[4] **viXra:1709.0439 [pdf]**
*submitted on 2017-09-30 10:53:14*

**Authors:** Bouetou Bouetou Thomas

**Comments:** 10 Pages. non

I have noticed a situation of plagiarist and want to draw the attention of the authors and readers.

**Category:** Geometry

[3] **viXra:1709.0144 [pdf]**
*replaced on 2018-12-29 04:20:55*

**Authors:** Pawan Kumar B.K.

**Comments:** 15 Pages.

Regular polygons are planar geometric structures that are used to a great extent in mathematics, engineering and physics. For all size of a regular polygon, the ratio of perimeter to the longest diagonal length is always constant and converges to the value of π as the number of sides of the polygon approaches to ∞. The purpose of this paper is to introduce Bishwakarma Ratio Formulas through mathematical explanations. The Bishwakarma Ratio Formulae calculate the ratio of perimeter of regular polygon to the longest diagonal length for all possible regular polygons. These ratios are called Bishwakarma Ratios- often denoted by short term BK ratios- as they have been obtained via Bishwakarma Ratio Formulae. The result has been shown to be valid by actually calculating the ratio for each polygon by using corresponding formula and geometrical reasoning. Computational calculations of the ratios have also been presented upto 30 and 50 significant figures to validate the convergence.

**Category:** Geometry

[2] **viXra:1709.0109 [pdf]**
*submitted on 2017-09-10 05:11:19*

**Authors:** Prashanth R. Rao

**Comments:** 3 Pages.

In this paper, we generate a special hexagon with two-fold symmetry by diagonally juxtaposing two squares of different dimensions so that they share exactly one common vertex and their adjacent sides are perpendicular to one another. We connect in specific pairs, the vertices adjacent to common vertex of both squares to generate a hexagon that is symmetrical about a line connecting the unconnected vertices. We show that this special hexagon must have one square whose points lie on its sides. With suitable modifications, it may be possible to use this technique to prove the Toeplitz conjecture for a simple closed curve generated by connecting the same six vertices of this special hexagon.

**Category:** Geometry

[1] **viXra:1709.0026 [pdf]**
*submitted on 2017-09-02 14:50:05*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

According to Toeplitz conjecture or the inscribed square conjecture, every simple closed curve in a plane must have atleast one set of four points on it that belong to a square. This conjecture remains unsolved for a general case although it has been proven for some special cases of simple closed curves. In this paper, we prove the conjecture for a special case of a simple closed curve derived from two simple closed curves, each of which have exactly only one set of points defining a square. The Toeplitz solution squares of two parent simple closed curves have the same dimensions and share exactly one common vertex and the adjacent sides of the two squares form a right angle. The derived simple closed curve is formed by eliminating this common vertex (that belonged to the two solutions squares to begin with) and connecting other available points on the parent curves. We show that this derived simple closed curve has atleast one solution square satisfying the Toeplitz conjecture.

**Category:** Geometry