Flat Polygons Research Articles

Association of two square difference identity to regular polygons and circles

GeoGebra is a dynamic software that is frequently used and of increasing importance in mathematics teaching processes in our digital age. Accordingly, in this study a new perspective has been brought to the proofs of the “two square difference identity” expressed for the square, which is a flat polygon, made with different approaches. With side lengths a, b, and a>b, it has been shown that the identity given by the equation (difference of area) a<sup>2</sup>-b<sup>2</sup>=(a-b)(a+b) is true for other regular polygons as well. In the study, direct proof method was used within the framework of the principle of conservation of measure, which is one of the basic principles of geometry teaching. GeoGebra program, which is a dynamic geometry software, was preferred for drawing geometric shapes used in proofs. In order to generalize the number n, a different fragmentation technique was preferred to the proofs made using different drawings for equilateral triangle and square, which are the simplest regular polygons. It has also been shown that this identity is true for circles viewed as polygons with an infinite number of sides.

Effects of cell morphology, physiology, biochemistry and CHS genes on four flower colors of Impatiens uliginosa.

Flower color is one of the important ornamental traits in the plants, which plays an active role in attracting pollinators to pollinate plants and reproduce their offspring. The flower color of Impatiens uliginosa is rich, there are four main flower colors in nature: deep red, red, pink, and white. However, it remains unclear whether on four different flower colors mechanism of I. uliginosa. We investigate colorimetric measurement, observation of epidermal cells, cellular pH determination, extraction and determination of total anthocyanins and flavonoid, semi-quantitative determination of pigment components, and gene cloning and qRT-PCR of CHS genes to study four flower colors of I. uliginosa. The L* and b* values were the highest in white flower, while the a* values were the highest in pink flower. The same shape of epidermal cells was observed in different flower colors, which was all irregular flat polygons, and there were partial lignification. Their cellular pH values were weakly acidic, while the pH values of the deep red flower was the highest and the white flower was the lowest. The highest pigment content of the four flower colors was total anthocyanin content. And malvidin-3-galactosidechloride (C23H25ClO12), cyanidin-3-O-glucoside (C21H21O11) and delphinidin (C15H11O7) were the main pigment components affecting the color of four different flower colors. The anthocyanin synthesis gene IuCHS was expressed in four flowers, and all three copies of it had the highest expression level in pink flower and the lowest expression level in white flower. These results revealed the influence of main internal factors on four different flower colors of I. uliginosa, and provided a basis for further understanding of the intracellular and molecular regulatory mechanisms of flower color variation, and laid a foundation for the improvement of flower color breeding of Impatiens.

Shape reconfiguring bistable structures using heat activated fibers

Engineering structures have increasingly begun to utilize mechanical instability in areas such as energy absorption, shape reconfiguration, and vibration attenuation. Most multi-stable mechanisms use 1D architectures such as pre-curved beams or inclined trusses to achieve multiple strain energy minima. These structures pose several limitations, (1) they generally require external support for functionality, (2) they often balance the competing demands of large stiffness and large amplitude in shape change, and (3) they are inherently less energetically dense compared to shell-based structures as there is negligible stretching in the material. In this work, we introduce a novel class of 4D printed bistable shell structures that morph from flat surfaces. We investigate both the formation of the structural geometry, and the mechanics of the shells once morphed. Following the precepts of 4D printing, the structural geometry is achieved in two stages. First 3D printing is used to fabricate flat polygon unit cells which are then morphed into symmetrical bistable shells. Utilizing Fused Filament Fabrication, a viscoelastic material is pre-strained as it is deposited on the print bed. By tuning the printing parameters, a predictable 2D to 3D shape morphing is triggered with heat, resulting in a height increase of up to 22.9 times the original size. Some resulting unit cells exhibit an elastic bistable response with a high peak force from a point indentation. Shape morphing and structural bistability is predicted using Finite-Element Analysis with prescribed pre-strain aligned with the fiber extrusion direction. The resulting geometries are analyzed for their mechanical behavior. Buckling simulations show good agreement with experimental results. To demonstrate an energy-dissipating metamaterial, we fabricate a planar meta-surface tiled using triangular unit cells to create an auxetic pattern. This sheet morphs to conform to a doubly curved surface globally, with each unit-cell transforming to a bistable shell locally. With this approach, we create a new generation of 4D printed bistable structures that can be applied to a variety of engineering applications.

Random sequential adsorption of aligned regular polygons and rounded squares: Transition in the kinetics of packing growth.

We study two-dimensional random sequential adsorption (RSA) of flat polygons and rounded squares aligned in parallel to find a transition in the asymptotic behavior of the kinetics of packing growth. Differences in the kinetics for RSA of disks and parallel squares were confirmed in previous analytical and numerical reports. Here, by analyzing the two classes of shapes in question we can precisely control the shape of the packed figuresand thus localize the transition. Additionally, we study how the asymptotic properties of the kinetics depend on the packing size. We also provide accurate estimations of saturated packing fractions. The microstructural properties of generated packings are analyzed in terms of the density autocorrelation function.

Sublimation-driven convection in Sputnik Planitia on Pluto.

Sputnik Planitia is a nitrogen-ice-filled basin on Pluto1. Its polygonal surface patterns2 have been previously explained as a result of solid-state convection with either an imposed heat flow3 or a temperature difference within the 10-km-thick ice layer4. Neither explanation is satisfactory, because they do not exhibit surface topography with the observed pattern: flat polygons delimited by narrow troughs5. Internal heating produces the observed patterns6, but the heating source in such a setup remains enigmatic. Here we report the results of modelling the effects of sublimation at the surface. We find that sublimation-driven convection readily produces the observed polygonal structures if we assume a smaller heat flux (~0.3 mW m-2) at the base of the ice layer than the commonly accepted value of 2-3 mW m-2 (ref. 7). Sustaining this regime with the latter value is also possible, but would require a stronger viscosity contrast (~3,000) than the nominal value (~100) considered in this study.

Классификация растительности массивов байджарахов в двух районах подзоны арктических тундр Сибирского сектора Российской Арктики

Классификация растительности массивов байджарахов в двух районах подзоны арктических тундр Сибирского сектора Российской Арктики

Fluid shear stress induces epithelial-mesenchymal transition (EMT) in Hep-2 cells.

Laryngeal squamous cell carcinoma (LSCC) is one of the most commonly diagnosed malignancies with high occurrence of tumor metastasis, which usually exposes to fluid shear stress (FSS) in lymphatic channel and blood vessel. Epithelial-mesenchymal transition (EMT) is an important mechanism that induces metastasis and invasion of tumors. We hypothesized that FSS induced a progression of EMT in laryngeal squamous carcinoma. Accordingly, the Hep-2 cells were exposed to 1.4 dyn/cm2 FSS for different durations. Our results showed that most of cells changed their morphology from polygon to elongated spindle with well-organized F-actin and abundant lamellipodia/filopodia in protrusions. After removing the FSS, cells gradually recovered their flat polygon morphology. FSS induced Hep-2 cells to enhance their migration capacity in a time-dependent manner. In addition, FSS down-regulated E-cadherin, and simultaneously up-regulated N-cadherin, translocated β-catenin into the nucleus. These results confirmed that FSS induced the EMT in Hep-2 cells, and revealed a reversible mesenchymal-epithelial transition (MET) process when FSS was removed. We further examined the time-expressions of signaling cascades, and demonstrated that FSS induces the EMT and enhances cell migration depending on integrin-ILK/PI3K-AKT-Snail signaling events. The current study suggests that FSS, an important biophysical factor in tumor microenvironment, is a potential determinant of cell behavior and function regulation.

Primitive Normality and Additivity of Free Projective and Strongly Flat Polygons

We study into monoids S over which the axiomatizable class of all free, projective, or strongly flat S-polygons is primitive normal or additive. It is proved that an axiomatizable class of all free, projective, or strongly flat S-polygons is primitive normal iff a monoid S primitive normal. Also it is shown that an axiomatizable class of all free, projective, or strongly flat S-polygons is not additive for any monoid S.

A Fractal Nature for Polymerized Laminin

Polylaminin (polyLM) is a non-covalent acid-induced nano- and micro-structured polymer of the protein laminin displaying distinguished biological properties. Polylaminin stimulates neuritogenesis beyond the levels achieved by ordinary laminin and has been shown to promote axonal regeneration in animal models of spinal cord injury. Here we used confocal fluorescence microscopy (CFM), scanning electron microscopy (SEM) and atomic force microscopy (AFM) to characterize its three-dimensional structure. Renderization of confocal optical slices of immunostained polyLM revealed the aspect of a loose flocculated meshwork, which was homogeneously stained by the antibody. On the other hand, an ordinary matrix obtained upon adsorption of laminin in neutral pH (LM) was constituted of bulky protein aggregates whose interior was not accessible to the same anti-laminin antibody. SEM and AFM analyses revealed that the seed unit of polyLM was a flat polygon formed in solution whereas the seed structure of LM was highly heterogeneous, intercalating rod-like, spherical and thin spread lamellar deposits. As polyLM was visualized at progressively increasing magnifications, we observed that the morphology of the polymer was alike independently of the magnification used for the observation. A search for the Hausdorff dimension in images of the two matrices showed that polyLM, but not LM, presented fractal dimensions of 1.55, 1.62 and 1.70 after 1, 8 and 12 hours of adsorption, respectively. Data in the present work suggest that the intrinsic fractal nature of polymerized laminin can be the structural basis for the fractal-like organization of basement membranes in the neurogenic niches of the central nervous system.

Purification and characterization of adult optic nerve head astrocytes

To establish a method of purifying and characterizing adult astrocytes from optic nerve head (ONH). Experimental study. The lamina cribrosa tissue from ONH of human eye was isolated under anatomic microscopy, and then 4 to 6 little explants were incubated in each culture plate containing culture medium DMEM/F12. After 8 to 10 weeks, the cells were removed by digesting cells with 0.25% trypsogen. Selective astrocyte culture medium is subsequently used. After two passages, astrocytes were identified by the observation of cell morphology and immunofluorescent staining of GFAP and NCAM. After 2 to 3 weeks of explants planting, cells showed an obvious migration procession by crawling in succession from the verge of the explants and rapidly splitting. Most cells displayed a flat star shape or polygon after digested with trypsogen. Several cells are long fusiformis. Almost all cells presented a flat star shape and simultaneously expressed GFAP and NCAM when the cells cultured with selective astrocyte culture medium. Cultured human ONH astrocytes can be obtained by precisely separating lamina cribrosa and placing the explants on the margin of culture medium, a method that promotes cell adherence. Using selective astrocyte culture medium is very effective and convenient in purifying primary astrocytes.

An analogue of the Liouville theorem for integrable Hamiltonian systems with incomplete flows

For integrable Hamiltonian systems with two degrees of freedom whose Hamiltonian vector fields have incomplete flows, an analogue of the Liouville theorem is established. A canonical Liouville fibration is defined by means of an "exact" 2-parameter family of flat polygons equipped with certain pairing of sides. For the integrable Hamiltonian systems given by the vector field $v=(-\partial f/\partial w, \partial f/\partial z)$ on ${\mathbb C}^2$ where $f=f(z,w)$ is a complex polynomial in 2 variables, geometric properties of Liouville fibrations are described.

A Face of a Projective Triangulation Removed for Its Geometric Realizability

Let M be a map on a surface F 2. A geometric realization of M is an embedding of F 2 into a Euclidian 3-space ℝ3 with no self-intersection such that each face of M is a flat polygon. In Bonnington and Nakamoto (Discrete Comput. Geom. 40:141–157, 2008), it has been proved that every triangulation G on the projective plane has a face f such that the triangulation G−f on the Mobius band obtained from G by removing the interior of f has a geometric realization. In this paper, we shall characterize such a face f of G.

On geometrically realizable Möbius triangulations

Let M be a map on a surface F2. A geometric realization of M is an embedding of F2 into a Euclidian 3-space R3 with no self-intersection such that each face of M is a flat polygon. In this paper, we characterize geometrically realizable triangulations on the Möbius band.

ON THE ANALYTIC-NUMERIC TREATMENT OF WEAKLY SINGULAR INTEGRALS ON ARBITRARY POLYGONAL DOMAINS

An alternative analytical approach to calculate the weakly singular free-space static potential integral associated to uniform sources is presented. Arbitrary oriented flat polygons are considered as integration domains. The technique stands out by its mathematical simplicity and it is based on a novel integral transformation. The presented formula is equivalent to others existing in literature, being also concise and suitable within a singularity subtraction framework. Generalized Cartesian product rules built on the double exponential formula are utilized to integrate numerically the resulting analytical 2D potential integral. As a consequence, drawbacks associated to endpoint singularities in the derivative of the potential are tempered. Numerical examples within a surface integral equation-Method of Moments framework are finally provided.

Geometric Realization of Möbius Triangulations

A Mobius triangulation is a triangulation on the Mobius band. A geometric realization of a map $M$ on a surface $\Sigma$ is an embedding of $\Sigma$ into a Euclidean 3-space $\mathbb{R}^3$ such that each face of $M$ is a flat polygon. In this paper, we shall prove that every 5-connected triangulation on the Mobius band has a geometric realization. In order to prove it, we prove that if $G$ is a 5-connected triangulation on the projective plane, then for any face $f$ of $G$, the Mobius triangulation $G-f$ obtained from $G$ by removing the interior of $f$ has a geometric realization.

A Numerical Construction Algorithm of Nash and Stackelberg Solutions for Two-person Non-zero Sum Linear Positional Differential Games

The report evolves a method, which uses the formalization and results of positional antagonistic differential games theory, developed by N. N. Krasovskii and his scientific school, for constructing solutions of a class of non-antagonistic differential games. The method transforms non-antagonistic game into so-called non-standard optimal control problem. Numerical solutions for Stackelberg games are constructed by an algorithm developed by S. Osipov. Numerical Nash solution construction algorithm based upon auxiliary bimatrix games sequence is presented. Used computational geometry algorithms include convex hull construction, union and intersection of polygons and a Minkowski sum for polygons. Results of numerical experiment for a material point motion in plane are presented. The point is moved by force formed by two players. Every player has his personal target point. Among the obtained results, there is a Nash solution such that along the corresponding trajectory the position of the game is non-antagonistic, at first, and then becomes globally antagonistic starting from some moment of time.

Geometric realization of a triangulation on the projective plane with one face removed

Abstract A geometric realization of a map on a surface is an embedding of the map into a 3-space such that each face is a flat polygon. In my talk, we prove that every triangulation G on the projective plane has a face f such that the triangulation G − f of the Mobius band obtained from G by removing the interior of f has a geometric realization. Moreover, we show that if G is 5-connected, then G − f has a geometric realization for any face f.

A geometric acoustics simulation proposal for curved geometry

Current methods for acoustical simulations based on geometrical acoustics are designed to ascertain the properties of rooms using models comprising large flat polygons. Typically these same methods are used on models of spaces with curved surfaces in which the curves are approximated using planar facets. In such cases, errors are introduced in the simulation when the infinitely varying normal of a curve is replaced with a finite number of piecewise constant normals, one for each facet. NURBS ‐ nonuniform rational B‐splines ‐ offer an alternative geometric representation that allows curves to be represented with precision. Using Rhinoceros, a commonly used NURBS‐based CAD program, as a platform for an acoustic simulation tool for models of both NURBS and polygon geometry, we can begin to discover whether it is possible to conceptualize a geometrical acoustics method that is more accurate for curved surfaces. This talk will cover the implementation and early testing of an acoustic‐simulation plug‐in for Rhinoceros.

Geometric Realization of a Triangulation on the Projective Plane with One Face Removed

Let M be a map on a surface F 2. A geometric realization of M is an embedding of F 2 into a Euclidean 3-space ℝ3 such that each face of M is a flat polygon. We shall prove that every triangulation G on the projective plane has a face f such that the triangulation of the Mobius band obtained from G by removing the interior of f has a geometric realization.

Animating real‐time explosions

AbstractAny sufficiently powerful explosion in air creates an expanding blast wave that propagates outwards from the source. The paper explores the physically based simulation of a blast wave impact on surrounding objects. The emphasis is on simplifying the underlying physical and chemical governing equations in order to achieve a visually believable result that performs in real time. A connected voxel model is used to represent objects, so that realistic solid debris is generated instead of flat polygons. The model permits arbitrary voxel shapes, which allow the creation of more complex objects with a lower number of voxels when compared to models using uniform voxel shapes. This model also overcomes certain limitations of the spring–mass particle model when it comes to representing rigid bodies. The paper also explores auxiliary visual effects caused by the blast wave, such as flame, smoke and dust. Each of these cues increases the visual plausibility of the explosion being simulated without being rigorously physically based or computationally intensive. Copyright © 2002 John Wiley & Sons, Ltd.