This paper describes the construction, properties and potential applications of a cartographic projection recently developed by the author, called the Zenithial Orthotriangular (ZOT) projection of an Octahedron. ZOT maps a planet to a plane by modelling it as an octahedron (a regular solid having 8 equilateral triangular facets), which is then unfolded and stretched to fit within a square. As d...

We study polyiamonds (polygons arising from the triangular grid) that fold into smallest yet unstudied platonic solid – octahedron. show a number of results. Firstly, we characterize foldable containing hole positive area, namely each but one polyiamond is foldable. Secondly, convex folds octahedron if and only it contains five polyiamonds. thirdly present sharp size bound: While there exist un...

2014 Regularly spaced singular lines appear when a cholesteric liquid crystal flows between glass plates inclined at a small wedge angle and remain after the flow ceases. The glass plates are treated to give homeotropic pinning at the surfaces, but the flow effectively creates a region of soft homogeneous pinning slightly away from the surfaces, thus simulating the conditions for development of...

This paper provides a partial generalization to matroid theory of the result of Halin and Jung that each simple graph with minimum vertex degree at least 4 has K5 or the octahedron K2,2,2 as a minor.

We give a simple proof based on symmetries that there are no geodesics from vertex to itself in the cube, tetrahedron, octahedron, and icosahedron.

We present an affine-invariant non-stationary subdivision scheme for the recursive refinement of any triangular mesh that is regular or has extraordinary vertices of valence 4. In particular, when applied to an arbitrary convex octahedron, it produces a G-continuous surface with a blob-like shape as the limit of the recursive subdivision process. In case of a regular octahedron, the subdivision...

Corpuscles are flexible geometric elements formed by regular triangles. Corpuscle elements can be joined, thus serving as building blocks for polyhedral chains and complex threedimensional structures. Here we describe a periodic three-dimensional corpuscle grid whose negative space consists of periodically arranged octahedra. As a recurrent local structure, the grid contains the octahedron-cent...

This study presents the first structural report of iranite, ideally CuPb10(CrO4)6(SiO4)2(OH)2 [copper decalead hexachromate bis(orthosilicate) dihydroxide], based on single-crystal X-ray diffraction data. Iranite is isomorphous with hemihedrite, with substitution of Cu for Zn and OH for F. The Cu atom is situated at the special position with site symmetry 1. The CrO4 and SiO4 tetrahedra and CuO...