Related papers: Algorithms and Geometric Constructions
Straightedge and compass construction problems are one of the oldest and most challenging problems in elementary mathematics. The central challenge, for a human or for a computer program, in solving construction problems is a huge search…
Geometry is essentially a global language, which is fully understood in different times, countries and cultures. The proof of a geometric theorem (e.g. the Pythagorean Theorem) or a geometric construction (e.g. the construction of an…
This article explores the limits of geometric construction using various tools, both classical and modern. Starting with ruler and compass constructions, we examine how adding methods such as origami, marked rulers (neusis), conic sections,…
It is well-known to be impossible to trisect an arbitrary angle and duplicate an arbitrary cube by a ruler and a compass. On the other hand, it is known from the ancient times that these constructions can be performed when it is allowed to…
We present a new kind of nontermination argument, called geometric nontermination argument. The geometric nontermination argument is a finite representation of an infinite execution that has the form of a sum of several geometric series.…
This note is purely expository. The statement of the Gauss theorem on the constructibility of regular polygons by means of compass and ruler is simple and well-known. However, its proofs given in most textbooks rely upon much unmotivated…
The proper Euclidean geometry is considered to be metric space and described in terms of only metric and finite metric subspaces (sigma-immanent description). Constructing the geometry, one does not use topology and topological properties.…
Impossible objects, geometric constructions that humans can perceive but that cannot exist in real life, have been a topic of intrigue in visual arts, perception, and graphics, yet no satisfying computer representation of such objects…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
It is well known that a center of a given circle cannot be constructed using only a straightedge and that this was proven by David Hilbert. Still it is not so clear what kind of object is proven to be non-existing. We analyze different…
We present a new kind of nontermination argument for linear lasso programs, called geometric nontermination argument. A geometric nontermination argument is a finite representation of an infinite execution of the form $(\vec{x} +…
Geometric predicates are a basic ingredient to implement a vast range of algorithms in computational geometry. Modern implementations employ floating point filtering techniques to combine efficiency and robustness, and state-of-the-art…
A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine (in the limit) topological spaces. If a pattern of the…
This is a simple way rigorously to construct Grassmann, Clifford and Geometric Algebras, allowing degenerate bilinear forms, infinite dimension, using fields or certain modules (characteristic 2 with limitation) - and characterize 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…
A problem that is simple to state in the context of spherical geometry, and that seems rather interesting, appears to have been unexamined to date in the mathematical literature. The problem can also be recast as a problem in the real…
We give a simple proof to the fact that it is impossible to use straightedge and compass to construct a triangle given the lengths of its internal bisectors, even if the triangle is isosceles.
While solving problems, if direct methods does not provide solution, indirect methods are explored. Today, we need an indirect method to solve the problem of angle trisection as the direct methods have been proved not to provide solutions.…
The problem of optimizing over random structures emerges in many areas of science and engineering, ranging from statistical physics to machine learning and artificial intelligence. For many such structures finding optimal solutions by means…
Orbifold equivalence is a notion of symmetry that does not rely on group actions. Among other applications, it leads to surprising connections between hitherto unrelated singularities. While the concept can be defined in a very general…