[3] **viXra:1709.0144 [pdf]**
*submitted on 2017-09-11 16:58:23*

**Authors:** Pawan Kumar Bishwakarma, Sumit Khadka

**Comments:** 16 Pages.

A regular polygon is a planar geometrical structure with all sides of equal length and all angles of equal magnitude. The ratio of perimeter of any regular polygon to the length of its longest diagonal is a constant term and the ratio converges to the value of as the number of sides of the polygon increases. The result has been shown to be valid by actually calculating the ratio for each polygon by using corresponding formula and geometrical reasoning. A computational calculation of the ratio has also been presented to validate the convergence. The values have been calculated up to 30 significant digits.

**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