Problems concerning complete structural characterization of quasicrystals involve locating the atomic positions as well as determining the distribution of elements at each site. Quasicrystalline approximants provide models for potential building units of quasicrystals, but a clear determination of the elemental decorations in such approximants also remains incomplete. We report experimental and...

A lattice tetrahedron T ⊂ R3 is a tetrahedron whose four vertices are all in the lattice Z3. Lattice tetrahedra are preserved by those affine linear maps of the form ~v 7→ A~v +~b, such that A is an element of GL(3,Z) and ~b is an element of the lattice Z3. Such affine linear maps are called unimodular maps. We say that a lattice tetrahedron whose barycentre is its only non-vertex lattice point...

We present a procedure for bisecting a tetrahedron T successively into an infinite sequence of tetrahedral meshes ¡Tü , ZTX , ST1, ... , which has the following properties: (1) Each mesh !Tn is conforming. (2) There are a finite number of classes of similar tetrahedra in all the ¿Tn , n > 0. (3) For any tetrahedron T? in ¿T" , n(T") > cxn(T), where n is a tetrahedron shape measure and cx is a c...

The 8-tetrahedra longest-edge (8T-LE) partition of any tetrahedron is defined in terms of three consecutive edge bisections, the first one performed by the longest-edge. The associated local refinement algorithm can be described in terms of the polyhedron skeleton concept using either a set of precomputed partition patterns or by a simple edgemidpoint tetrahedron bisection procedure. An effecti...

In order to investigate the displacement of the IO4 tetrahedron in the ferroelectric phase II of pyridinium periodate in detail, we performed X-ray structural analysis with the use of the estimated site occupancies of carbon and nitrogen atoms at many temperatures. We confirmed the continuous increase of the rotation angle of the IO4 tetrahedron about the a-axis with decreasing temperature in t...

In this series of papers, we construct operators that extend certain given functions on the boundary of a tetrahedron into the interior of the tetrahedron, with continuity properties in appropriate Sobolev norms. These extensions are novel in that they have certain polynomial preservation properties important in the analysis of high order finite elements. This part of the series is devoted to i...

Abstract: Tetrahedron is the only 3-simplex convex polyhedron having four faces, and its shape has a wide application in science and technology. In this article, using graph theory and combinatorics, a study on a special type of tetrahedron called coxeter Andreev’s tetrahedron has been facilitated and it has been found that there are exactly one, four and thirty coxeter Andreev’s tetrahedrons h...

Two tetrahedra are called orthologic if the lines through vertices of one and perpendicular to corresponding faces of the other are intersecting. This is equivalent to the orthogonality of non-corresponding edges. We prove that the additional assumption of intersecting non-corresponding edges (“orthosecting tetrahedra”) implies that the six intersection points lie on a sphere. To a given tetrah...

We prove an embedding theorem that says a convex body can pass through a triangular hole∆ if and only if the convex body can be congruently embedded in a right triangular prism with base ∆. Combining this with a known result on congruent embeddings of a regular tetrahedron in a triangular prism, we show that a regular tetrahedron with unit edge can pass through an equilateral triangular hole in...

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, parallelo...