Micro Reuleaux Polygons Geometric Models

Abstract

In geometry, a Reuleaux polygon is a curve of constant width made up of circular arcs of constan  radius. These shapes are named after their prototypical example, the Reuleaux triangle, which in turn is named after 19th-century German engineer Franz Reuleaux.The Reuleaux triangle can be constructed from an equilateral triangle by connecting each pair of adjacent vertices with a circular arc centered on the opposing vertex, and Reuleaux polygons can be formed by a similar construction from any regular polygon with an odd number of sides as well as certain irregular polygons. Every curve of constant width can be accurately approximated by Reuleaux polygons. The Reuleaux polygons based on regular polygons are the only curves of constant width whose boundaries are formed by finitely many circular arcs of equal length. Sometimes in Reuleaux polygons, each curve of constant width may not have the same width but may have similar widths. This creates an irregular Reuleaux polygon. In other words; every  curve of constant width can be approximated arbitrarily closely by a (possibly irregular) Reuleaux polygon of the same width. It is possible to come across examples   polygons of Reuleaux in nature. According to Plateau's laws, the circular arcs in two-dimensional soap bubble clusters meet at 120° angles, the same angle found at the corners of a Reuleaux triangle. Based on this fact, it is possible to construct clusters in which some of the bubbles take the form of a Reuleaux triangle. The shape was first isolated in crystal form in 2014 as Reuleaux triangle disks. Basic bismuth nitrate disks with the Reuleaux triangle shape were formed from the hydrolysis and precipitation of bismuth nitrate in an ethanol–water system in the presence of 2,3-bis(2-pyridyl)pyrazine. In this study, we investigated the examples of Reuleaux polygons geometry models that we can easily see with the naked eye in nature  in micro structures. We have detected these geometric models in the cells where we can observe with the help of microscope. We saw that Some of these cells we examine are regular and inregular  reuleaux polygon. We took microscopic photos of these geometric models.