tends to become dense in the smooth surface, whereas the cut locus from the same point on the smooth surface is also a tree, but of finite length. We introduce a method for avoiding this problem. The method involves introducing a minimal angular resolution and discarding those points of the cut locus on the triangulation for which the angle measured between the shortest geodesic curves meeting at these points is smaller than the given angular resolution. We also describe software based upon this method that allows one to visualize the cut locus from a point on a surface of the form (x/a) n +(y/b)n +(z/c) n = 1, where n is a positive even integer. We use the software to support a new conjecture that the cut locus of a general ellipsoid is a subarc of a curvature line of the ellipsoid.

Affine equivalent classes of parallelohedra

In the paper the affine equivalence relation in the set of parallelohedra is studied. One proves the uniqueness theorem for a wide class of d-dimensional parallelohedra. From here it follows that for every d (≥2) the space of affine equivalent classes of d-dimensional primitive parallelohedra has dimension d(d+1)/2−1.

Acute triangulations of flat tori

In this paper, we investigate the acute triangulations of the family of flat tori. We prove that every flat torus can be triangulated into at most 16 acute triangles.

A Sard theorem for the distance function

Abstract. In this article, we prove that the set of all critical values of the distance function from a submanifold of a complete Riemannian manifold whose dimension is less than 5 is of Lebesgue measure zero.

Continuous flattening of platonic polyhedra

We prove that each Platonic polyhedron P can be folded into a flat multilayered face of P by a continuous folding process for polyhedra.

Continuous Flattening of Convex Polyhedra

A flat folding of a polyhedron is a folding by creases into a multilayered planar shape. It is an open problem of E. Demaine et al., that every flat folded state of a polyhedron can be reached by a continuous folding process. Here we prove that every convex polyhedron possesses infinitely many continuous flat folding processes. Moreover, we give a sufficient condition under which every flat folded state of a convex polyhedron can be reached by a continuous folding process.

A Gauss-Bonnet-type formula on Riemann-Finsler surfaces with nonconstant indicatrix volume

We prove a Gauss-Bonnet type formula for Riemann-Finsler surfaces of non-constant indicatrix volume and with regular piecewise smooth boundary. We give a Hadamard type theorem for N-parallels of a Landsberg surface.