Related papers: Exploring Mathematics with Curvagon Tiles
We look at sets of tiles that can tile any region of size greater than 1 on the square grid. This is not the typical tiling question, but relates closely to it and therefore can help solve other tiling problems -- we give an example of…
Although they are sometimes considered problematic to grasp by students, the concepts behind non-Euclidean geometry can be taught using astronomical images. By using photographs of the Moon taken with a smartphone through a simple…
Geometric Algebra and Calculus are mathematical languages encoding fundamental geometric relations that theories of physics seem to respect. We propose criteria given which statistics of expressions in geometric algebra are computable in…
Three types of geometric structure---grid triangulations, rectangular subdivisions, and orthogonal polyhedra---can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an…
Open cell foams have diverse industrial applications e.g. heat exchangers, structured reactors, filtration due to their unique properties such as high porosity and high specific surface area. In order to theoretically determine the…
The universal building block which is an essential part of all atomic structures on (1 1 0) silicon and germanium surfaces and their vicinals is proposed from first-principles calculations and a comparison of results with available…
We introduce a concept of an embedding of a quadratic space in an associative algebra. The general properties of such embeddings are analyzed by linking it to the Clifford algebra. Conversely, there isa simple description of the standard…
We discuss the art and science of producing conformally correct euclidean and hyperbolic tilings of compact surfaces. As an example, we present a tiling of the Chmutov surface by hyperbolic (2, 4, 6) triangles.
We ask the question of how small a self-assembling set of tiles can be yet have interesting computational behaviour. We study this question in a model where supporting walls are provided as an input structure for tiles to grow along: we…
Many mathematicians find mathematics aesthetically beautiful and even comparable to art forms such as music or painting. On the other hand, every year a great number of school students leave mathematics with total disillusionment and…
We present structures comprised of identical convex polyhedra which are interlocked geometrically. These sets cannot be disassembled by removing individual polyhedra by translations and/or rotations. The shapes that permit interlocking…
A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions,…
Infographics are a form of data visualization combining data, information, and statistics. Over the last ten years, infographics have become a popular method for displaying concise information, where infographics are a useful tool for…
This article is intended to an introductory lecture in material physics, in which the modern computational group theory and the electronic structure calculation are in collaboration. The effort of mathematicians in field of the group…
A combinatorial tiling of the sphere is naturally given by an embedded graph. We study the case that each tile has exactly five edges, with the ultimate goal of classifying combinatorial tilings of the sphere by geometrically congruent…
This paper proves the following statement: If a convex body can form a three or fourfold translative tiling in the three-dimensional space, it must be a parallelohedron. In other words, it must be a parallelotope, a hexagonal prism, a…
Constructivists (and intuitionists in general) asked what kind of mental construction is needed to convince ourselves (and others) that some mathematical statement is true. This question has a much more practical (and even cynical)…
Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the…
We consider tiles of some fixed size, with an associated weighting on the shapes of tile, of total mass 1. We study the pressure, $p$, of tilings with those tiles; the pressure, one over the volume times the logarithm of the partition…
In this paper we make some considerations about using a dynamic geometry software for teaching history of mathematica at university level. After a short introduction to the software GeoGebra, we discuss four activities. The first is an…