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In this paper, we investigate the Veronesean arc, the non-Veronesean arc by converting a point expressed in Cartesian coordinates to homogeneous coordinates in left nearfield plane of order 9 where a k--arc in a finite projective or affine plane is a set of k points no three of which are collinear. And also, we examine that whether founded complete (7,2)-Non-Veronesean arc satisfy Pascal's Theorem in the left nearfield projective plane of order 9. Six of (7,2)-Non-Veronesean arc's all points which are {1,4,11,21,35,38} points line on same conic. But it is determined that (7,2)-Non-Veronesean arc does not satisfy Pascal's Theorem. [ABSTRACT FROM AUTHOR]

Utilizing first principles calculations within PW91 exchange-correlation method, we investigated a boron sheet that exhibits related electronic properties. The 2- dimensional boron sheet is flattened and has an atomic structure where the pair cores of every three ordered hexagons within the hexagonal network are loaded up by extra atoms, which saves the triangular lattice symmetry. The boron sheet takes possession of intrinsic metal properties and the electronic bands are comparable to the bands of the graphene that are close to the Fermi level. The real and imaginary parts of the dielectric function show a metallic or semiconductor behaviour, depending on the electric field direction. [ABSTRACT FROM AUTHOR]

Let K denote a nonsingular conic in the complex projectiveplane. Given six distinct points a,b,c,d,e,f on K, Pascal's theorem saysthat the intersection points ab ∩ de, bc ∩ ef, cd ∩ af are collinear. The linecontaining them is called the Pascal line of the hexagonabcdef. We get sixtysuch lines by permuting the vertices. These lines satisfy some incidencetheorems, which lead to an extremely rich combinatorial structure calledthehexagrammum mysticum. In 1877, Veronese discovered a procedure tocreate an infinite sequence of such structures via successive mutations; thissequence is called themultimysticum. In this paper we prove a strong rigiditytheorem which shows that the multimysticum contains 300 projective rangesthat are absolutely invariant. The two interspersed sequences of cross-ratioswhich encode this invariance turn out to be the alternate convergents in thecontinued fraction expansions of 1±1/√3. [ABSTRACT FROM AUTHOR]

Problem 3845 in Crux Mathematicorum is a nice configuration about eight circles with many special cases. We sketch a proof of this problem and rename it as eight circles theorem. Using Miquel's six circles theorem, the bundle theorem and the eight circles theorem, we give a proof of the dual of the eight circles theorem and its converse. [ABSTRACT FROM AUTHOR]

Given six points on a conic, Pascal's theorem gives rise to a configuration called the hexagrammum mysticum. It contains 20 Steiner points and 20 Cayley-Salmon lines. By a classical theorem due to von Staudt, the Steiner points fall into 10 conjugate pairs with reference to the conic; but this is not true of the C-S lines for a general choice of six points. We show that the C-S lines are pairwise conjugate precisely when the original sextuple is tri-symmetric. The variety of tri-symmetric sextuples turns out to be arithmetically Cohen-Macaulay of codimension two. We determine its SL2-equivariant minimal resolution. [ABSTRACT FROM AUTHOR]

In this article we check, with the Mizar system [2], Pascal's theorem in the real projective plane (in projective geometry Pascal's theorem is also known as the Hexagrammum Mysticum Theorem). Pappus' theorem is a special case of a degenerate conic of two lines. For proving Pascal's theorem, we use the techniques developed in the section "Projective Proofs of Pappus' Theorem" in the chapter "Pappus' Theorem: Nine proofs and three variations" [11]. We also follow some ideas from Harrison's work. With HOL Light, he has the proof of Pascal's theorem. For a lemma, we use PROVER9 and OTT2MIZ by Josef Urban [12, 6, 7]. We note, that we don't use Skolem/Herbrand functions (see "Skolemization" in [1]). [ABSTRACT FROM AUTHOR]

Recently the authors presented a matrix representation approach to real Appell polynomials essentially determined by a nilpotent matrix with natural number entries. It allows to consider a set of real Appell polynomials as solution of a suitable first order initial value problem. The paper aims to confirm that the unifying character of this approach can also be applied to the construction of homogeneous Appell polynomials that are solutions of a generalized Cauchy–Riemann system in Euclidean spaces of arbitrary dimension. The result contributes to the development of techniques for polynomial approximation and interpolation in non-commutative Hypercomplex Function Theories with Clifford algebras. [ABSTRACT FROM AUTHOR]

Purpose Cardinal points are used for ray tracing through Gaussian systems. Anti-principal and anti-nodal points (which we shall refer to as the anti-cardinal points), along with the six familiar cardinal points, belong to a much larger set of special points. The purpose of this paper is to obtain a set of relationships and resulting equalities among the cardinal and anti-cardinal points and to illustrate them using Pascal's ring. Methods The methodology used relies on Gaussian optics and the transference T. We make use of two equations, obtained via the transference, which give the locations of the six cardinal and four anti-cardinal points with respect to the system. We obtain equalities among the cardinal and anti-cardinal points. We utilise Pascal's ring to illustrate which points depend on frequency and their displacement with change in frequency. Results Pascal described a memory schema in the shape of a hexagon for remembering equalities among the points and illustrating shifts in these points when an aspect of the system changes. We modify and extend Pascal's ring to include the anti-cardinal points. We make use of Pascal's ring extended to illustrate which points are dependent on the frequency of light and the direction of shift of the equalities with change in frequency. For the reduced eye the principal and nodal points are independent of frequency, but the focal points and the anti-cardinal points depend on frequency. For Le Grand's four-surface model eye all six cardinal and four anti-cardinal points depend on frequency. This has implications for definitions, particularly of chromatic aberrations of the eye, that make use of cardinal points and that themselves depend on frequency. Conclusions Pascal's ring and Pascal's ring extended are novel memory schema for remembering the equalities among the cardinal and anti-cardinal points. The rings are useful for illustrating changes among the equalities and direction of shift of points when an aspect of a system changes. Care should be taken when defining concepts that rely on cardinal points that depend on frequency. [ABSTRACT FROM AUTHOR]

Existence of non-abelian representations of the near hexagon Q(5, 2) ⊗ Q(5, 2)In [5], a new combinatorial model with four types of points and nine types of lines of the slim dense near hexagon Q(5, 2)⊗Q(5, 2) was provided and it was then used to construct a non-abelain representation of Q(5, 2) ⊗ Q(5, 2) in the extraspecial 2-group 21+18-. In this paper, we give a direct proof for the existence of a non-abelian representation of Q(5, 2) ⊗ Q(5, 2) in 21+18-. [ABSTRACT FROM AUTHOR]

The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. The topics include Desargues's Theorem, harmonic conjugates, projectivities, involutions, conics, Pascal's Theorem, poles and polars. The axioms used for the synthetic treatment are constructive versions of the traditional axioms. The analytic construction is used to verify the consistency of the axiom system; it is based on the usual model in three-dimensional Euclidean space, using only constructive properties of the real numbers. The methods of strict constructivism, following principles put forward by Errett Bishop, reveal the hidden constructive content of a portion of classical geometry. A number of open problems remain for future studies. [ABSTRACT FROM AUTHOR]

In Sturmfels and Whiteley (J Symb Comput 11(5):439-453, ), it is proven that any multihomogenous bracket polynomial with integer coefficients can be interpreted by a ruler construction when introducing simple non-degeneracy conditions. This problem is called (generalized) Cayley factorization. This allows for geometrically interpreting many projective invariant properties. We reprove the above statement in rank 3 giving a better bound on the size of the non-degeneracy conditions. The constant factor in the bound is essentially reduced from 105 to 9. The algorithm described is concise enough to be implemented on a computer. With this algorithm, interpreting a common condition for six points to lie on a common conic is very close to Pascal's construction (see Apel, in: The geometry of brackets and the area principle, Dissertation, TU München, ). [ABSTRACT FROM AUTHOR]

The article discusses proof of Poncelet's theorem using real projective plane based on Pascal's theorem. Topics discussed include Poncelet's theorem related to n-sided polygon in nondegenerate conics and vertices, intersections of circles, Abelian integrals theory, interpretation of elliptic billiards, Poncelet's theorem being consequence of Pascal's theorem and a theorem proving sides of triangles being tangent if triangles are inscribed in a conic.

We consider whether it is possible to recover the three dimensional strain field tomographically from neutron and x-ray diffraction data for polycrystalline materials. We show that the distribution of strain transverse to a ray cannot be deduced from one diffraction pattern accumulated along that path, but that a certain moment of that data corresponds to the transverse ray transform of the strain tensor and so may be recovered by inverting that transform given sufficient data. We show that the whole strain tensor can be reconstructed from diffraction data measured using rotations about six directions that do not lie on a projective conic. In addition we give an inversion formula for complete data for the transverse ray transform. We also show that Bragg edge transmission data, which has been suggested for strain tomography with polychromatic data, cannot provide the strain distribution within the material but only the average along the ray path. [ABSTRACT FROM AUTHOR]