[3] **viXra:1306.0221 [pdf]**
*submitted on 2013-06-26 22:32:47*

**Authors:** John Frederick Sweeney

**Comments:** 21 Pages.

The ancient Chinese divination method called Qi Men Dun Jia is mathematically based on the Clifford Clock, which is a ring of Clifford Algebras related through the Bott Periodicity Theorem. On top of this lies the icosahedra or A5 with its sixty or one hundred twenty elements. A5 in the model functions to account for Time, as well as the Five Elements and the Na Jia, additional elements of Chinese metaphysics used in divination. This icosahedra is composed of three Golden Rectangles. The Golden Rectangles are related to three Fano Planes and the octonions, which are in turn related to the Golden Section. This provides the basis for the Lie Algebra and lattice of E8, again with its own Golden Section properties.

**Category:** Topology

[2] **viXra:1306.0192 [pdf]**
*replaced on 2013-09-03 08:53:52*

**Authors:** Liu Ran

**Comments:** 38 Pages.

The method and basic theory are far from traditional graph theory. Maybe they are the key factor of success. K4 regions (every region is adjacent to other 3 regions) are the max adjacent relationship, four-color theorem is true because more than 4 regions, there must be a non-adjacent region existing. Non-adjacent regions can be color by the same color and decrease color consumption.
Another important three-color theorem is that the border of regions can be colored by 3 colors. Every region has at least 2 optional colors, which can be permuted.

**Category:** Topology

[1] **viXra:1306.0187 [pdf]**
*submitted on 2013-06-21 08:24:13*

**Authors:** Kamran Alam Khan

**Comments:** 4 Pages. Published in International Journal of Mathematical Archive (IJMA)

The notion of a bitopological space as a triple (X,I_1,I_2), where X is a set and I_1and I_2are topologies on X, was first formulated by J.C.Kelly [5]. In this paper our aim is to introduce and study the notion of an N-topological space (X,I_1,I_2,………I_N). We first generalize the notion of an ordinary metric to n variables. This metric will be called K-metric. Then the notion of a quasi-pseudo-K-metric will be introduced. We then follow the approach of Kelly to introduce and study the notion of an N-topological space. An example for such a space is produced using chain topology. And finally we define and study some of the possible separation properties for N-topological spaces.

**Category:** Topology