Related papers: Iterative Designs with Similar Triangles
This work presents a recursive construction for simple $t$-designs using resolutions of the ingredient designs. The result extends a construction of $t$-designs in our recent paper [39]. Essentially, the method in [39] describes the blocks…
Graphs are a central object of study in various scientific fields, such as discrete mathematics, theoretical computer science and network science. These graphs are typically studied using combinatorial, algebraic or probabilistic methods,…
There are four characteristic circles for each triangle on a plane. All for are tangential to the three straight lines containing the triangles' three sides. Three are exterior circles, the fourth is the in-circle. When the triangle is…
Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…
We prove that almost every triangle can be dissected only into $n^2$ triangles which have to be equal one another. Moreover, such a dissection is unique for every $n$. It turns out that to solve this "simple" problem it is convenient to use…
We use some fundamental ideas from complex analysis to create symmetric images and animations. Using a domain coloring algorithm, we generate mappings to the entire complex plane or the hyperbolic upper half-plane. The resulting designs can…
This work discusses an approach to solving geometric construction problems in which the given figure is included in a set ordered by construction steps. The flow of information is carried through the chain, allowing the original problem to…
The two basic equations satisfied by the parameters of a block design define a three-dimensional affine variety $\mathcal{D}$ in $\mathbb{R}^{5}$. A point of $\mathcal{D}$ that is not in some sense trivial lies on four lines lying in…
A matrix approach to continuous iteration is proposed for general formal series. It leads, in particular, to an order{to{order iteration of the exponential function, and consequently to an algorithmic approach to tetration. Lower{order…
A common representation of a three dimensional object in computer applications, such as graphics and design, is in the form of a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy…
3D shape creation and modeling remains a challenging task especially for novice users. Many methods in the field of computer graphics have been proposed to automate the often repetitive and precise operations needed during the modeling of…
Transformations in the field of computer graphics and geometry are one of the most important concepts for efficient manipulation and control of objects in 2-dimensional and 3-dimensional space. Transformations take many forms each with…
A tiling is a decomposition of a polygon into finitely many non-overlapping triangles. We prove that if a regular n-gon, $n \geq 5$, $n \neq 28$, can be tiled with similar right triangles, then one of the angles of these triangles is in…
One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry…
Consider the problem of constructing an experimental design, optimal for estimating parameters of a given statistical model with respect to a chosen criterion. To address this problem, the literature usually provides a single solution.…
We give three new proofs of the triangle inequality in Euclidean Geometry. There seems to be only one known proof at the moment. It is due to properties of triangles, but our proofs are due to circles or ellipses. We aim to prove the…
A geometric conception is a method of a geometry construction. The Riemannian geometric conception and a new T-geometric one are considered. T-geometry is built only on the basis of information included in the metric (distance between two…
Increasing reuse opportunities is a well-known problem for software designers as well as for hardware designers. Nonetheless, current software and hardware engineering practices have embraced different approaches to this problem. Software…
Buildings are beautiful mathematical objects tying a variety of subjects in algebra and geometry together in a very direct sense. They form a natural bridge to visualising more complex principles in group theory. As such they provide an…
This is a case study of teaching 3D design and 3D printing in a project-based computing course for undergraduate math majors. This article discusses content organization, implementation, project grading, and includes a personal reflection.…