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Let K denote a nonsingular conic in the projective plane. Pascal's theorem says that, given six distinct points A, B, C, D, E, F on K , the three intersection points A E ∩ B F , A D ∩ C F , B D ∩ C E are collinear. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal's hexagrammum mysticum. [ABSTRACT FROM AUTHOR]

Pascal's wager faces serious criticisms and is generally considered unconvincing. We argue that it can make a comeback powered by an unlikely ally: postmodernism. If one denies the existence of objective facts (e.g. about God or His relation to the world), then various non-theological considerations should come to the fore when considering the rationality of religious commitment and the choice of education for one's children. In fact, we shall argue that, if one genuinely cares about one's children, then – in many Western countries – one cannot consistently be both secular and post-modernist. [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]

Projective invariants are not only important objects in mathematics especially in geometry, but also widely used in many practical applications such as in computer vision and object recognition. In this work, we show a projective invariant named as characteristic number, from which we obtain an intrinsic property of an algebraic hypersurface involving the intersections of the hypersurface and some lines that constitute a closed loop. From this property, two high-dimensional generalizations of Pascal's theorem are given, one establishing the connection of hypersurfaces of distinct degrees, and the other concerned with the intersections of a hypersurface and a simplex. [ABSTRACT FROM AUTHOR]

The article focuses on the concept of classical mysticum by French mathematician Blaise Pascal. It states that the Pascal lines are defined by the classical mysticum which starts from the 45 meeting points. It mentions that the Polar Pascal theorem is hard to prove because the standard method of applying Desargues’s theorem to earlier lines and points does not work.

In this article we present a simple and elegant algebraic proof of Pascal's hexagon theorem which requires only knowledge of basics on conic sections without theory of projective transformations. Also, we provide an efficient algorithm for finding an equation of the conic containing five given points and a criterion for verification whether a set of points is a subset of the conic. [ABSTRACT FROM AUTHOR]

The article discusses some apparently new theorems in projective geometry that are similar in spirit to Pascal's Theorem and Briancon's Theorem. It is informed that Pappus's Theorem goes back about 1700 years. In 1639, Blaise Pascal discovered a generalization in which there was six points that lie on a conic section.

`Riders' in geometry are always a pleasure to tackle , and this pleasure is doubled when on e finds connections between plane geometry and algebra . This three -part article is about such connection s. In Parts 1 and 2 , we explore some connections between plane geometry and the algebra of conics and cubics; in P art 1 we give algebraic proofs of results such as Pascal's Theorem and the Butterfly. Theorem , and in Part 2 we study some group theoretic and number theoretic aspects of cubic curves. In Part 3 we look at the role of mappings and transformation groups in plane geometry. [ABSTRACT FROM AUTHOR]

Abstract   Erdős asked whether every sufficiently large set of points in general position in the plane contains six points that form a convex hexagon without any points from the set in its interior. Such a configuration is called an empty convex hexagon. In this paper, we answer the question in the affirmative. We show that every set that contains the vertex set of a convex 9-gon also contains an empty convex hexagon. [ABSTRACT FROM AUTHOR]

By a transfer principle, Pascal's Theorem is equivalent to a theorem about point pairs on the real line. It appears that Pascal's Theorem is equivalent to the vanishing of a common invariant of six quadratic forms. Using the q-deformed invariant theory of Leitenberger ( J. Algebra 222 (1999), 82), we construct corresponding quantum invariants by a computer calculation. [ABSTRACT FROM AUTHOR]