The orthogonal projections of the Voronoi and Delone cells of root lattice A n onto the Coxeter plane display various rhombic and triangular prototiles including thick and thin rhombi of Penrose, Amman–Beenker tiles, Robinson triangles, and Danzer triangles to name a few. We point out that the symmetries representing the dihedral subgroup of order 2 h involving the Coxeter element of order h = n + 1 of the Coxeter–Weyl group a n play a crucial role for h -fold symmetric tilings of the Coxeter plane. After setting the general scheme we give samples of patches with 4-, 5-, 6-, 7-, 8-, and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice A 3 , whose Wigner–Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an h = 4 -fold symmetry.

In this work we evaluated the performance of first-year students of the College of Science and the College of Engineering at Sultan Qaboos University (SQU), Oman on the concepts of electricity and magnetism using the Brief Electricity and Magnetism Assessment (BEMA). The BEMA pre-test and post-test were implemented at the beginning of semester and toward the end of the semester respectively. We investigated the correlation between pre-test and post-test BEMA scores to offer a deeper insight into the students’ understanding of concepts. This study also examined the gain in individual subtopics and compared the BEMA results with the students’ final course grades. The findings indicate that students perform better on questions related to electostatics, struggle more with those related to magnetostatic and have significant difficulties with Faraday’s law. Additionally, a gender-based analysis was conducted, revealing no statistical difference in performance between male and female students.

Two Groups 2^3.〖PSL〗_2 (7) and 2^3:〖PSL〗_2 (7) of Order 1344

It is well known that the point group of the root lattice D₆ admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H₃, its roots, and weights are determined in terms of those of D₆. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D₆ determined by a pair of integers (m₁, m₂) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of H₃, and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers, which are linear combinations of the integers (m₁, m₂) with coefficients from the Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the <ABCK> octahedral tilings in 3D space with icosahedral symmetry H₃, and those related transformations in 6D space with D₆ symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in D₆. The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron”, and “C-polyhedron” generated by the icosahedral group have been discussed.