[10] **viXra:1903.0566 [pdf]**
*submitted on 2019-03-31 15:59:03*

**Authors:** Saburou Saitoh

**Comments:** 19 Pages. In this paper, we will introduce the division by zero calculus in triangles and trigonometric functions as the first stage in order to see the elementary properties.

In this paper, we will introduce the division by zero calculus in triangles and trigonometric functions as the first stage in order to see the elementary properties.

**Category:** Geometry

[9] **viXra:1903.0560 [pdf]**
*replaced on 2019-05-06 19:59:05*

**Authors:** Anamitra Palit

**Comments:** 5 Pages.

The direct sum decomposition of a vector space has been explored to bring out a conflicting feature in the theory. We decompose a vector space using two subspaces. Keeping one subspace fixed we endeavor to replace the other by one which is not equal to the replaced subspace. Proceeding from such an effort we bring out the conflict. From certain considerations it is not possible to work out the replacement with an unequal subspace. From alternative considerations an unequal replacement is possible.

**Category:** Geometry

[8] **viXra:1903.0433 [pdf]**
*submitted on 2019-03-24 19:08:38*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

The following conjecture is refuted: "[An] n-dimensional Euclidean geometry can be embedded into (n+1)-dimensional hyperbolic non Euclidean geometry. Therefore hyperbolic non Euclidean geometry and Euclidean geometry are equally consistent, that is, either both are consistent or both are inconsistent." Hence, the conjecture is a non tautologous fragment of the universal logic VŁ4.

**Category:** Geometry

[7] **viXra:1903.0317 [pdf]**
*replaced on 2019-03-18 18:09:25*

**Authors:** Colin James III

**Comments:** 2 Pages.

We prove two parallel lines are tautologous in Euclidean geometry. We next prove that non Euclidean geometry of Lobachevskii is not tautologous and hence not consistent. What follows is that Riemann geometry is the same, and non Euclidean geometry is a segment of Euclidean geometry, not the other way around. Therefore non Euclidean geometries are a non tautologous fragment of the universal logic VŁ4.

**Category:** Geometry

[6] **viXra:1903.0244 [pdf]**
*submitted on 2019-03-12 10:13:23*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in english

We propose a 3-form in differential geometry which depends only of a connection over the tangent fiber bundle.

**Category:** Geometry

[5] **viXra:1903.0241 [pdf]**
*submitted on 2019-03-12 13:24:51*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 10 Pages.

In this note, we give an application of the Method of the Repère Mobile to the Ellipsoid of Reference in Geodesy using a symplectic approach.

**Category:** Geometry

[4] **viXra:1903.0126 [pdf]**
*submitted on 2019-03-07 10:25:27*

**Authors:** Colin James III

**Comments:** 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the classical logic section on set theory, we evaluate definitions of the atom and primitive set. None is tautologous. From the quantum logic and topology section on set theory, we evaluate the disjoint union (as equivalent to the XOR operator) and variances in equivalents for the AND and OR operators. None is tautologous. This reiterates that set theory and quantum logic are not bivalent, and hence non-tautologous segments of the universal logic VŁ4. The assertion of Riemannian geometry as generalization of Euclidean geometry is not supported.

**Category:** Geometry

[3] **viXra:1903.0100 [pdf]**
*submitted on 2019-03-07 05:46:58*

**Authors:** Johan Noldus

**Comments:** 67 Pages.

Non commutative geometry is developed from the point of view of an extension of quantum logic. We provide for an example of a non-abelian simplex as well as a non-abelian curved Riemannian space.

**Category:** Geometry

[2] **viXra:1903.0082 [pdf]**
*submitted on 2019-03-05 20:48:15*

**Authors:** Colin James III

**Comments:** 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the area and dimensions of an outer triangle, the height point of an inner triangle implies the minimum distance to the outer triangle. This proves the solution of Bellman's Lost in the forest problem for triangles. By extension, it is the general solution proof for other figures.

**Category:** Geometry

[1] **viXra:1903.0023 [pdf]**
*submitted on 2019-03-01 09:40:43*

**Authors:** Helmut Söllinger

**Comments:** 10 Pages. language: German

The paper analyses the issue of optimised packaging of spheres of the same size. The question is whether a linear packaging of spheres in the shape of a sausage or a spatial cluster of spheres can minimise the volume enveloping the spheres. There is an assumption that for less than 56 spheres the linear packaging is denser and for 56 spheres the cluster is denser, but the question remains how a cluster of 56 spheres could look like. The paper shows two possible ways to build such a cluster of 56 spheres. The author finds clusters of 59, 62, 65, 66, 68, 69, 71, 72, 74, 75, 76, 77, 78, 79 and 80 spheres - using the same method - which are denser than a linear packaging of the same number and gets to the assumption that all convex clusters of spheres of sufficient size are denser than linear ones.

**Category:** Geometry