Related papers: Morikawa's Unsolved Problem

Between 17th and 19th centuries, mathematically orientated votive tablets appeared in Shinto shrines and Buddhist temples all over Japan. Known as sangaku, they contained problems of a largely geometrical nature. In the 17th century, the…

Eighteenth century Japan was a time of isolation and peace, where education and the arts blossomed. Originally posted before 1749 by an unknown author, the sangaku (mathematical tablet) that became known as the Gion Shrine problem, has…

This paper demonstrates how a nineteenth century Japanese votive temple problem known as a sangaku from Okayama prefecture can be solved using traditional mathematical methods of the Japanese Edo (1603-1868 CE). We compare a modern solution…

This work presents for the first time a solution to the 1821 unsolved Sawa Masayoshi's problem, giving an explicit and algebraically exact solution for the symmetric case (particular case b = c, i.e., ABC \equiv right-angled isosceles…

Let $\mu(r)$ denote the minimal side length of a square inscribed in the curvilinear triangular region formed by two tangent circles of radii $1$ and $r \ge 1$ together with their common tangent line. The problem of finding a closed-form…

We give a new solution to the famous Gion shrine geometry problem from eighteenth-century Japan. Like the classical Japanese solution, ours is given in the form of a degree ten equation. However, our polynomial has the advantage of being…

The inverse problem of Galois Theory was developed in the early 1800 s as an approach to understand polynomials and their roots. The inverse Galois problem states whether any finite group can be realized as a Galois group over Q (field of…

As Jordan observed in 1870, just as univariate polynomials have Galois groups, so do problems in enumerative geometry. Despite this pedigree, the study of Galois groups in enumerative geometry was dormant for a century, with a systematic…

While geometry with transcendental curves, like the Quadratrix of Hippias and the Spiral of Archimedes, played a significant role in our modern developments of geometry and algebra. The investigation has fallen off in the modern era despite…

Details for known solutions of some geometric and algebraic problems with the help of origami are presented: two theorems of Haga, the general cubic equation, especially the heptagon equation, doubling the cube as well as the trisection of…

This article is concerned with an example of complex planar geometry arising from flat origami challenges. The complexity of solution algorithms is illustrated, depending on the depth of the initial analysis of the problem, starting from…

In this piece, we examine one variant of the infamous 15 Tile Puzzle and develop a mathematical backing behind why it is unsolvable. Using concepts of permutations, bijectivity, and cycle transpositions, we not only prove how to model this…

Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Starting from the same core idea…

We generalize problems in Wasan geometry which involve no folded figures but are related to Haga's fold in origami. Using the tangent circles appeared in those problems we give a parametric representation of the generalized Haga's fold…

I give another proof of the geometric Satake equivalence from Mirkovic-Vilonen over a separably closed field. Using Galois descent, I obtain a canonical construction of the Galois form of the full $L$-group.

In this paper, we prove the non-existence of certain semistable Galois representations of a number field. Our consequence can be applied to some geometric problems. For example, we prove a special case of a Conjecture of Rasmussen and…

Fifteen years after their discovery, ample fields now stand at the center of research in contemporary Galois theory and attract more and more attention also from other areas of mathematics. This survey gives an introduction to the theory of…

In this paper we give a unified approach in categorical setting to the problem of finding the Galois closure of a finite cover, which includes as special cases the familiar finite separable field extensions, finite unramified covers of a…

We prove a variant of the Sylvester-Gallai theorem for cubics (algebraic curves of degree three): If a finite set of sufficiently many points in $\mathbb{R}^2$ is not contained in a cubic, then there is a cubic that contains exactly nine of…

Galois theory is developed using elementary polynomial and group algebra. The method follows closely the original prescription of Galois, and has the benefit of making the theory accessible to a wide audience. The theory is illustrated by a…