When we stare into our complex surroundings, we see objects of different shapes and sizes. However, the shape that is always present, regardless of the complexity of the object, is the circle. The circle is arguably t...When we stare into our complex surroundings, we see objects of different shapes and sizes. However, the shape that is always present, regardless of the complexity of the object, is the circle. The circle is arguably the most fascinating shape in the universe. A circle is defined as the set of all points equidistant from a given point, which, therefore, lies at the center of the circle. One of the various properties of circles is that it has infinite inscribed squares. This is because it is a continuous function, therefore if any point in the circle is transitioned by a specific factor, the other related points in the square would be shifted by the same factor. An ellipse is a similar shape with several inscribed squares. But does any closed curve have an inscribed square? This question was proposed by Otto Toeplitz in 1911 and to this day it is not answered. Another version of this problem is the inscribed rectangle problem which will be discussed in this paper.展开更多
A pair of selenanthrene-bridged molecular cages have been constructed through a one-step cyclization reaction of a tetrakis(iodo) crown ether with selenium powder. The tubular belt-shaped cage has an intrinsic cavity ...A pair of selenanthrene-bridged molecular cages have been constructed through a one-step cyclization reaction of a tetrakis(iodo) crown ether with selenium powder. The tubular belt-shaped cage has an intrinsic cavity which can adaptively transform to accommodate electron-deficient guests forming [2]pseudorotaxane complexes. The other product was determined to be an isomeric cage featuring a Mobius strip structure, which exhibits slower twist-migration dynamics than its thianthrene counterpart. The success of using selenanthrene as joints enables an alternative way to structural design and property regulation of molecular cages.展开更多
文摘When we stare into our complex surroundings, we see objects of different shapes and sizes. However, the shape that is always present, regardless of the complexity of the object, is the circle. The circle is arguably the most fascinating shape in the universe. A circle is defined as the set of all points equidistant from a given point, which, therefore, lies at the center of the circle. One of the various properties of circles is that it has infinite inscribed squares. This is because it is a continuous function, therefore if any point in the circle is transitioned by a specific factor, the other related points in the square would be shifted by the same factor. An ellipse is a similar shape with several inscribed squares. But does any closed curve have an inscribed square? This question was proposed by Otto Toeplitz in 1911 and to this day it is not answered. Another version of this problem is the inscribed rectangle problem which will be discussed in this paper.
基金National Natural Science Foundation of China(Nos.21971268,22171295)the Program for Guangdong Introducing Innovative and Entrepreneurial Teams(No.2017ZT07C069)+1 种基金Pearl River Talent Program(No.2017GC010623)the Starry Night Science Fund of Zhejiang University Shanghai Institute for Advanced Study(No.SN-ZJU-SIAS-006)for financial support.
文摘A pair of selenanthrene-bridged molecular cages have been constructed through a one-step cyclization reaction of a tetrakis(iodo) crown ether with selenium powder. The tubular belt-shaped cage has an intrinsic cavity which can adaptively transform to accommodate electron-deficient guests forming [2]pseudorotaxane complexes. The other product was determined to be an isomeric cage featuring a Mobius strip structure, which exhibits slower twist-migration dynamics than its thianthrene counterpart. The success of using selenanthrene as joints enables an alternative way to structural design and property regulation of molecular cages.
