[5] **viXra:1812.0226 [pdf]**
*submitted on 2018-12-12 06:35:09*

**Authors:** Edgar Valdebenito

**Comments:** 108 Pages.

This note presents a collection of elementary fractals.

**Category:** Geometry

[4] **viXra:1812.0206 [pdf]**
*submitted on 2018-12-11 21:37:57*

**Authors:** James A. Smith

**Comments:** 5 Pages.

Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to solve one of the beautiful \emph{sangaku} problems from 19th-Century Japan. Among the GA operations that prove useful is the rotation of vectors via the unit bivector i.

**Category:** Geometry

[3] **viXra:1812.0090 [pdf]**
*submitted on 2018-12-05 19:29:23*

**Authors:** Colin James III

**Comments:** 3 Pages. © Copyright 2016-2018 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the axioms of the title. The axiom of identity of betweenness and axiom Euclid are tautologous, but the others are not. The commonplace expression of the axiom of Euclid does not match its other two variations which is troubling. This effectively refutes the planar R-geometry.

**Category:** Geometry

[2] **viXra:1812.0085 [pdf]**
*submitted on 2018-12-04 07:19:40*

**Authors:** Hannes Hutzelmeyer

**Comments:** 93 Pages.

Geometries of O adhere to Ockham's principle of simplest possible ontology: the only individuals are points, there are no straight lines, circles, angles etc. , just as it was was laid down by Tarski in the 1920s, when he put forward a set of axioms that only contain two relations, quaternary congruence and ternary betweenness. However, relations are not as intuitive as functions when constructions are concerned. Therefore the planar geometries of O contain only functions and no relations to start with. Essentially three quaternary functions occur: appension for line-joining of two pairs of points, linisection representing intersection of straight lines and circulation corresponding to intersection of circles. Functions are strictly defined by composition of given ones only. Both, Euclid and Lobachevsky planar geometries are developed using a precise notation for object-language and metalanguage, that allows for a very broad area of mathematical systems up to theory of types. Some astonishing results are obtained, among them: (A) Based on a special triangle construction Euclid planar geometry can start with a less powerful ontological basis than Lobachevsky geometry. (B) Usual Lobachevsky planar geometry is not complete, there are nonstandard planar Lobachevsky geometries. One needs a further axiom, the 'smallest' system is produced by the proto-octomidial-axiom. (C) Real numbers can be abandoned in connection with planar geometry. A very promising conjecture is put forward stating that the Euclidean Klein-model of Lobachevsky planar geometry does not contain all points of the constructive Euclidean unit-circle.

**Category:** Geometry

[1] **viXra:1812.0061 [pdf]**
*submitted on 2018-12-03 06:42:06*

**Authors:** Edgar Valdebenito

**Comments:** 119 Pages.

This note presents a collection of elementary Fractals.

**Category:** Geometry