Related papers: A Hierarchy of Geometric Constructions
It is well known that the set of origami constructible numbers is larger than the classical straight-edge and compass constructible numbers. However, the Huzita-Justin-Hatori origami constructible numbers remain algebraic so that the…
We give a hierarchial set of axioms for mathematical origami. The hierachy gives the fields of Pythagorean numbers, first discussed by Hilbert, the field of Euclidean constructible numbers which are obtained by the usual constructions of…
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…
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…
It is well known that several classical geometry problems (e.g., angle trisection) are unsolvable by compass and straightedge constructions. But what kind of object is proven to be non-existing by usual arguments? These arguments refer to…
Origami is an ancient art that continues to yield both artistic and scientific insights to this day. In 2012, Buhler, Butler, de Launey, and Graham extended these ideas even further by developing a mathematical construction inspired by…
Methods were developed in Ref. [1] for constructing reference metrics (and from them differentiable structures) on three-dimensional manifolds with topologies specified by suitable triangulations. This note generalizes those methods by…
Geometric number systems, obtained by extending the real number system to include new anticommuting square roots of +1 and -1, provide a royal road to higher mathematics by largely sidestepping the tedious languages of tensor analysis and…
Computing the crossing number of a graph is one of the most classical problems in computational geometry. Both it and numerous variations of the problem have been studied, and overcoming their frequent computational difficulty is an active…
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…
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…
The purpose of this short manuscript is to show that all point constructions that can be done via ruler and compass, can also be done with compass exclusively. If we are using compass and ruler the way we construct new points is by first…
Let $G$ be a connected undirected graph on $n$ vertices with no loops but possibly multiedges. Given an arithmetical structure $(\textbf{r}, \textbf{d})$ on $G$, we describe a construction which associates to it a graph $G'$ on $n-1$…
We introduce a new method of generating Computer Aided Design (CAD) profiles via a sequence of simple geometric constructions including curve offsetting, rotations and intersections. These sequences start with geometry provided by a…
Kirigami, the creative art of paper cutting, is a promising paradigm for mechanical metamaterials. However, to make kirigami-inspired structures a reality requires controlling the topology of kirigami to achieve connectivity and rigidity.…
In the making of origami, one starts with a piece of paper, and through a series of folds along seed points one constructs complicated three-dimensional shapes. Mathematically, one can think of the complex numbers as representing the piece…
We develop a theoretical study of non-terminating hypergeometric summations with one free parameter. Composing various methods in complex and asymptotic analysis, geometry and arithmetic of certain transcendental curves and rational…
After a short review of the Method of Recursive Counting we introduce a general algebraic description of recursive lattice building. This provides a rigorous framework for discussion of method's limitations.
We describe first steps towards a system for automated triangle constructions in absolute and hyperbolic geometry. We discuss key differences between constructions in Euclidean, absolute and hyperbolic geometry, compile a list of primitive…
All constructive methods employed in modern mathematics produce only countable sets, even when designed to transcend countability. We show that any constructive argument for uncountability -- excluding diagonalization techniques --…