The last chapter of the Disquisitiones of Gauss

This exposition reviews what exactly Gauss asserted and what did he prove in the last chapter of {\sl Disquisitiones Arithmeticae} about dividing the circle into a given number of equal parts. In other words, what did Gauss claim and actually prove concerning the roots of unity and the construction of a regular polygon with a given number of sides. Some history of Gauss's solution is briefly recalled, and in particular many relevant classical references are provided which we believe deserve to be better known.

Stress Superposition Method and Mechanical Analysis Properties of Regular Polygon Membranes

In this paper, the stress superposition method (SSM) is proposed to solve the stress distribution of regular polygon membranes. The stress-solving coefficient and the calculation formula of arbitrary point stress of regular polygon membrane are derived. The accuracy of the SSM for calculating stresses in regular polygonal membranes is verified by comparing the calculation results of the SSM with the finite element simulation results. This article is the first to propose a method to investigate the response of the arch height of the membrane curved edge to the membrane’s mechanical properties while keeping the effective area constant. It is found that the equivalent stress and the second principal stress at the midpoint of the membrane curved edge are effectively increased with the increase of the arch height of the curved edge. The second principal stress at the edge region of the membrane is relatively small, leading to the occurrence of wrinkles. When the stress at the midpoint of the curved edge is equal to that at the center of the membrane, the membrane plane attains the maximum stiffness and reduces the possibility of wrinkling at the edge.

Regulation of 2D DNA Nanostructures by the Coupling of Intrinsic Tile Curvature and Arm Twist

The overwinding and underwinding of duplex segments between junctions have been used in designing both left-handed and right-handed DNA origami nanostructures. For a variety of DNA tubes obtained from self-assembled tiles, only a theoretical approach of the intrinsic curvature of the DNA tile (specified as the intrinsic tile curvature) has been previously used to explain their formation. Details regarding the quantitative and structural descriptions of the tile curvature and its evolution in DNA tubes by the coupling of the twist of the inter-tile arm (specified as the arm twist) have never been addressed. In this work, we designed three types of tile cores built around a circular 128 nucleotide scaffold by using longitudinal weaving (LW), bridged longitudinal weaving (bLW) and transverse weaving (TW). Joining the tiles with inter-tile arms having the length of an odd number of DNA half-turns (termed O-tiling) almost resulted into planar 2D lattices, whereas joining the tiles with the arms having the length of an even number of DNA half-turns (termed E-tiling) nearly generated tubes. Streptavidin bound to biotin was used as a labeling technique to characterize the inside and outside surfaces of the E-tiling tubes and thereby the conformations of their component tiles with addressable concave and convex curvatures. When the arms have the normal winding at the relaxed B-form of DNA, the intrinsic tile curvature deter-mines the chirality of the E-tiling tubes. By regulating the arm length and the sticky end length of the bLW-Ep/q (E-tiling of the bLW cores with the arm length of p-bp and the sticky end length of q-nt) assemblies, the arm can be overwound, resulting in a left-handed twist, and can also be underwound, resulting in a right-handed twist. Chiral bLW-Ep/q tubes with either a right-handed curvature or a left-handed curvature can also be formed by the coupling of the intrinsic tile curvature and the arm twist. We were able to assign the chiral indices (n,m) to each tube using high-resolution AFM images, and therefore were able to estimate the tile curvature using a regular polygon model that approximated the transverse section of the tube. A deeper understanding of the integrated actions of dif-ferent types of twisting forces on the DNA tubes will be extremely helpful in engineering more elaborate DNA nanostructures in the future.

Rectangles and spirals

Every reader knows about the Golden Rectangle (see [1, pp. 85, 119], [2, 3]), and that it can be subdivided into a square and a smaller copy of itself, and that this process can be continued indefinitely, converging towards the intersection point of diagonals of any two successive rectangles in the sequence. The circumscribed logarithmic spiral passing through the vertices and converging to the same point is also familiar (see [3, 4]), and is analogous to the circumcircle of a regular polygon or a triangle. The approximate logarithmic spiral obtained by drawing a quarter-circle inside each of the squares is equally well known [3, p. 64]. Perhaps slightly less familiar is the inscribed spiral, which is tangential to a side of every rectangle, like the incircle of a triangle or a regular polygon. It does not (quite) coincide with the spiral passing through the point of subdivision of each side, as discussed in [3, pp. 73-77]. The Golden Rectangle, its subdivisions, and the circumscribed and inscribed spirals are illustrated in Figure 1.

Common Neighborhood Energy of Commuting Graphs of Finite Groups

The commuting graph of a finite non-abelian group G with center Z(G), denoted by Γc(G), is a simple undirected graph whose vertex set is G∖Z(G), and two distinct vertices x and y are adjacent if and only if xy=yx. Alwardi et al. (Bulletin, 2011, 36, 49-59) defined the common neighborhood matrix CN(G) and the common neighborhood energy Ecn(G) of a simple graph G. A graph G is called CN-hyperenergetic if Ecn(G)>Ecn(Kn), where n=|V(G)| and Kn denotes the complete graph on n vertices. Two graphs G and H with equal number of vertices are called CN-equienergetic if Ecn(G)=Ecn(H). In this paper we compute the common neighborhood energy of Γc(G) for several classes of finite non-abelian groups, including the class of groups such that the central quotient is isomorphic to group of symmetries of a regular polygon, and conclude that these graphs are not CN-hyperenergetic. We shall also obtain some pairs of finite non-abelian groups such that their commuting graphs are CN-equienergetic.

Applying 5E teaching model in recognizing regular polygons and rotations with the help of GeoGebra software

The 5E teaching model is a teaching model that gives students positive motivation. The 5E teaching model was proposed by Rodger W. Bybee and colleagues while working in the educational organization "Research on the curriculum framework for biology" in Colorado, the USA in 1987. The model is built based on the constructivist theory of learning, knowledge is formed through the experience and practice of learners. 5E stands for 5 words: Engagement - Exploration - Explanation - Elaboration - Evaluation, representing the five steps of the 5E teaching process. Our study aims to apply the 5E teaching model in regular polygon recognition and rotation with the help of GeoGebra software. With the theoretical research method and the experimental method for 102 students, including 52 students in the experimental class and 50 students in the control class, the study has shown that students who learn according to the 5E teaching model have better results than students who do not study this model. In addition, the research results also show that the 5E teaching model with the support of GeoGebra software in teaching regular polygon and rotation recognition has many advantages, developing learners' capacity and quality. Learners show much more interest and excitement than traditional teaching. Students develop their self-study and self-discovery abilities. The findings from the research results applying the 5E model are of great significance to teaching today.

Asymptotics of the Spectrum of a Periodic Boundary ValueProblem for an Odd-Order Differential Operator

The spectrum of a differential operator of high odd order with periodic boundary conditions is studied. The asymptotics of the fundamental system of solutions of the differential equation defining the operator are obtained by the method of successive Picard approximations. With the help of this fundamental system of solutions the periodic boundary conditions are studied. As a result, the equation for the eigenvalues of the differential operator under study is obtained, which is a quasi-polynomial. The indicator diagram of this equation, which is a regular polygon, is investigated. In each of the sectors of the complex plane, defined by the indicator diagram, the asymptotics of the eigenvalues of the operator under study is found. An equation for the eigenvalues of the differential operator under study is derived. The indicator diagram of this equation has been studied. The asymptotics of the eigenvalues of the studied operator in different sectors of the indicator diagram is found.

Regulation of 2D DNA Nanostructures by the Coupling of Tile Curvatures and Arm Twists

DNA overwinding and underwinding between adjacent Holliday junctions have been applied in DNA origami constructs to design both left-handed and right-handed nanostructures. For a variety of DNA tubes assembled from small tiles, only a theoretical approach of the intrinsic tile curvature was previously used to explain their formation. Details regarding the quantitative and structural descriptions of the intrinsic tile curvature and its evolution in DNA tubes by coupling with arm twists were missing. In this work, we designed three types of tile cores from a circular 128 nucleotide scaffold by longitudinal weaving (LW), bridging longitudinal weaving (bLW), and transverse weaving (TW) and assembled their 2D planar or tubular nanostructures via inter-tile arms with a distance of an odd or even number of DNA half-turns. The biotin/streptavidin (SA) labeling technique was applied to define the tube configuration with addressable inside and outside surfaces and thus their component tile conformation with addressable concave and convex curvatures. Both chiral tubes possessing left-handed and right-handed curvatures could be generated by finely tuning p and q in bLW-E_p/q designs (bLW tile cores joined together by inter-tile arms of an even number of half-turns with the arm length of p base pairs (bp) and the sticky end length of q nucleotides (nt)). We were able to assign the chiral indices (n,m) to each specific tube from the high-resolution AFM images, and thus estimated the tile curvature angle with a regular polygon model that approximates each tube’s transverse section. We attribute the curvature evolution of bLW-E_p/q tubes composed of the same tile core to the coupling of the intrinsic tile curvature and different arm twists. A better understanding of the integrated actions of different types of twisting forces on DNA tubes will be much more helpful in engineering DNA nanostructures in the future.

Regulation of 2D DNA Nanostructures by the Coupling of Tile Cur-Vatures and Arm Twists

DNA overwinding and underwinding between adjacent Holliday junctions have been applied in DNA origami constructs to design both left-handed and right-handed nanostructures. For a variety of DNA tubes assembled from small tiles, only an abstract concept of the intrinsic tile curvature was previously used to explain their formation. Details regarding the quantitative and structural descriptions of the intrinsic tile curvature and its evolution in DNA tubes by coupling with arm twists have been lacking. In this work, we designed three types of tile cores from a circular 128 nucleotide scaffold by longitudinal weaving (LW), bridging longitudinal weaving (bLW), and transverse weaving (TW) and assembled their 2D planar or tubular nanostructures via inter-tile arms with a distance of an odd or even number of DNA half-turns. The biotin/streptavidin (SA) labeling technique was applied to define the tube configuration with addressable inside and outside surfaces and thus their component tile conformation with addressable concave and convex curvatures. Both chiral tubes possessing left-handed and right-handed curvatures could be generated by finely tuning p and q in bLW-E_p/q designs (bLW tile cores joined together by inter-tile arms of even number of half-turns with the arm length of p base pairs (bp) and the sticky end length of q nucleotides (nt)). We were able to assign the chiral indices (n,m) to each specific tube from the high-resolution AFM images, and thus estimated the tile curvature angle with a regular polygon model that approximates each tube’s transverse section. We attribute the curvature evolution of bLW-E_p/q tubes composed of the same tile core to the coupling of the intrinsic tile curvature and different arm twists. A better understanding of integrated actions of different types of twisting forces on DNA tubes will be much more helpful in engineering DNA nanostructures in the future.

Regulation of 2D DNA Nanostructures by the Coupling of Tile Cur-Vatures and Arm Twists

DNA overwinding and underwinding between adjacent Holliday junctions have been applied in DNA origami constructs to design both left-handed and right-handed nanostructures. For a variety of DNA tubes assembled from small tiles, only an abstract concept of the intrinsic tile curvature was previously used to explain their formation. Details regarding the quantitative and structural descriptions of the intrinsic tile curvature and its evolution in DNA tubes by coupling with arm twists have been lacking. In this work, we designed three types of tile cores from a circular 128 nucleotide scaffold by longitudinal weaving (LW), bridging longitudinal weaving (bLW), and transverse weaving (TW) and assembled their 2D planar or tubular nanostructures via inter-tile arms with a distance of an odd or even number of DNA half-turns. The biotin/streptavidin (SA) labeling technique was applied to define the tube configuration with addressable inside and outside surfaces and thus their component tile conformation with addressable concave and convex curvatures. Both chiral tubes possessing left-handed and right-handed curvatures could be generated by finely tuning p and q in bLW-E_p/q designs (bLW tile cores joined together by inter-tile arms of even number of half-turns with the arm length of p base pairs (bp) and the sticky end length of q nucleotides (nt)). We were able to assign the chiral indices (n,m) to each specific tube from the high-resolution AFM images, and thus estimated the tile curvature angle with a regular polygon model that approximates each tube’s transverse section. We attribute the curvature evolution of bLW-E_p/q tubes composed of the same tile core to the coupling of the intrinsic tile curvature and different arm twists. A better understanding of integrated actions of different types of twisting forces on DNA tubes will be much more helpful in engineering DNA nanostructures in the future.