[4] **viXra:1512.0414 [pdf]**
*submitted on 2015-12-23 16:24:57*

**Authors:** Jose Carlos Tiago de Oliveira

**Comments:** 6 Pages.

ABSTRACT
Mario Markus, a Chilean scientist and artist from Dortmund Max Planck Institute, has exposed a large set of
images of Lyapunoff exponents for the logistic equation modulated through rhythmic oscillation of parameters. The
pictures display features like foreground/background contrast, visualizing superstability, structural instability and, above
all, multistability, in a way visually analogous to three-dimensional representation.
See, for instance, http://www.mariomarkus.com/hp4.html.
The present papers aims to classify, through codification of numbers in the unit interval, the ensemble of images
thus generated. The above is intended as a part of a still unfulfilled work in progress, the classification of style in visual
fractal images-a common endeavour to Art and Science.

**Category:** Geometry

[3] **viXra:1512.0403 [pdf]**
*submitted on 2015-12-23 03:32:54*

**Authors:** Martin Erik Horn

**Comments:** 1 Page. The complete paper can be found at http://www.phydid.de (Wuppertal 2015)

An overview over all possible elementary reflections is given. It shows that a quarter of all reflections are negative.

**Category:** Geometry

[2] **viXra:1512.0303 [pdf]**
*replaced on 2015-12-18 20:37:40*

**Authors:** Robert B. Easter

**Comments:** 30 Pages.

This paper introduces the differential operators in the G(8,2) Geometric Algebra, called the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA). The differential operators are three x, y, and z-direction bivector-valued differential elements and either the commutator product or the anti-commutator product for multiplication into a geometric entity that represents the function to be differentiated. The general form of a function is limited to a Darboux cyclide implicit surface function. Using the commutator product, entities representing 1st, 2nd, or 3rd order partial derivatives in x, y, and z can be produced. Using the anti-commutator product, entities representing the anti-derivation can be produced from 2-vector quadric surface and 4-vector conic section entities. An operator called the pseudo-integral is defined and has the property of raising the x, y, or z degree of a function represented by an entity, but it does not produce a true integral. The paper concludes by offering some basic relations to limited forms of vector calculus and differential equations that are limited to using Darboux cyclide implicit surface functions. An example is given of entity analysis for extracting the parameters of an ellipsoid entity using the differential operators.

**Category:** Geometry

[1] **viXra:1512.0233 [pdf]**
*submitted on 2015-12-06 05:52:26*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 6 Pages. In French.

In this paper, we give the equations of the geodesics of the tori and the integration of it.

**Category:** Geometry