# Geometry

## 1908 Submissions

[5] **viXra:1908.0366 [pdf]**
*submitted on 2019-08-17 17:05:58*

### 1.a Surprising Note About EUCLID’S Parallel Axiom

**Authors:** SzÉkely Endre

**Comments:** 5 Pages. see in "Other instructions"

In this short note an important consequence of my previous paper5 is investigated.
We reinvestigate EUCLID’S 5th postulate

**Category:** Geometry

[4] **viXra:1908.0095 [pdf]**
*submitted on 2019-08-05 10:29:36*

### A Closed Form Solution to the Snub Dodecahedron

**Authors:** Mark Adams

**Comments:** 14 Pages.

We consider the volume of the unit edge length Snub Dodecahedron.

**Category:** Geometry

[3] **viXra:1908.0074 [pdf]**
*replaced on 2019-08-12 04:48:42*

### Reconsideration of X^3 - dx - a = 0 Based on the Cubic Equation X^3 = 15x + 4 Solved by Rafael Bombelli

**Authors:** Atsushi Koike

**Comments:** 14 Pages.

According to Pierre Wantzel’s proof of 1837 that the trisection of 60 degree is impossible, because the cubic equation of x^3 - 3x - 1 = 0 had a absence of a rational solution. And his proof reached already a consensus as a general opinion. I learned from Nobukazu Shimeno’s introductory book on complex numbers that Rafael Bombelli got a rational solution x= 4 from the cubic equation x^3 = 15x + 4 based on the Cardinal formula. x^3 = 15x + 4 is x^3 - 15x - 4 = 0, which can be further replaced by x^3 - dx - a = 0. Kentaro Yano, who introduces the trisection of angles, says that the basic equation of x^3 - 3x - 1 = 0 is x^3 - dx - a = 0. In other words, the equation that Rafael Bombelli obtained a rational number solution is the same as the equation of the trisection of the angle. On the other hand, Yano raises x^3 - 3x = 0 as an example when the angle can be divided into three equal parts. Needless to say, x^3 - 3x - 0 = 0 and x^3 - dx - a = 0. Therefore, comparing Rafael Bombelli’s solution and the equation where Yano’s angle trisection is impossible and possible, the equation for angle trisection is x = a. And I found that if x = a = 2 then the solution is obtained at all angles. This paper proves that.

**Category:** Geometry

[2] **viXra:1908.0020 [pdf]**
*submitted on 2019-08-01 10:58:49*

### A Note on Circle Chains Associated with the Incircle of a Triangle

**Authors:** Hiroshi Okumura

**Comments:** 2 Pages.

We generalize a problem in Wasan geometry involving
the incircle of a triangle.

**Category:** Geometry

[1] **viXra:1908.0004 [pdf]**
*submitted on 2019-08-01 03:51:30*

### The Connections for Exterior Forms

**Authors:** Antoine Balan

**Comments:** 1 page, written in english

Here is defined a generalization of connections with help of exterior forms.

**Category:** Geometry