Related papers: Trisections and Totally Real Origami
In this article we give explicit descriptions of the multiplicities of some classes of monomial ideals. For instance, we give a formula for the multiplicities of all codimension 1 monomial ideals, and another formula for the multiplicities…
We present an algorithm for the asymmetric traveling salesman problem on instances which satisfy the triangle inequality. Like several existing algorithms, it achieves approximation ratio O(log n). Unlike previous algorithms, it uses…
Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction…
It is often the case in mathematical analysis that solving an open problem can be facilitated by finding a new set of coordinates which may illumniate the known difficulties. In this article, we illustrate how to derive an assortment…
In this paper we prove that all irrational numbers from totally real cubic number fields are well approximable by rationals (i.e. the partial quotients in the continued fraction expansion of such a number are unbounded). This settles the…
In this article, we introduce an algorithm for automatic generation and categorization of triangle geometry theorems.
We describe an approach to the question of finding real solutions to problems of enumerative geometry, in particular the question of whether a problem of enumerative geometry can have all of its solutions be real. We give some methods 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…
We establish a Pythagorean theorem for the absolute values of the blocks of a partitioned matrix. This leads to a series of remarkable operator inequalities.
We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class). In fact, we prove a general upper bound on the number of perfect matchings in a…
We prove the complete intersection theorem and complete nontrivial-intersection theorem for systems of set partitions
Determining the number of (complex) realisations of a rigid graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we provide two new tools for determining realisation numbers in arbitrary…
Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line…
We prove some identities for the squares of generalized Tribonacci numbers. Various summation identities involving these numbers are derived.
In 2022, Olivier Longuet, a French mathematics teacher, created a game called the \textit{calissons puzzle}. Given a triangular grid in a hexagon and some given edges of the grid, the problem is to find a calisson tiling such that no input…
We show how Coxeter's work implies a bijection between complex reflection groups of rank two and real reflection groups in $O(3)$. We also consider this magic square of reflections and rotations in the framework of Clifford algebras: we…
We prove several hardness results on folding origami crease patterns. Flat-folding finite crease patterns is fixed-parameter tractable in the ply of the folded pattern (how many layers overlap at any point) and the treewidth of an…
In this paper, we extend the work of \cite{Chahal} in several directions. We first determine all Heron triangles that tightly circumscribe the unit circle and the associated $\tau$-congruent numbers generated by them. We then characterize…
Triangular decomposition is one of the standard ways to represent the radical of a polynomial ideal. A general algorithm for computing such a decomposition was proposed by A. Szanto. In this paper, we give the first complete bounds for the…
Magic squares are well-known arrangements of integers with common row, column, and diagonal sums. Various other magic shapes have been proposed, but triangles have been somewhat overlooked. We introduce certain triangular arrangements of…