Search

People are noted as intrigued for centuries by interplay between algebra and geometry with many important advances viewed to have come down through some sort of linking of the two. Examples are given of advantages to learning and discovery that can be found in an investigative approach combining them. (Author/MP)

We investigate whether a series ∑∞ n=1 Xn with terms defined recursively by the relation xn+1 = f (Xn), for all n ⩾1, converges or diverges. For the examples studied, the ratio test is inconclusive. We provide a theorem to decide whether such series are either convergent or divergent under some natural analytic conditions. [ABSTRACT FROM AUTHOR]

John Nash initiated the study of arcs in algebraic geometry in a quite unusual paper written in 1968, at the beginning of his long illness. This paper raised many interesting and still-open questions that can be investigated using elementary methods. The tone of the Nash article is discursive, and through a series of examples and comments it reports on "some interesting possible truths which were encountered. "We also follow this approach and present the intricacies and beauties of this subject using concrete examples. In addition we discuss some recent results and unsolved problems that are accessible to undergraduates. [ABSTRACT FROM AUTHOR]

The article presents a research on Napoleon polygon by mathematicians including Titu Andreescu, Vladimir Georgiev and Oleg Mushkarov. It highlights geometric characterization of Napoleon polygon and offers proof of Barlotti-Greber Theorem. It also presents a generalized theorem by analytical characterization of Napoleon polygons .

The article discusses the study conducted to generalization of Routh's triangle theorem using the unification of Ceva and Menelaus theorems and informs about its connection with Feyman's triangle. Topics discussed include modulus space of triangles and iterative processes, using linear algebra and analytic geometry for the generalization, and determination of a triangle by three non-collinear points.

The article presents the computation of Σ 1/n2 by solving triangles. A geometric method for determining the exact value of Σ 1/n2 is through basic trigonometry using the sine rule for triangles in combination with area-preserving maps. It presents a computation that uses the bipolar coordinates and the (co)sine rule, and spreading and piling of squares. Theorems include the Jacobian determinant of the area-preserving map, area-preserving bijective map permutes the amoeba subsets, and others.

The article reports on enumerative algebraic geometry of conics. It discusses solutions to problems on conics which include one provided by Jakob Steiner who was a professor of geometry at the University of Berlin. Steiner's problem asks how many conics are tangent to all five given conics in a plane. The problems presented involve algebraic geometry and depend in the conics, lines, and points. Definitions for conic are presented as well as a moduli space for conics. The article also argues that Steiner's solution the problem was incorrect and proposes a different method. An overview of the report is provided.

The article presents a discussion about the Koch Curve. It states that the mathematically popular Koch curve is usually used as an example to illustrate a continuous but not differentiable function and as an example of a non-rectifiable curve. However, it emphasizes that only the fact that it is not rectifiable is proved, and the proof that it is a curve and that no point does this curve have a tangent line is disregarded. It offers a mathematical and simple geometric proof that the Koch curve is indeed an arc and that it at no point has a tangent line.

The article demonstrates that the two consequences of tetrahedrons circumscribing a sphere also holds true for more general solids called circumsolids whose faces can be both planar and curved. Any plane through the center of the inscribed sphere divides the circumsolid into two smaller solids whose surface areas are equal if and only if they have equal volumes. The centroid of the boundary surface of a circumsolid and the centroid of its volume are also collinear always with the center of the inscribed sphere at distances in the ratio of four is to three from the center.

Considers the conditions when the position vector of a space curve always lie in the rectifying plane. Characteristics of rectifying curves; Proof that a twisted curve is congruent to a rectifying curve.

Presents mathematical ways to determine the identity of 2 polyhedra. Use of planes and vertices; Analytical geometry.

Discusses several theorems on rectangles tiled by rectangles. Theorem 1 proved by Dehn in 1903; Proof of the same theorem by Brooks in 1940; Several theorems obtained through modifications in Dehn's and Brooks' approaches; Proofs of the different theorems.