How Does Topology Help Solve the Inscribed Rectangle Problem by Proving that Every Jordan Curve Has 4 Vertices that Form a Rectangle?

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.

The Big Bang as the Creative Force of the Creation of the Universe

The paper considers the mechanism of the Big Bang energy influence on the creation of space-time fields of four structures of the Universe from the 1st type Ether (the Main Field and three spheres of the Relic). It explains how the Big Bang energy leads to the processes of “melting” in these structures, generating emergent properties that are different from their properties before the Big Bang. The key role of the Big Bang in completing the process of formation of 70% of DE is emphasized. It is shown that the Big Bang preceded the emergence of the furcation point, which chose several directions for the creation of cosmic matter—it was the combined efforts of these directions that created the visible worlds. The principle of dynamic equilibrium is considered the main criterion of the space-time field, in contrast to other physical fields, which is a necessary prerequisite for the quantization of the gravitational field. A spin particle is introduced, capable of emitting special particles—spitons, the characteristics of which are associated with the topology of the Mobius strip and determine the spinor properties of gravitational fields. The mechanism of interaction of particles of the 2nd type of Ether with the fields of space-time is described, allowing the creation of matter first and then the materiality of visible worlds. At the same time, the role of the “matter-negotiator” in the creation process of visible worlds of the Universe is especially highlighted. Since the new properties of gravitational fields go beyond Einstein’s standard theory of gravity, it is proposed to build a new theory of space-time that generalizes it and has a clear geometric interpretation. The proposed theory is based on the action built on a full set of invariants of the Ricci tensor. Within the framework of the Poincaré theory, the classification of furcation points is considered. The processes at the furcation point are described by the Gauss-Laplace curve, for which the principle of conservation of probability density is introduced when considering the transition at the furcation point to four different directions of development.

Synthesis,structure,and host-guest chemistry of a pair of isomeric selenanthrene-bridged molecular cages

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.